A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM’S NEW KIND...
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Tutorials and Reviews International Journal of Bifurcation and Chaos, Vol. 13, No. 9 (2003) 2377–2491 c World Scientific Publishing Company A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM’S NEW KIND OF SCIENCE. PART II: UNIVERSAL NEURON LEON O. CHUA, VALERY I. SBITNEV and SOOK YOON Department of Electrical Engineering and Computer Sciences, University of California at Berkeley, Berkeley, CA, 94720, USA Received March 1, 2003; Revised May 15, 2003 Wolfram’s celebrated three-input Cellular Automata is further developed and extended from the perspective of neural networks. A single explicit formula involving two nested absolute-value functions and eight adjustable parameters called synaptic weights, is presented. Such a neuron is proved to be universal by specifying the synaptic weights of all 256 local rules. Applying the nonlinear dynamics concepts developed from Part I of this multipart series of papers, we present the rational for partitioning the entire set of 256 local rules into 16 distinct gene families, each composed of 16 gene siblings. Such a partitioning allows us to explain, if not predict, the pattern features generated from each local rule. Finally, these 16 gene families of Cellular Automata rules are encoded onto a new compact and insightful representation called the “double-helix torus.” Keywords : Cellular Automata; cellular neural networks; CNN; universal neuron; universal Turing machine; double-helix torus. 1. Firing and Quenching Patterns The 256 local rules studied incisively by Wolfram in A New Kind of Science [Wolfram, 2002] for the one-dimensional binary Cellular Automata (1D CA) shown in Fig. 1 are encoded onto the vertices of 256 corresponding Boolean cubes and listed in Table 1 of [Chua et al., 2002]. This table is redrawn in the following four pages (Table 1) to emphasize the correlation between the red vertices of each Boolean cube with the corresponding firing patterns of an ar- tificial neuron whose “sphere of influence” consists of its two nearest neighbors [Chua, 1998]. Each local rule “N ” of a 1D CA can be de- coded from the corresponding Boolean cube “N ” in Table 1 by associating each RED vertex of this cube with the logic state 1, and each BLUE vertex with the logic state −1 (or logic state 0 when {0, 1} are chosen as the binary states). It is important to remember that, for rea- sons already articulated in Part I, we will con- tinue to use {−1, 1} as the logic states in this paper, unless specified otherwise. The only ex- ception is when one wishes to translate the last column (γ 7 γ 6 γ 5 γ 4 γ 3 γ 2 γ 1 γ 0 ) of the truth table (lower-right corner in Table 1) into a binary for- mat (β 7 β 6 β 5 β 4 β 3 β 2 β 1 β 0 ), in which case we simply change each γ j = −1 to β j = 0. Observe that β j and γ j are related via the one-to-one mapping γ j =2β j − 1, or equivalently, β j = 1 2 (γ j + 1). The number N below each Boolean cube in Ta- ble 1 is the decimal expression of the corresponding binary number (β 7 β 6 β 5 β 4 β 3 β 2 β 1 β 0 ); namely, N = β 7 • 2 7 + β 6 • 2 6 + β 5 • 2 5 + β 4 • 2 4 + β 3 • 2 3 + β 2 • 2 2 + β 1 • 2 1 + β 0 • 2 0 = 7 k=0 β k 2 k . (1) 2377
Transcript of A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM’S NEW KIND...
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Tutorials and Reviews
International Journal of Bifurcation and Chaos, Vol. 13, No. 9 (2003) 2377–2491c© World Scientific Publishing Company
A NONLINEAR DYNAMICS PERSPECTIVE OFWOLFRAM’S NEW KIND OF SCIENCE.
PART II: UNIVERSAL NEURON
LEON O. CHUA, VALERY I. SBITNEV and SOOK YOONDepartment of Electrical Engineering and Computer Sciences,
University of California at Berkeley,Berkeley, CA, 94720, USA
Received March 1, 2003; Revised May 15, 2003
Wolfram’s celebrated three-input Cellular Automata is further developed and extended fromthe perspective of neural networks. A single explicit formula involving two nested absolute-valuefunctions and eight adjustable parameters called synaptic weights, is presented. Such a neuronis proved to be universal by specifying the synaptic weights of all 256 local rules.
Applying the nonlinear dynamics concepts developed from Part I of this multipart series ofpapers, we present the rational for partitioning the entire set of 256 local rules into 16 distinctgene families, each composed of 16 gene siblings. Such a partitioning allows us to explain, if notpredict, the pattern features generated from each local rule. Finally, these 16 gene families ofCellular Automata rules are encoded onto a new compact and insightful representation calledthe “double-helix torus.”
Keywords : Cellular Automata; cellular neural networks; CNN; universal neuron; universal Turingmachine; double-helix torus.
1. Firing and Quenching Patterns
The 256 local rules studied incisively by Wolframin A New Kind of Science [Wolfram, 2002] for theone-dimensional binary Cellular Automata (1D CA)shown in Fig. 1 are encoded onto the vertices of 256corresponding Boolean cubes and listed in Table 1of [Chua et al., 2002]. This table is redrawn in thefollowing four pages (Table 1) to emphasize thecorrelation between the red vertices of each Booleancube with the corresponding firing patterns of an ar-tificial neuron whose “sphere of influence” consistsof its two nearest neighbors [Chua, 1998].
Each local rule “N” of a 1D CA can be de-coded from the corresponding Boolean cube “N” inTable 1 by associating each RED vertex of this cubewith the logic state 1, and each BLUE vertex withthe logic state −1 (or logic state 0 when {0, 1} arechosen as the binary states).
It is important to remember that, for rea-sons already articulated in Part I, we will con-tinue to use {−1, 1} as the logic states in thispaper, unless specified otherwise. The only ex-ception is when one wishes to translate the lastcolumn (γ7γ6γ5γ4γ3γ2γ1γ0) of the truth table(lower-right corner in Table 1) into a binary for-mat (β7β6β5β4β3β2β1β0), in which case we simplychange each γj = −1 to βj = 0. Observe thatβj and γj are related via the one-to-one mappingγj = 2βj − 1, or equivalently, βj = 1
2(γj + 1).The number N below each Boolean cube in Ta-
ble 1 is the decimal expression of the correspondingbinary number (β7β6β5β4β3β2β1β0); namely,
N = β7 • 27 + β6 • 26 + β5 • 25 + β4 • 24
+ β3 • 23 + β2 • 22 + β1 • 21 + β0 • 20
=7∑
k=0
βk2k . (1)
2377
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2378 L. O. Chua et al.
Cell
(N-1)
Cell
(N-2)Cell
N Cell
0
Cell
1
Cell
2
Cell
(i-1)
Cell
i
Cell
(i+1)
(a)
1iu −
iu
1iu +
stateix
output
iy
input
(b)
Fig. 1. (a) A one-dimensional Cellular Automata (CA) made of (N+1) identical cells with a periodic boundary condition.Each cell “i” is coupled only to its left neighbor cell (i–1) and right neighbor cell (i+1). (b) Each cell “i” has a state variablexi(t), an output variable yi(t), and three constant binary inputs ui−1, ui and ui+1.
Observe that each vertex k of the Booleancube is associated with a corresponding multipli-cation factor 2k (depicted in the lower-left corner inTable 1). Consequently, we can calculate the deci-mal number N associated with each Boolean cubeby simply adding the multiplication factors asso-ciated with all RED vertices. For example, theBoolean cube 110 in Table 1 has five RED ver-tices (6, 5, 3, 2, 1), whose corresponding multiplica-tion factors are 26 = 64, 25 = 32, 23 = 8, 22 = 4,and 21 = 2. Hence, N = 64 + 32 + 8 + 4 + 1 = 110as expected.
Recall from [Chua et al., 2002] that the Booleancube ID number N is printed in Red in Table 1if its associated local rule N is linearly sepa-rable. Otherwise, N is printed in Blue. Observethat only 104 out of 256 local rules are linearlyseparable.
It is very important to remember that the dec-imal number N in Table 1 plays two roles: As anidentification (ID) number in decimal format, or asa code for specifying the truth table (local rule) whenN is translated into its binary format [Chua, 1998;Wolfram, 2002].
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A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part II 2379
Table 1. Encoding 256 local rules defining a binary 1D CA onto 256 corresponding “Boolean Cubes”.
2 3
1
5
0
4
6 7
76543210
15141312111098
2322212019181716
3130292827262524
3938373635343332
4746454443424140
5554535251504948
6362616059585756
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
N = decimal equivalent of binary number
Rule N
111
-111
1-11
-1-11
11-1
-11-1
1-1-1
-1-1-1
vertexk
0
1
2
3
4
5
6
7
1
t
iu −t
iu 1
t
iu +1t
iu +
0γ
1γ
2γ
3γ
4γ
5γ
6γ
7γ
vertex1
1 1( , , ) 1t t t t
i i i iu u u u+− + =k
1
1 1( , , ) 1t t t t
i i i iu u u u+− + = −k
1kβ =
0kβ =
6β5β 4β 3β
2β 1β 0β7β
4
6 7
5
1
32
0 21 = 2
27 = 12826 = 64
22 = 4 23 = 8
24 = 16
20 = 1
25 = 32(1,-1,1)
(-1,-1,1)
(1,1,1)
(-1,1,1)(-1,1,-1)
(1,1,-1)
(-1,-1,-1)
(1,-1,-1)
1
t
iu −
1
t
iu +
t
iu
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2380 L. O. Chua et al.
Table 1. (Continued)
2 3
1
5
0
4
6 7
7170696867666564
7978777675747372
8786858483828180
9594939291908988
10310210110099989796
111110109108107106105104
119118117116115114113112
127126125124123122121120
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
N = decimal equivalent of binary number
Rule N
111
-111
1-11
-1-11
11-1
-11-1
1-1-1
-1-1-1
vertexk
0
1
2
3
4
5
6
7
1
t
iu −t
iu 1
t
iu +1t
iu +
0γ
1γ
2γ
3γ
4γ
5γ
6γ
7γ
vertex1
1 1( , , ) 1t t t t
i i i iu u u u+− + =k
1
1 1( , , ) 1t t t t
i i i iu u u u+− + = −k
1kβ =
0kβ =
6β5β 4β 3β
2β 1β 0β7β
4
6 7
5
1
32
0 21 = 2
27 = 12826 = 64
22 = 4 23 = 8
24 = 16
20 = 1
25 = 32(1,-1,1)
(-1,-1,1)
(1,1,1)
(-1,1,1)(-1,1,-1)
(1,1,-1)
(-1,-1,-1)
(1,-1,-1)
1
t
iu −
1
t
iu +
t
iu
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A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part II 2381
Table 1. (Continued)
2 3
1
5
0
4
6 7
135134133132131130129128
143142141140139138137136
151150149148147146145144
159158157156155154153152
167166165164163162161160
175174173172171170169168
183182181180179178177176
191190189188187186185184
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
N = decimal equivalent of binary number
Rule N
111
-111
1-11
-1-11
11-1
-11-1
1-1-1
-1-1-1
vertexk
0
1
2
3
4
5
6
7
1
t
iu −t
iu 1
t
iu +1t
iu +
0γ
1γ
2γ
3γ
4γ
5γ
6γ
7γ
vertex1
1 1( , , ) 1t t t t
i i i iu u u u+− + =k
1
1 1( , , ) 1t t t t
i i i iu u u u+− + = −k
1kβ =
0kβ =
6β5β 4β 3β
2β 1β 0β7β
4
6 7
5
1
32
0 21 = 2
27 = 12826 = 64
22 = 4 23 = 8
24 = 16
20 = 1
25 = 32(1,-1,1)
(-1,-1,1)
(1,1,1)
(-1,1,1)(-1,1,-1)
(1,1,-1)
(-1,-1,-1)
(1,-1,-1)
1
t
iu −
1
t
iu +
t
iu
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2382 L. O. Chua et al.
Table 1. (Continued)
2 3
1
5
0
4
6 7
199198197196195194193192
207206205204203202201200
215214213212211210209208
223222221220219218217216
231230229228227226225224
239238237236235234233232
247246245244243242241240
255254253252251250249248
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
2 3
1
5
0
4
6 7
N = decimal equivalent of binary number
Rule N
111
-111
1-11
-1-11
11-1
-11-1
1-1-1
-1-1-1
vertexk
0
1
2
3
4
5
6
7
1
t
iu −t
iu 1
t
iu +1t
iu +
0γ
1γ
2γ
3γ
4γ
5γ
6γ
7γ
vertex1
1 1( , , ) 1t t t t
i i i iu u u u+− + =k
1
1 1( , , ) 1t t t t
i i i iu u u u+− + = −k
1kβ =
0kβ =
6β5β 4β 3β
2β 1β 0β7β
4
6 7
5
1
32
0 21 = 2
27 = 12826 = 64
22 = 4 23 = 8
24 = 16
20 = 1
25 = 32(1,-1,1)
(-1,-1,1)
(1,1,1)
(-1,1,1)(-1,1,-1)
(1,1,-1)
(-1,-1,-1)
(1,-1,-1)
1
t
iu −
1
t
iu +
t
iu
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A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part II 2383
Finally, observe that a three-pixel colored se-quence, henceforth called a neighborhood pattern Pk,is printed next to each vertex ©k in the lower-leftcorner of Table 1. The color sequence of each neigh-borhood pattern Pk is defined by the coordinates(ut
i−1, uti, u
ti+1) of vertex ©k in the truth table
(bottom-right of Table 1), where “1” is encoded inred, and “−1” is encoded in blue. For example, theneighborhood pattern P3 corresponding to vertex ©3of every Boolean cube in Table 1 has the same colorsequence (blue, red, red).
A neighborhood pattern Pk is said to be a fir-ing pattern for a local rule N iff the vertex ©k of thecorresponding Boolean cube N in Table 1 is coloredin RED. Otherwise, it is called a quenching pattern.
The significance of the firing patterns is thata cell Ci in any one-dimensional input pattern (seeFig. 1) will be RED in the next generation if, andonly if, the color pattern of the three contiguouscells (Ci−1, Ci, Ci+1) coincides with one of the fir-ing patterns of the Boolean cube. For example, forthe local rule 110, only the five firing patterns shownin Table 2 will result in a red cell at Ci in the nextgeneration. In neural network parlance, a neuron Ci
emulating local rule 110 will “fire” (by switching to1 in a CNN [Chua, 1998], or by initiating an ac-tion potential in a real neuron), if, and only if, itrecognizes one of the five firing patterns in Table 2.
Table 3 shows the quenching patterns associ-ated with the local rule 110. Note that they corre-spond to the blue vertices in the Boolean cube 110in Table 1. Just like the synapses of a neuron canbe either excitatory, or inhibitory, the neighborhoodpatterns associated with a local rule can be either“firing” or “quenching”. It is the subtle combinationof the “firing” and “quenching” patterns of a localrule which gives rise to its information processingcapabilities [Chua & Roska, 2002].
2. A Universal Neuron
The main result from [Chua et al., 2002] is atheorem asserting that each of the 256 local rulesstudied in [Wolfram, 2002] for the 1D CA shownin Fig. 1 can be generated from a single scalarnonlinear differential equation with at most eightparameters. This 8-parameter family of differentialequations is given explicitly by:
xi =(− xi +
(∣∣xi + 1∣∣ − ∣∣xi − 1
∣∣))
+{z2 + c2
∣∣∣(z1 + c1∣∣(z0 + b1ui−1 + b2ui + b3ui+1)
∣∣)∣∣∣}xi(0) = 0
i = 0, 1, 2, . . . , N .1
(2)
For each local rule N = 0, 1, 2, . . . , 255 from Table 1, there exists at least one parameter vector[c2, c1, z2, z1, z0, b1, b2, b3] ∈ R
8 such that Eq. (2) converges to an equilibrium point xi(Q) for each cell Ci.Moreover, the output
yi(t)∆=
12(|xi(t) + 1| − |xi(t) − 1|) (3)
of cell Ci in Fig. 1 converges to either yi(Q) = 1 or yi(Q) = −1 as prescribed by the Boolean cube N foreach of the eight distinct inputs (ui−1, ui, ui+1) from the truth table (bottom-right corner of Table 1).
In particular,
yi(Q) = sgn{z2 + c2
∣∣∣(z1 + c1∣∣(z0 + b1ui−1 + b2ui + b3ui+1)
∣∣)∣∣∣} (4)
1We have abused our notation by using N to denote either the number of cells in Fig. 1, or the ID number of a local rule.The choice will be clear from the context.
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2384 L. O. Chua et al.
Table 2. Firing Patterns for local rule 110.
P3(0,1,1)(-1,1,1)
P5(1,0,1)(1,-1,1)
P6(1,1,0)(1,1,-1)
P2(0,1,0)(-1, 1,-1)
P1(0,0,1)(-1,-1, 1)
Firing pattern Pk
Equivalent
Boolean code
Decimal
code
Vertex
number
1
2
3
5
6
Table 3. Quenching Patterns for local rule 110.
P7(1,1,1)(1,1,1)
P4(1,0,0)(1,-1,-1)
P0(0,0,0)(-1,-1,-1)
Quenching pattern Pk
Equivalent
Boolean code
Decimal
code
Vertex
number
0
4
7
where2
sgn[x] ∆={
1 , x > 0−1 , x < 0
(5)
It follows from Eq. (4) that the discrete-time evolution
ut+1i = F
(ut
i−1, ut
i, ut
i+1
)t = 0, 1, 2, . . . ,∞i = 0, 1, 2, . . . , N
(6)
2Although sgn[0] is undefined, we can always choose the parameters such that the argument for sgn[•] in Eq. (4) is never zero.
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A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part II 2385
of the 1D CA in Fig. 1 can be specified explicitlyvia a single formula:
Universal CA Map
F(ut
i−1, uti, u
ti+1)
= sgn{
z2 + c2
∣∣∣(z1 + c1∣∣(z0
+ b1uti−1 + b2u
ti + b3u
ti+1)
∣∣)∣∣∣} (7)
Observe that since uti−1 ∈ {−1, 1}, ut
i ∈{−1, 1}, and ut
i+1 ∈ {−1, 1}, the nonlinear map
F : {−1, 1} × {−1, 1} × {−1, 1} → {−1, 1} (8)
is a discrete map.The eight parameters {z2, c2, z1, c1, z0, b1, b2, b3}
in Eq. (7) for each local rule N are real numbersbelonging to a non-empty subregion R(N) ⊂ R
8
of the parameter space, where N = 0, 1, 2, . . . , 255.Since each βk in Eq. (1) can assume only the value“0” or “1”, whereas each of the eight parameters{z2, c2, z1, c1, z0, b1, b2, b3} in Eq. (7) can assumeany real number from −∞ to ∞, one can expectto find many points in R
8 which map into thesame local rule N . From extensive statistical sim-ulations we have found the parameter subregionR(N) to be relatively large, at least for all of thelocal rules we have examined so far. This empiricalobservation suggests that the map F is quite ro-bust, relative to the choice of the eight real param-eters {z2, c2, z1, c1, z0, b1, b2, b3}, henceforth calledsynaptic weights in view of its close analogy to thesynapses characterizing the task to be performedby an artificial neuron [Chua, 1998].
For future reference, as well as a constructiveproof that the map F can realize any one of the 256local rules, one set of synaptic weights (among in-finitely many other valid weights) is given in Table 4for each local rule. It would be instructive for thereader to substitute the eight synaptic weights inthis table, for each local rule N , into Eq. (7) andverify that all of the 256 Boolean cubes in Table 1can be so generated.
From the perspective of information process-ing, we can use Eqs. (2) and (3) to define acontinuous-time neuron, or use Eqs. (6) and (7)
to define a discrete-time neuron, each with threeinputs (ui−1(t), ui(t), ui+1(t)) and (ut
i−1, uti, u
ti+1),
respectively. We call such neurons universal becauseby tuning only eight synaptic weights, it is possiblefor the neuron to implement any one of the 256Boolean Functions of three inputs. Moreover, ouruniversal neuron is the simplest possible realizationin the sense that it requires only eight adjustableparameters, which is equal to the number of bitsto specify a local rule (bottom-right of Table 1). Apractical implementation of this universal neuronvia a CMOS integrated circuit is given in [Dogaruet al., 2003].
Note that our neuron is also universal in thesense of a Universal Turing machine because atleast one of the 256 local rules which it emulates(say, rule 110) is capable of universal computation[Wolfram, 2002].
As a final remark for this section, we note thatsince the discrete map F is defined via an explicitformula in Eq. (7), rather than by an algorithm, wecan, at no extra cost, allow the three input variablesut
i−1, uti, u
ti+1 to be any real numbers. In the special
case where each input is restricted to assume onlythe value 1 or −1, as in a CA, the map F will auto-matically output a 1 or a −1. One bonus we get forextending the input space to include all real num-bers, namely, ut
i−1 ∈ R, uti ∈ R and ut
i+1 ∈ R, isthat it allows us to exploit and extend the resultsin [Sbitnev et al., 2001] and [Sbitnev & Chua, 2002]to the study of cellular automata where virtuallyno analytical tools are applicable. In particular, itis more enlightening and mathematically tractableto view the discrete-time evolution of Eq. (6) as ascalar nonlinear difference equation.
3. Gallery of One-DimensionalCellular Automata
For future reference, some useful information as-sociated with each local rule N = 0, 1, 2, . . . , 255are listed in Table 5, where each local rule occupiesone-fourth of a page. In particular, each quadrantcontains the following data3:
1. Top row: The local rule number N and its eight-bit Boolean function representation are given inthe upper left-hand corner. The complexity index
3The bottom pattern for each local rule N is identical to that given in [Wolfram, 2002], and in [Chua et al., 2002]. We takethis opportunity to point out an error in the latter reference (p. 2705) where the red and blue colors of all lines except theinitial condition in the pattern given for local rule 159 should be changed to blue and red, respectively.
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2386 L. O. Chua et al.
Table 4. Parameters for the universal CA map.
b3b2b1z0c1z1c2z2
-2-2-24-1-31-623
-2-55-11-41-624
-64-2-1-11-1325
-6-26-3-121-626
62-53-151-427
34-42-1-11-728
3-5-6-2-14-1329
11-311-5-1230
1143-12-1431
-4-3-2-313-1622
-2-15-31-6-1221
4-1-14-1-2-1320
63-1-41-61-519
-56-56-1-4-1518
640-21-51-617
-4-112-111-616
0-46-61-31-415
34-661-51-514
3-16-41-21-513
04-2-6-1-2-1312
-113-41-11-511
20-461-51-610
-3-3-2-21-6-119
11-61-141-48
316-5-141-47
-4-466-1-1-136
4-14-3-151-45
-43-42-151-64
6-1-2-3-16-113
3-3-34-131-62
-4-56-1-10-131
-3-53-21-3-1-30
Synaptic weightsLocal
Rule
N b3b2b1z0c1z1c2z2
-16-1-21-31-255
-2-6411-31-556
2-5-3-410-1557
-64-151-5-1258
35-1-4-111-359
14-601-31-360
-6-2-261-61-561
-1-56-41-11-462
3555-13-1663
-35-3-4-111-554
2-5-4-1-14-1353
-46-6-1-141-452
2-6-16-121-651
-6-3-65-16-1350
13-2-3111-549
0-4421-51-448
3-4-5-61-3-1647
3-2-14-131-246
2-2-6-11-3-1345
-6-146-16-1244
-212-5-111-443
6-5-561-51-642
-63641-1-1541
3-6-5-4-10-1340
-6-2541-5-1339
4632-1-2-1638
-5-2611-4-1337
-64-6-2-161-636
-1-53-4-11-1335
-2-3-6-1-16-1134
2-51-51-3-1133
3531-12-1132
Synaptic weightsLocal
Rule
N
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A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part II 2387
Table 4. (Continued)
b3b2b1z0c1z1c2z2
-4-63-41-6-1487
-5-3-61-11-1388
-3144-161-389
42-50-111-490
52-3-1-111-391
-51-3-61-5-1392
16-621-61-493
6351-12-1694
3-2561-5-1695
41-511-6-1486
-4603-161-285
6-1-3-41-31-484
3-65-2-15-1483
-32-41-10-1382
-3-6-14-151-481
-21-53-10-1280
25-4-41-5-1379
32-561-51-478
3-225-1-1-1677
-4-6-421-5-1276
-1-4341-5-1275
-5-1-21-11-1374
-46-5-511-1473
262-610-1372
3-154-14-1271
4-2-51-151-470
-1-2211-2-1169
-350-61-2-1168
-34-6-1-11-1467
4-16013-1566
46-64-1-1-1365
3-2-3-61-44-764
Synaptic weightsLocal
Rule
N b3b2b1z0c1z1c2z2
2-613101-3119
33-4-3101-6120
56-2-6-131-3121
-3-2-52-13-14122
-456-5-131-2123
1-5-461-6-13124
-2-443-101-2125
-2-1-2-6-16-14126
-2-3641-51-1127
34-5-51-51-3118
6-31-5-1-21-6117
62-35-14-13116
125-3-161-2115
-23-64-15-13114
-1-11-4-1-1-15113
22-4-61-41-3112
2-46-6-121-2111
-4-5-111-4-14110
5-34-6-111-4109
-34-23-101-5108
5-4-55-1-41-7107
-433-1101-4106
-333-31-6-15105
-5-6-45-11-12104
55-41-13-13103
-6504-101-6102
3-22-1-16-13101
364-41-5-13100
-351-31-11-499
-261-1-131-498
-5-53-51-2-1397
-3-3-1-11-2-1196
Synaptic weightsLocal
Rule
N
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2388 L. O. Chua et al.
Table 4. (Continued)
b3b2b1z0c1z1c2z2
434-4101-2151
-352-210-13152
5-51110-14153
5-12-210-15154
-44-6-31-6-14155
25-23-101-5156
642-3101-4157
-6-46-4-101-6158
-1-2-66141-6159
-43-341-61-3150
3-252-16-13149
4-4-45-1-1-15148
121-1111-3147
-44-25-1-1-15146
-26-44-13-12145
-5-3-4-41-61-5144
-1-14-5-1-11-3143
44-2-6-12-13142
-4-1-32-131-2141
26-361-31-6140
3-462-15-13139
-6-22-61-41-6138
-3-3551-5-14137
64141-41-6136
436-21-21-4135
-5-66613-15134
426-21-21-5133
1-5-35-12-11132
-4335-15-13131
-52-4511-14130
1-610-10-15129
-3-4-3-1-151-5128
Synaptic weightsLocal
Rule
N b3b2b1z0c1z1c2z2
-343-5-161-2183
-5-2-3-4-141-3184
5431-121-2185
-61-561-3-14186
-4-10-31-6-15187
1-6-52-14-15188
3-3-4-11-2-14189
56-62-131-1190
433-2-1-61-7191
5-655131-6182
3-2-421-2-14181
2-3-431-2-13180
-1-5-241-11-2179
3-22-41-2-12178
652-31-41-3177
-23-4-6-121-6176
-60-3-1111-4175
4-66-51-61-4174
5-4-6-2-14-14173
-63421-5-13172
6234-1-31-7171
-504-51-41-1170
-6-2-5-4-111-5169
-3-1-1-51-11-6168
-6-1531-3-15167
6-255101-7166
-6-3501-1-14165
-4-6-25-16-12164
-254-61-51-3163
6-3151-41-4162
43411-31-2161
4-154-151-6160
Synaptic weightsLocal
Rule
N
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A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part II 2389
Table 4. (Continued)
b3b2b1z0c1z1c2z2
-3-4-41-1-41-7215
46-31-141-3216
-5-6-1-31-21-2217
4-3-6-11-11-4218
-4-6-20-1-51-7219
-1425-1-11-4220
4-1041-4-14221
-15-251-41-2222
62-2-3-1-21-4223
-4-254-161-2214
-542-5-13-14213
-433-6-12-15212
14-461-5-13211
1-25-21-2-13210
-431-51-4-12209
-4256-121-6208
0-3-521-11-2207
-536-31-51-3206
-4-6-4-21-31-2205
1-524-12-13204
4-5631-4-15203
5-2-631-41-3202
2-3261-5-13201
2425-131-4200
-2-432-11-15199
-36541-31-4198
-524-61-5-13197
-6-46-4-15-14196
05-6212-16195
4-5-6-21-41-4194
-4-6-6-41-51-6193
03641-21-7192
Synaptic weightsLocal
Rule
N b3b2b1z0c1z1c2z2
35-35-15-16247
-1-254-1-21-5248
13-5-5-1-21-5249
6-24-51-5-15250
-154-2111-2251
02-3-1-15-14252
-11-5-11-11-2253
6-41-1-151-2254
-35-4-3131-3255
5-633-1-31-7246
4051131-4245
1-4-65 -11-16244
13-431-3-14243
-44-55-13-16242
-13-6-5-141-2241
162-2-16-12240
-31-42101-1239
-4-5-14-12-15238
3-43-5111-3237
-25-24131-7236
-641-6-1-11-5235
-4-1-1-5131-5234
-5-6-3-61-31-4233
4112-131-2232
46-3-11-3-16231
34-1-21-2-13230
-6-26-4-13-14229
3-2-461-5-13228
-1-53-2-11-15227
6-135-151-3226
55621-11-6225
-15241-51-4224
Synaptic weightsLocal
Rule
N
![Page 14: A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM’S NEW KIND …nonlinear.eecs.berkeley.edu/CellularAutomata/PartII.pdf · in A New Kind of Science [Wolfram, 2002] for the one-dimensionalbinaryCellular](https://reader034.fdocuments.us/reader034/viewer/2022051604/5ff5c0f683eff57983406834/html5/thumbnails/14.jpg)
2390 L. O. Chua et al.
Table 5. Discrete time evolution of 1D cellular automata in Fig. 1 via a nonlinear difference equation for local rulesN = 0, 1, 2, . . . , 255.
1= 1=Rule 0 :
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
0
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
0
Rule :
1= 1=Rule 2 : Rule 3 :
1
1 1sgn 0 0 0 1[ ]t t t t
i i i iu u u u+− += ⋅ + ⋅ + ⋅ −[ 1
1 1sgn 2[ ]t t t t
i i i iu u u u+− += −− − −
1
1 1sgn 2[ ]t t t t
i i i iu u u u+− += − − + −[ 1
1sgn 1[ ]t t t
i i iu u u+−= − − −
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
t
iu −t
iu 1
t
iu +1t
iu +
-1
-1
-1
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-1
-1
-1
-1
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
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1
1
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1
1
t
iu −t
iu 1
t
iu +1t
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1
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2
3
4
6
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1
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1
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1
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1
1
t
iu −t
iu 1
t
iu +1t
iu +
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1
-1
-1
-1
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2
3
4
6
7
5
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1
1
t
iu −t
iu 1
t
iu +1t
iu +
1
1
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Difference Equation Difference Equation
Difference Equation Difference Equation
![Page 15: A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM’S NEW KIND …nonlinear.eecs.berkeley.edu/CellularAutomata/PartII.pdf · in A New Kind of Science [Wolfram, 2002] for the one-dimensionalbinaryCellular](https://reader034.fdocuments.us/reader034/viewer/2022051604/5ff5c0f683eff57983406834/html5/thumbnails/15.jpg)
A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part II 2391
Table 5. (Continued)
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2392 L. O. Chua et al.
Table 5. (Continued)
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A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part II 2393
Table 5. (Continued)
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2394 L. O. Chua et al.
Table 5. (Continued)
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
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0
2
3
4
6
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5
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4
6
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5
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1 1sgn 2[ ]t t t t
i i i iu u u u+− += − − − ] 1
1sgn 1[ ]t t t
i i iu u u++= − −− ]
1
1 121 2sgn[ ]t t t t
i i i iu u u u+− ++ + +−= | | ]1
1 1sgn 2 1[ ]t t t t
i i i iu u u u+− += − − − −
1= 1=Rule 6 :
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
0
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red pixel :
2 red pixels :
3 red pixels :
4
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
0
Rule 7 :
2= 1=Rule 8 : Rule 9 :
2 3
5
0
4
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5
0
4
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5
0
4
6 7
2 3
5
0
4
6 7
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
vertex
1
t
iu −t
iu 1
t
iu +1t
iu +1
t
iu −t
iu 1
t
iu +1t
iu +
1
t
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t
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iu +1
t
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t
iu +1t
iu +
![Page 19: A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM’S NEW KIND …nonlinear.eecs.berkeley.edu/CellularAutomata/PartII.pdf · in A New Kind of Science [Wolfram, 2002] for the one-dimensionalbinaryCellular](https://reader034.fdocuments.us/reader034/viewer/2022051604/5ff5c0f683eff57983406834/html5/thumbnails/19.jpg)
A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part II 2395
Table 5. (Continued)
vertex
n
0
2
3
4
6
7
5
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-1
-1
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1
1
1
1
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6
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1 11 1sgn 2 2[ ]t t t t
i i i iu u u u+− +− − +− += | | ] 1
1 1sgn 2 1[ ]t t t t
i i i iu u u u+− += − − − −
1
1 1 11sgn[ ]t t t t
i i i iu u u u+− ++ + +−= | | ] 1
1 1sgn[ ]t t t t
i i i iu u u u+− += − − − ]
1=
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Firing Patterns
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red pixel :
2 red pixels :
3 red pixels :
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red pixel :
2 red pixels :
3 red pixels :
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0
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red pixel :
2 red pixels :
3 red pixels :
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0
2=Rule 20 : Rule 2 :
2=Rule 22 : Rule 23 :
2 3
5
0
4
6 7
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5
0
4
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5
0
4
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5
0
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Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
vertex
1
t
iu −t
iu 1
t
iu +1t
iu +1
t
iu −t
iu 1
t
iu +1t
iu +
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iu 1
t
iu +1t
iu +1
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iu +1t
iu +
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2396 L. O. Chua et al.
Table 5. (Continued)
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
-1
1
1
-1
-1
-1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
-1
1
1
-1
-1
-1
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
1
-1
-1
-1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
1
-1
1
1
-1
-1
-1
1
1 12sgn[ ]t t t t
i i i iu u u u+− +− − −+= | | ] 1
1 13 2 4 1sgn[ ]t t t t
i i i iu u u u+− +− +− += | |
1
1 13 2 4 1sgn[ ]t t t t
i i i iu u u u+− ++ −−= −| | ]1
1 1sgn 2 4 332[ ]t t t t
i i i iu u u u+− += − +− + − − −| |( )| |
2=
3=
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
3
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
3
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
0
3
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
0
3
2=Rule 24 : Rule 25 :
2=Rule 26 : Rule 27 :
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
vertex
1
t
iu −t
iu 1
t
iu +1t
iu +1
t
iu −t
iu 1
t
iu +1t
iu +
1
t
iu −t
iu 1
t
iu +1t
iu +1
t
iu −t
iu 1
t
iu +1t
iu +
![Page 21: A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM’S NEW KIND …nonlinear.eecs.berkeley.edu/CellularAutomata/PartII.pdf · in A New Kind of Science [Wolfram, 2002] for the one-dimensionalbinaryCellular](https://reader034.fdocuments.us/reader034/viewer/2022051604/5ff5c0f683eff57983406834/html5/thumbnails/21.jpg)
A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part II 2397
Table 5. (Continued)
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
1
1
1
-1
-1
-1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
1
1
1
-1
-1
-1
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
1
1
1
-1
-1
-1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
1
1
1
1
-1
-1
-1
1
1 12 4 1sgn 3[ ]t t t t
i i i iu u u u+− ++ − −−= | | ]1
1 132sgn 2 4 3[ ]t t t t
i i i iu u u u+− += − + − −− − | |( )| |
1
1 122 1sgn[ ]t t t t
i i i iu u u u+− +− − −− −= | | ] 1
1 1sgn 3 2[ ]t t t t
i i i iu u u u+− += +− − − ]
3=
1=
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
3
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
3
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
0
3
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
0
3
2=Rule 28 : Rule 29 :
2=Rule 30 : Rule 3 :
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
vertex
1
t
iu −t
iu 1
t
iu +1t
iu +1
t
iu −t
iu 1
t
iu +1t
iu +
1
t
iu −t
iu 1
t
iu +1t
iu +1
t
iu −t
iu 1
t
iu +1t
iu +
![Page 22: A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM’S NEW KIND …nonlinear.eecs.berkeley.edu/CellularAutomata/PartII.pdf · in A New Kind of Science [Wolfram, 2002] for the one-dimensionalbinaryCellular](https://reader034.fdocuments.us/reader034/viewer/2022051604/5ff5c0f683eff57983406834/html5/thumbnails/22.jpg)
2398 L. O. Chua et al.
Table 5. (Continued)
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+− +− − + + += � �
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A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part II 2399
Table 5. (Continued)
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
1
-1
-1
1
-1
-1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
1
-1
-1
1
-1
-1
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
1
-1
-1
1
-1
-1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
1
1
-1
-1
1
-1
-1
1
1 12sgn[ ]t t t t
i i i iu u u u+− ++− −+= | ]|||| 1
1 13 4 1sgn 2[ ]t t t t
i i i iu u u u+− +− − −−= | | ]
1
1 13 2 4 1sgn[ ]t t t t
i i i iu u u u+− +− − + + −= | | ]1
1 1sgn 4 2 332[ ]t t t t
i i i iu u u u+− += − +− + − − −| |( )| |
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
5
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
5
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
0
5
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
0
5
Rule 36 : Rule 37 :
Rule 38 : Rule 39 :
2= 2=
2= 3=
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
vertex
1
t
iu −t
iu 1
t
iu +1t
iu +1
t
iu −t
iu 1
t
iu +1t
iu +
1
t
iu −t
iu 1
t
iu +1t
iu +1
t
iu −t
iu 1
t
iu +1t
iu +
![Page 24: A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM’S NEW KIND …nonlinear.eecs.berkeley.edu/CellularAutomata/PartII.pdf · in A New Kind of Science [Wolfram, 2002] for the one-dimensionalbinaryCellular](https://reader034.fdocuments.us/reader034/viewer/2022051604/5ff5c0f683eff57983406834/html5/thumbnails/24.jpg)
2400 L. O. Chua et al.
Table 5. (Continued)
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
-1
1
-1
1
-1
-1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
-1
1
-1
1
-1
-1
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
1
-1
1
-1
1
-1
-1
1
1 11 2 2 1sgn[ ]t t t t
i i i iu u u u+− +− − +− −= | | ] 1
1 11 1sgn[ ]t t t t
i i i iu u u u+− +− − +− −= | | ]
1
1 1sgn 2 1[ ]t t t t
i i i iu u u u+− += − − + − ] 1
1 1sgn[ ]t t t t
i i i iu u u u+− += +− − ]
2=
1=
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
3 5
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
3 5
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
0
3 5
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
0
3 5
Rule 40 : Rule 4 :
1=Rule 42 : Rule 43 :
2=
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
vertex
1
t
iu −t
iu 1
t
iu +1t
iu +1
t
iu −t
iu 1
t
iu +1t
iu +
1
t
iu −t
iu 1
t
iu +1t
iu +1
t
iu −t
iu 1
t
iu +1t
iu +
![Page 25: A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM’S NEW KIND …nonlinear.eecs.berkeley.edu/CellularAutomata/PartII.pdf · in A New Kind of Science [Wolfram, 2002] for the one-dimensionalbinaryCellular](https://reader034.fdocuments.us/reader034/viewer/2022051604/5ff5c0f683eff57983406834/html5/thumbnails/25.jpg)
A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part II 2401
Table 5. (Continued)
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
1
1
-1
1
-1
-1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
1
1
-1
1
-1
-1
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
1
1
-1
1
-1
-1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
1
1
1
-1
1
-1
-1
1
1 143 2 1sgn[ ]t t t t
i i i iu u u u+− +− − +− −= | | ]1
1 12 1sgn 2[ ]t t t t
i i i iu u u u+− +− − +− −= | |
1
1 1sg 2 33n 42[ ]t t t t
i i i iu u u u+− +− −= − − +| |( )| | ] 1
1 1sgn 3 2[ ]t t t t
i i i iu u u u+− += +− − + ]
1=
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
3 5
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
3 5
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
0
3 5
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
0
3 5
Rule 44 : Rule 45 :
Rule 46 : Rule 47 :
2= 2=
3=
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
vertex
1
t
iu −t
iu 1
t
iu +1t
iu +1
t
iu −t
iu 1
t
iu +1t
iu +
1
t
iu −t
iu 1
t
iu +1t
iu +1
t
iu −t
iu 1
t
iu +1t
iu +
![Page 26: A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM’S NEW KIND …nonlinear.eecs.berkeley.edu/CellularAutomata/PartII.pdf · in A New Kind of Science [Wolfram, 2002] for the one-dimensionalbinaryCellular](https://reader034.fdocuments.us/reader034/viewer/2022051604/5ff5c0f683eff57983406834/html5/thumbnails/26.jpg)
2402 L. O. Chua et al.
Table 5. (Continued)
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A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part II 2403
Table 5. (Continued)
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2404 L. O. Chua et al.
Table 5. (Continued)
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
-1
1
1
1
-1
-1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
-1
1
1
1
-1
-1
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
1
1
-1
-1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
1
-1
1
1
1
-1
-1
1
1 13 2 4 1sgn[ ]t t t t
i i i iu u u u+− ++ − +−= | | ] 1
1 122 1sgn[ ]t t t t
i i i iu u u u+− +− − +− −= | | ]
1
1 1sgn 2 33 42[ ]t t t t
i i i iu u u u+− +−= −− + − +| |( )| | ] 1
1 1sgn 3 2[ ]t t t t
i i i iu u u u+− += +− − + ]
2=
1=
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
3 5
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
3 5
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
0
3 5
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
0
3 5
Rule 56 : Rule 57 :
Rule 58 : Rule 59 :3=
2=
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
vertex
1
t
iu −t
iu 1
t
iu +1t
iu +1
t
iu −t
iu 1
t
iu +1t
iu +
1
t
iu −t
iu 1
t
iu +1t
iu +1
t
iu −t
iu 1
t
iu +1t
iu +
![Page 29: A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM’S NEW KIND …nonlinear.eecs.berkeley.edu/CellularAutomata/PartII.pdf · in A New Kind of Science [Wolfram, 2002] for the one-dimensionalbinaryCellular](https://reader034.fdocuments.us/reader034/viewer/2022051604/5ff5c0f683eff57983406834/html5/thumbnails/29.jpg)
A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part II 2405
Table 5. (Continued)
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
1
1
1
1
-1
-1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
1
1
1
1
-1
-1
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
1
1
1
1
-1
-1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
1
1
1
1
1
-1
-1
1
11sgn[ ]t t t
i i iu u u+−−= +| | ] 1
1 13 2 2 1sgn[ ]t t t t
i i i iu u u u+− +− − +− −= | |
1
1 13 2 2 1sgn[ ]t t t t
i i i iu u u u+− +− − −− −= | | ] 1 sgn 1[ ]t t
i iu u+ = +− []
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
3 5
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
3 5
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
0
3 5
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
0
3 5
Rule 60 : Rule 6 :
Rule 62 : Rule 63 :
2= 2=
2= 1=
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
vertex
1
t
iu −t
iu 1
t
iu +1t
iu +1
t
iu −t
iu 1
t
iu +1t
iu +
1
t
iu −t
iu 1
t
iu +1t
iu +1
t
iu −t
iu 1
t
iu +1t
iu +
![Page 30: A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM’S NEW KIND …nonlinear.eecs.berkeley.edu/CellularAutomata/PartII.pdf · in A New Kind of Science [Wolfram, 2002] for the one-dimensionalbinaryCellular](https://reader034.fdocuments.us/reader034/viewer/2022051604/5ff5c0f683eff57983406834/html5/thumbnails/30.jpg)
2406 L. O. Chua et al.
Table 5. (Continued)
1
1 1sgn 2[ ]t t t t
i i i iu u u u
+− += + − − [] 1
1 11 1sgn 2 2[ ]t t t t
i i i iu u u u
+− +− + +−= | | []
1
1 12sgn[ ]t t t t
i i i iu u u u
+− +− + − −= +| | [] 1
1 13 1sgn 2 4[ ]t t t t
i i i iu u u u
+− +− − −−= | | []
Difference Equation Difference Equation
Difference Equation Difference Equation
1=Rule 64 :
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
0
6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
0
6
Rule 65 :
Rule 66 : Rule 67 :
2=
2= 2=
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
t
iu −
t
iu 1
t
iu +
1t
iu
+
-1
-1
-1
-1
-1
-1
1
-1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
-1
-1
-1
-1
-1
1
-1
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
t
iu −
t
iu 1
t
iu +
1t
iu
+
-1
1
-1
-1
-1
-1
1
-1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
1
-1
-1
-1
-1
1
-1
vertex
vertex
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A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part II 2407
Table 5. (Continued)
1
1sgn 1[ ]t t t
i i iu u u
++= − − [] 1
1 1sgn 2 1[ ]t t t t
i i i iu u u u
+− += − + − − [
1
1 13 4 2 1sgn[ ]t t t t
i i i iu u u u
+− +−− + + −= | | []1
1 132sgn 4 2 3[ ]t t t t
i i i iu u u u
+− += − + − −− − | |( )| | [
1=
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
0
6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
0
6
1=Rule 68 : Rule 69 :
2=Rule 70 : Rule 7 : 3=
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
1
-1
-1
-1
1
-1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
1
-1
-1
-1
1
-1
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
1
-1
-1
-1
1
-1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
1
1
-1
-1
-1
1
-1
vertex
vertex
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
![Page 32: A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM’S NEW KIND …nonlinear.eecs.berkeley.edu/CellularAutomata/PartII.pdf · in A New Kind of Science [Wolfram, 2002] for the one-dimensionalbinaryCellular](https://reader034.fdocuments.us/reader034/viewer/2022051604/5ff5c0f683eff57983406834/html5/thumbnails/32.jpg)
2408 L. O. Chua et al.
Table 5. (Continued)
1
1 11 2 2 1sgn[ ]t t t t
i i i iu u u u
+− +− − + − −= | | []1
1 1 11sgn[ ]t t t t
i i i iu u u u
+− +− + +−= | | []
1
1 13 2 4 1sgn[ ]t t t t
i i i iu u u u
+− ++ +−= −| | []1
1 12 (2 1)sgn[ ]t t t t
i i i iu u u u
+− +− + +−= | | [
2=
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
3 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
3 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
0
3 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
0
3 6
Rule 72 : Rule 73 :
Rule 74 : Rule 75 :
2=
2=2=
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
-1
1
-1
-1
1
-1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
-1
1
-1
-1
1
-1
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
1
-1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
1
-1
1
-1
-1
1
-1
vertex
vertex
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
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A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part II 2409
Table 5. (Continued)
1
1 1sgn 2 1[ ]t t t t
i i i iu u u u
+− += − + − − [] 1
1 1sgn[ ]t t t t
i i i iu u u u
+− += − + − []
1
1 1sgn 3 2 432[ ]t t t t
i i i iu u u u
+− += − + ++− − |( ) []||||| || || || | 1
1 1sgn 3 2[ ]t t t t
i i i iu u u u
+− += +− + − [
1=
1=
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
3 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
3 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
0
3 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
0
3 6
1=Rule 76 : Rule 77 :
Rule78 : Rule 79 :3=
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
1
1
-1
-1
1
-1
n
0
2
3
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7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
1
1
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-1
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vertex
n
0
2
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-1
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1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
1
1
-1
-1
1
-1
n
0
2
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-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
1
1
1
-1
-1
1
-1
vertex
vertex
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
![Page 34: A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM’S NEW KIND …nonlinear.eecs.berkeley.edu/CellularAutomata/PartII.pdf · in A New Kind of Science [Wolfram, 2002] for the one-dimensionalbinaryCellular](https://reader034.fdocuments.us/reader034/viewer/2022051604/5ff5c0f683eff57983406834/html5/thumbnails/34.jpg)
2410 L. O. Chua et al.
Table 5. (Continued)
1
1 1sgn 1[ ]t t t
i i iu u u
+− += − − [] 1
1 1sgn 2 1[ ]t t t t
i i i iu u u u
+− += − − − []
1
1 13 4 1sgn 2[ ]t t t t
i i i iu u u u
+− +− − − − −= []| || || || | 1
1 1sgn 2 3 3 4 2 1[ ]t t t t
i i i iu u u u
+− +− −= − + − +( ) [| || || || || || || || |
1= 1=Rule 80 :
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
0
6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
0
6
Rule 8 :
Rule 82 : Rule 83 :2= 3=
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
-1
-1
1
-1
1
-1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
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-1
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-1
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1
-1
-1
-1
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-1
1
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vertex
n
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-1
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1
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-1
1
1
-1
1
-1
1
-1
1
-1
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-1
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-1
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1
-1
1
-1
n
0
2
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-1
-1
-1
-1
1
1
1
1
-1
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1
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-1
1
-1
1
1
1
-1
-1
1
-1
1
-1
vertex
vertex
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
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A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part II 2411
Table 5. (Continued)
1
1 1sgn 2 1[ ]t t t t
i i i iu u u u
+− += −+ − [] 1
1sgn[ ]t t
i iu u
++= − []
1
1 122 1sgn[ ]t t t t
i i i iu u u u
+− +− − − − −= []| || || || | 1
1 1sgn 3 2[ ]t t t t
i i i iu u u u
+− += +− − − []
1=
1=
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
0
6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
0
6
1=Rule 84 : Rule 85 :
2=Rule 86 : Rule 87 :
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
1
-1
1
-1
1
-1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
1
-1
1
-1
1
-1
vertex
n
0
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-1
-1
-1
-1
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1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
1
-1
1
-1
1
-1
n
0
2
3
4
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5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
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-1
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-1
1
-1
1
1
1
1
-1
1
-1
1
-1
vertex
vertex
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
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2412 L. O. Chua et al.
Table 5. (Continued)
1
1 13 4 2 1sgn[ ]t t t t
i i i iu u u u
+− +− − − − += []| || || || | 1
1 122 1sgn[ ]t t t t
i i i iu u u u
+− +− + +−= | | []
1
1 11sgn[ ]t t t
i i iu u u
+− ++−= []| || || || | 1
1 13 2 2 1sgn[ ]t t t t
i i i iu u u u
+− +− − + − −= | []||||
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
3 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
3 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
0
3 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
0
3 6
Rule 88 : Rule 89 :
Rule 90 : Rule 9 :
2= 2=
2= 2=
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
-1
1
1
-1
1
-1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
-1
1
1
-1
1
-1
vertex
n
0
2
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6
7
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-1
-1
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1
1
-1
-1
1
1
-1
1
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1
-1
1
-1
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-1
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-1
1
1
-1
1
-1
n
0
2
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7
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-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
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-1
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1
-1
1
-1
1
1
1
-1
1
1
-1
1
-1
vertex
vertex
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
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A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part II 2413
Table 5. (Continued)
1
1 1sg 2 43n 32[ ]t t t t
i i i iu u u u
+− +− + +−= −( ) []| || || || || || || || | 1
1 1sgn 3 2[ ]t t t t
i i i iu u u u
+− += +− + − []
1
1 13 2 2 1sgn[ ]t t t t
i i i iu u u u
+− +− − − − −= []| || || || | 1
1 1sgn 1[ ]t t t
i i iu u u
+− += +− − []
1=
1=
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
3 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
3 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
0
3 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
0
3 6
Rule 92 : Rule 93 :
Rule 94 : Rule 95 :
3=
2=
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
1
1
1
-1
1
-1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
1
1
1
-1
1
-1
vertex
n
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-1
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-1
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1
1
1
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-1
1
-1
1
1
1
1
1
1
-1
1
-1
vertex
vertex
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
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2414 L. O. Chua et al.
Table 5. (Continued)
1
1 121 2sgn[ ]t t t t
i i i iu u u u
+− +− + +− += []| || || || | 1
1 11 1sgn[ ]t t t t
i i i iu u u u
+− +− + +− += []| || || || |
1
1 13 2 4 1sgn[ ]t t t t
i i i iu u u u
+− +− − − − += []| || || || | 1
1 12 1sgn 2[ ]t t t t
i i i iu u u u
+− ++ +−= − + []| || || || |
Rule 96 :
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
0
5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
0
5 6
Rule 97 :
Rule 98 : Rule 99 :
2=2=
2= 2=
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
-1
-1
-1
1
1
-1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
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-1
1
-1
1
1
-1
-1
-1
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-1
vertex
n
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1
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-1
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-1
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-1
-1
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1
-1
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-1
-1
-1
-1
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1
1
-1
-1
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-1
-1
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-1
1
-1
1
1
1
-1
-1
-1
1
1
-1
vertex
vertex
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
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A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part II 2415
Table 5. (Continued)
1
1 13 4 2 1sgn[ ]t t t t
i i i iu u u u
+− +−− − − += []| || || || | 1
1 12 2 2 1sgn[ ]t t t t
i i i iu u u u
+− +− − −−= []| || || || |
1
11sgn[ ]t t t
i i iu u u
+++−= []| || || || | 1
1 13 2 2 1sgn[ ]t t t t
i i i iu u u u
+− +− + += − + []| || || || |
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
0
5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
0
5 6
Rule 00 : Rule 0 :
Rule 02 : Rule 03 :
2= 2=
2= 2=
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
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-1
-1
1
-1
-1
1
1
-1
n
0
2
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4
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-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
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1
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1
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1
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1
-1
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vertex
n
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-1
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1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
1
-1
-1
1
1
-1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
1
1
-1
-1
1
1
-1
vertex
vertex
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
![Page 40: A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM’S NEW KIND …nonlinear.eecs.berkeley.edu/CellularAutomata/PartII.pdf · in A New Kind of Science [Wolfram, 2002] for the one-dimensionalbinaryCellular](https://reader034.fdocuments.us/reader034/viewer/2022051604/5ff5c0f683eff57983406834/html5/thumbnails/40.jpg)
2416 L. O. Chua et al.
Table 5. (Continued)
1
1 11 1sgn[ ]t t t t
i i i iu u u u
+− +− − −− += []| || || || | 1
1 1sg 1 12n[ ]t t t t
i i i iu u u u
+− += − + − +− −( ) [| || || || || || || || |
1
1 122 1sgn[ ]t t t t
i i i iu u u u
+− ++ + −−= []| || || || | 1
1 11 1sgn[ ]t t t t
i i i iu u u u
+− +− − +− + += []| || || || |
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
3 5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
3 5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
0
3 5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
0
3 5 6
Rule 04 : Rule 05 :
Rule 06 : Rule 07 :
2=
2=2=
3=
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
-1
1
-1
1
1
-1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
-1
1
-1
1
1
-1
vertex
n
0
2
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-1
-1
-1
-1
1
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-1
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1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
1
-1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
1
-1
1
-1
1
1
-1
vertex
vertex
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
![Page 41: A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM’S NEW KIND …nonlinear.eecs.berkeley.edu/CellularAutomata/PartII.pdf · in A New Kind of Science [Wolfram, 2002] for the one-dimensionalbinaryCellular](https://reader034.fdocuments.us/reader034/viewer/2022051604/5ff5c0f683eff57983406834/html5/thumbnails/41.jpg)
A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part II 2417
Table 5. (Continued)
1
1 122 1sgn[ ]t t t t
i i i iu u u u
+− +− − −− += []| || || || | 1
1 11 1sgn[ ]t t t t
i i i iu u u u
+− +− + −− + += []| || || || |
1
1 12 32 1sgn[ ]t t t t
i i i iu u u u
+− +− + −+ −= []| || || || | 1
1 11 2 1sgn 2[ ]t t t t
i i i iu u u u
+− +− + −+= − []| || || || |
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
3 5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
3 5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
0
3 5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
0
3 5 6
Rule 08 : Rule 09 :
Rule 0 : Rule :
2= 2=
2=2=
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
1
1
-1
1
1
-1
n
0
2
3
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5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
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1
-1
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1
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-1
1
1
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1
1
-1
vertex
n
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2
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-1
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-1
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-1
1
1
-1
-1
1
1
-1
1
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1
-1
1
-1
1
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1
1
1
-1
1
1
-1
n
0
2
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-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
1
1
1
-1
1
1
-1
vertex
vertex
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
![Page 42: A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM’S NEW KIND …nonlinear.eecs.berkeley.edu/CellularAutomata/PartII.pdf · in A New Kind of Science [Wolfram, 2002] for the one-dimensionalbinaryCellular](https://reader034.fdocuments.us/reader034/viewer/2022051604/5ff5c0f683eff57983406834/html5/thumbnails/42.jpg)
2418 L. O. Chua et al.
Table 5. (Continued)
1
1 1sgn 2 1[ ]t t t t
i i i iu u u u
+− += − − − [] 1
1 1sgn[ ]t t t t
i i i iu u u u
+− += − − []
1
1 132sgn 4 2 3[ ]t t t t
i i i iu u u u
+− += − + − +− −( ) []| || || || || || || || | 1
1 1sgn 3 2[ ]t t t t
i i i iu u u u
+− += − − + []
1=Rule 2 :
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
0
5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
0
5 6
Rule 3 :
1=Rule 4 : Rule 5 :
1=
3=
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
-1
-1
1
1
1
-1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
-1
-1
1
1
1
-1
vertex
n
0
2
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-1
-1
-1
-1
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1
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-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
1
1
1
-1
n
0
2
3
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7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
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1
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-1
1
-1
1
1
1
-1
-1
1
1
1
-1
vertex
vertex
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
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A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part II 2419
Table 5. (Continued)
1
1 132sgn 4 2 3[ ]t t t t
i i i iu u u u
+− += − − + +− −( ) []| || || || || || || || | 1
1 1sgn 3 2[ ]t t t t
i i i iu u u u
+− += − − + []
1
1 13 2 2 1sgn[ ]t t t t
i i i iu u u u
+− +− − − − −= []| || || || | 1
1sgn 1[ ]t t t
i i iu u u
++= +− − []
1=
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
0
5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
0
5 6
Rule 6 : Rule 7 :
2=Rule 8 : Rule 9 :
3=3=
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
1
-1
1
1
1
-1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
1
-1
1
1
1
-1
vertex
n
0
2
3
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6
7
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-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
1
-1
1
1
1
-1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
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-1
1
-1
1
-1
1
1
1
1
-1
1
1
1
-1
vertex
vertex
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
![Page 44: A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM’S NEW KIND …nonlinear.eecs.berkeley.edu/CellularAutomata/PartII.pdf · in A New Kind of Science [Wolfram, 2002] for the one-dimensionalbinaryCellular](https://reader034.fdocuments.us/reader034/viewer/2022051604/5ff5c0f683eff57983406834/html5/thumbnails/44.jpg)
2420 L. O. Chua et al.
Table 5. (Continued)
1
1 122 1sgn[ ]t t t t
i i i iu u u u
+− ++ + −−= []| || || || | 1
1 11 1sgn[ ]t t t t
i i i iu u u u
+− +− + +− + −= []| || || || |
1
1 13 2 2 1sgn[ ]t t t t
i i i iu u u u
+− +− − − − += []| || || || | 1
1 121 2 1sgn[ ]t t t t
i i i iu u u u
+− +− ++ − −= []| || || || |
2=
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
3 5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
3 5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
0
3 5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
0
3 5 6
Rule 20 : Rule 2 :
Rule 22 : Rule 23 :
2=
2=2=
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
-1
1
1
1
1
-1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
-1
1
1
1
1
-1
vertex
n
0
2
3
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6
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5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
1
1
1
-1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
1
-1
1
1
1
1
-1
vertex
vertex
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
![Page 45: A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM’S NEW KIND …nonlinear.eecs.berkeley.edu/CellularAutomata/PartII.pdf · in A New Kind of Science [Wolfram, 2002] for the one-dimensionalbinaryCellular](https://reader034.fdocuments.us/reader034/viewer/2022051604/5ff5c0f683eff57983406834/html5/thumbnails/45.jpg)
A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part II 2421
Table 5. (Continued)
1
1 13 2 2 1sgn[ ]t t t t
i i i iu u u u
+− ++ + −−= []| || || || | 1
1 121 2sgn[ ]t t t t
i i i iu u u u
+− +− + − + + −= []| || || || |
1
1 12sgn[ ]t t t t
i i i iu u u u
+− +− − − −= []| || || || | 1
1 1sgn 2[ ]t t t t
i i i iu u u u
+− += +− − − []
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
3 5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
3 5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
0
3 5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
0
3 5 6
Rule 24 : Rule 25 :
Rule 26 : Rule 27 :
2= 2=
2= 1=
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
1
1
1
1
1
-1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
1
1
1
1
1
-1
vertex
n
0
2
3
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6
7
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-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
1
1
1
1
1
-1
n
0
2
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-1
-1
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-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
1
1
1
1
1
1
-1
vertex
vertex
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
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2422 L. O. Chua et al.
Table 5. (Continued)
1=Rule 28 :
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
0
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
0
Rule 29 :
Rule 30 : Rule 3 :
2=
2= 2=
7
7
7
7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
1
1 1sgn 2[ ]t t t t
i i i iu u u u
+− += + + − [] 1
1 12sgn[ ]t t t t
i i i iu u u u
+− +− + ++= []| || || || |
1
1 121 2sgn[ ]t t t t
i i i iu u u u
+− +− − + − += []| || || || | 1
1 13 4 2 1sgn[ ]t t t t
i i i iu u u u
+− +− − +−= []| || || || |
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
-1
-1
-1
-1
-1
1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
-1
-1
-1
-1
-1
1
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
-1
-1
-1
1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
1
-1
-1
-1
-1
-1
1
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
vertex
vertex
![Page 47: A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM’S NEW KIND …nonlinear.eecs.berkeley.edu/CellularAutomata/PartII.pdf · in A New Kind of Science [Wolfram, 2002] for the one-dimensionalbinaryCellular](https://reader034.fdocuments.us/reader034/viewer/2022051604/5ff5c0f683eff57983406834/html5/thumbnails/47.jpg)
A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part II 2423
Table 5. (Continued)
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
0
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
0
Rule 32 : Rule 33 :
Rule 34 : Rule 35 :
2= 2=
2= 2=
7
7
7
7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
1
1 11 2 2 1sgn[ ]t t t t
i i i iu u u u
+− +− − +−= []| || || || | 1
1 13 2 4 1sgn[ ]t t t t
i i i iu u u u
+− +− − − + += [| || || || |
1
1 1 11sgn[ ]t t t t
i i i iu u u u
+− +− − +−= []| || || || | 1
1 122 1sgn[ ]t t t t
i i i iu u u u
+− +− − +−= []| || || || |
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
1
-1
-1
-1
-1
1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
1
-1
-1
-1
-1
1
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
1
-1
-1
-1
-1
1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
1
1
-1
-1
-1
-1
1
vertex
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
vertex
vertex
![Page 48: A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM’S NEW KIND …nonlinear.eecs.berkeley.edu/CellularAutomata/PartII.pdf · in A New Kind of Science [Wolfram, 2002] for the one-dimensionalbinaryCellular](https://reader034.fdocuments.us/reader034/viewer/2022051604/5ff5c0f683eff57983406834/html5/thumbnails/48.jpg)
2424 L. O. Chua et al.
Table 5. (Continued)
2=
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
3
7
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
3
7
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
0
3
7
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
0
3
7
1=Rule 36 : Rule 37 :
1=Rule 38 : Rule 39 : 3=
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
1
1sgn 1[ ]t t t
i i iu u u
++= + − [] 1
1 13 2 4 1sgn[ ]t t t t
i i i iu u u u
+− +− + −= − − []| || || || |
1
1 1sgn 2 1[ ]t t t t
i i i iu u u u
+− += − + + − [] 1
1 1sgn 2 43 32[ ]t t t t
i i i iu u u u
+− += − − + −−− +( ) [| || || || || || || || |
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
-1
1
-1
-1
-1
1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
-1
1
-1
-1
-1
1
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
-1
1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
1
-1
1
-1
-1
-1
1
vertex
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
vertex
vertex
![Page 49: A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM’S NEW KIND …nonlinear.eecs.berkeley.edu/CellularAutomata/PartII.pdf · in A New Kind of Science [Wolfram, 2002] for the one-dimensionalbinaryCellular](https://reader034.fdocuments.us/reader034/viewer/2022051604/5ff5c0f683eff57983406834/html5/thumbnails/49.jpg)
A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part II 2425
Table 5. (Continued)
1=
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
3
7
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
3
7
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
0
3
7
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
0
3
7
1=Rule 40 : Rule 4 :
1=Rule 42 : Rule 43 :
3=
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
1
1 1sgn 2 1[ ]t t t t
i i i iu u u u
+− += − + + − []1
1 1sgn 2 3 2 4 3 1[ ]t t t t
i i i iu u u u
+− +− −= − − + +( ) [| || || || || || || || |
1
1 1sgn[ ]t t t t
i i i iu u u u
+− += − ++ [] 1
1 1sgn 3 2[ ]t t t t
i i i iu u u u
+− += + +− + [
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
1
1
-1
-1
-1
1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
1
1
-1
-1
-1
1
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
1
1
-1
-1
-1
1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
1
1
1
-1
-1
-1
1
vertex
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
vertex
vertex
![Page 50: A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM’S NEW KIND …nonlinear.eecs.berkeley.edu/CellularAutomata/PartII.pdf · in A New Kind of Science [Wolfram, 2002] for the one-dimensionalbinaryCellular](https://reader034.fdocuments.us/reader034/viewer/2022051604/5ff5c0f683eff57983406834/html5/thumbnails/50.jpg)
2426 L. O. Chua et al.
Table 5. (Continued)
Rule 44 :
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
7
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
0
7
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
7
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
0
7
Rule 45 :
Rule 46 : Rule 47 :
2= 2=
2= 2=
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
1
1 121 2sgn[ ]t t t t
i i i iu u u u
+− +− + −− += []| || || || | 1
1 13 2 4 1sgn[ ]t t t t
i i i iu u u u
+− +− + += − − []| || || || |
1
1 1 11sgn[ ]t t t t
i i i iu u u u
+− +− + −−= []| || || || | 1
1 122 1sgn[ ]t t t t
i i i iu u u u
+− +− + −−= []| || || || |
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
-1
-1
1
-1
-1
1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
-1
-1
1
-1
-1
1
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
1
-1
-1
1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
1
-1
-1
1
-1
-1
1
vertex
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
vertex
vertex
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A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part II 2427
Table 5. (Continued)
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
7
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
7
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
0
7
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
0
7
Rule 48 : Rule 49 :
Rule 50 : Rule 5 :
2= 2=
2=3=
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
1
1 11 1sgn[ ]t t t t
i i i iu u u u
+− +− − +− += []| || || || | 1
1 122 1sgn[ ]t t t t
i i i iu u u u
+− +− − − + += []| || || || |
1
1 1sgn 3 7 4 2 4 3[ ]t t t t
i i i iu u u u
+− +− −= − − + −( ) []| || || || || || || || | 1
1 11 1sgn[ ]t t t t
i i i iu u u u
+− +− − −− + += []| || || || |
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
1
-1
1
-1
-1
1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
1
-1
1
-1
-1
1
vertex
n
0
2
3
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7
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-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
1
-1
1
-1
-1
1
n
0
2
3
4
6
7
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-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
1
1
-1
1
-1
-1
1
vertex
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
vertex
vertex
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2428 L. O. Chua et al.
Table 5. (Continued)
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
3
7
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
3
7
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
0
3
7
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
0
3
7
Rule 52 : Rule 53 :
Rule 54 : Rule 55 :
2= 2=
2= 2=
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
1
1 13 4 2 1sgn[ ]t t t t
i i i iu u u u
+− ++ −− −= []| || || || | 1
11sgn[ ]t t t
i i iu u u
++− +−= []| || || || |
1
1 122 1sgn[ ]t t t t
i i i iu u u u
+− ++ −−= − []| || || || | 1
1 13 2 2 1sgn[ ]t t t t
i i i iu u u u
+− +− − + − += []| || || || |
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
-1
1
1
-1
-1
1
vertex
n
0
2
3
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-1
-1
-1
-1
1
1
1
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-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
1
-1
-1
1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
1
-1
1
1
-1
-1
1
vertex
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
vertex
vertex
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A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part II 2429
Table 5. (Continued)
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
3
7
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
3
7
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
0
3
7
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
0
3
7
Rule 56 : Rule 57 :
Rule 58 : Rule 59 :
2= 2=
2= 2=
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
4
1
1 122 1sgn[ ]t t t t
i i i iu u u u
+− ++ − −−= | | [] 1
1 13 2 2 1sgn[ ]t t t t
i i i iu u u u
+− ++ −− −= [| || || || |
1
1 11 1sgn[ ]t t t t
i i i iu u u u
+− +− −− + −= []| || || || | 1
1 1sg 21n 2[ ]t t t t
i i i iu u u u
+− +− + + +− += [| || || || |
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
1
1
1
-1
-1
1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
1
1
1
-1
-1
1
vertex
n
0
2
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-1
-1
-1
-1
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1
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-1
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1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
1
1
1
-1
-1
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n
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2
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-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
1
1
1
1
-1
-1
1
vertex
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
vertex
vertex
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2430 L. O. Chua et al.
Table 5. (Continued)
1=Rule 60 :
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
5
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
0
5
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
5
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
0
5
Rule 6 :
1=Rule 62 : Rule 63 :
2=
3=
7
7
7
7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
1
1 1sgn 1[ ]t t t
i i iu u u
+− += + − [] 1
1 13 4 1sgn 2[ ]t t t t
i i i iu u u u
+− ++ −−= + []| || || || |
1
1 1sgn 2 1[ ]t t t t
i i i iu u u u
+− += − + − [] 1
1 1sgn 2 3 3 4 2 1[ ]t t t t
i i i iu u u u
+− +− − −−= −( ) []| || || || || || || || |
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
-1
-1
-1
1
-1
1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
-1
-1
-1
1
-1
1
vertex
n
0
2
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-1
-1
-1
-1
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1
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-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
-1
1
-1
1
n
0
2
3
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7
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-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
1
-1
-1
-1
1
-1
1
vertex
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
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iu −
t
iu 1
t
iu +
1t
iu
+
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A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part II 2431
Table 5. (Continued)
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
5
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
5
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
0
5
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
0
5
Rule 64 : Rule 65 :
Rule 66 : Rule 67 :
2= 2=
2= 2=
7
7
7
7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
1
1 1sg 13n 2 2[ ]t t t t
i i i iu u u u
+− +−− + + − −= []| || || || | 1
1 11sgn[ ]t t t
i i iu u u
+− +− − += []| || || || |
1
1 122 1sgn[ ]t t t t
i i i iu u u u
+− +− − +−= []| || || || | 1
1 13 2 2 1sgn[ ]t t t t
i i i iu u u u
+− +− − + + −= []| || || || |
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
1
-1
-1
1
-1
1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
1
-1
-1
1
-1
1
vertex
n
0
2
3
4
6
7
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-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
1
-1
-1
1
-1
1
n
0
2
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7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
1
1
-1
-1
1
-1
1
vertex
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
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vertex
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2432 L. O. Chua et al.
Table 5. (Continued)
2=
1=
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
3 5
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
3 5
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
0
3 5
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
0
3 5
Rule 68 : Rule 69 :
1=Rule 70 : Rule 7 :
7
7
7
7
1=
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
1
1 1sgn 2 1[ ]t t t t
i i i iu u u u
+− += + + − [] 1
1 122 1sgn[ ]t t t t
i i i iu u u u
+− ++ − +−= []| || || || |
1
1sgn[ ]t t
i iu u
++= [] 1
1 1sgn 2 1[ ]t t t t
i i i iu u u u
+− += +− − + []
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
-1
1
-1
1
-1
1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
-1
1
-1
1
-1
1
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
1
-1
1
-1
1
-1
1
vertex
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
vertex
vertex
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A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part II 2433
Table 5. (Continued)
1=
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
3 5
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
3 5
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
0
3 5
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
0
3 5
Rule 72 : Rule 73 :
Rule 74 : Rule 75 :
2=3=
1=
7
7
7
7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
1
1 1sgn 2 3 3 4 2 1[ ]t t t t
i i i iu u u u
+− +− −= − − + +( ) []| || || || || || || || | 1
1 13 2 2 1sgn[ ]t t t t
i i i iu u u u
+− +−− − + −= [| || || || |
1
1 1sgn 2 1[ ]t t t t
i i i iu u u u
+− += +− + + [] 1
1 1sgn 1[ ]t t t
i i iu u u
+− += +− + []
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
1
1
-1
1
-1
1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
1
1
-1
1
-1
1
vertex
n
0
2
3
4
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7
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-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
1
1
-1
1
-1
1
n
0
2
3
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5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
1
1
1
-1
1
-1
1
vertex
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
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vertex
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2434 L. O. Chua et al.
Table 5. (Continued)
1=Rule 76 :
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
5
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
0
5
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
5
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
0
5
Rule 77 :
1= 1=Rule 78 : Rule 79 :
3=
7
7
7
7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
1
1 1sgn 2 1[ ]t t t t
i i i iu u u u
+− += − + − [] 1
1 1sgn 4 2 332 )[ ]t t t t
i i i iu u u u
+− += − − −− − +( [| || || || || || || || |
1
1 1sgn[ ]t t t t
i i i iu u u u
+− += − + [] 1
1 1sgn 3 2[ ]t t t t
i i i iu u u u
+− += +− + []
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
-1
-1
1
1
-1
1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
-1
-1
1
1
-1
1
vertex
n
0
2
3
4
6
7
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-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
1
1
-1
1
n
0
2
3
4
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7
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-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
1
-1
-1
1
1
-1
1
vertex
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
vertex
vertex
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A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part II 2435
Table 5. (Continued)
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
5
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
5
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
0
5
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
0
5
Rule 80 : Rule 8 :
2=Rule 82 : Rule 83 :
2= 2=
2=
7
7
7
7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
1
1 122 1sgn[ ]t t t t
i i i iu u u u
+− +− − +− += []| || || || | 1
1 13 2 2 1sgn[ ]t t t t
i i i iu u u u
+− ++ − −−= [| || || || |
1
1 11 1sgn[ ]t t t t
i i i iu u u u
+− +− + −− + −= []| || || || | 1
1 11 2 2sgn[ ]t t t t
i i i iu u u u
+− +− +− + += [| || || || |
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
1
-1
1
1
-1
1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
1
-1
1
1
-1
1
vertex
n
0
2
3
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-1
-1
-1
-1
1
1
1
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-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
1
-1
1
1
-1
1
n
0
2
3
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7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
1
1
-1
1
1
-1
1
vertex
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
vertex
vertex
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2436 L. O. Chua et al.
Table 5. (Continued)
2=
1=
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
3 5
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
3 5
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
0
3 5
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
0
3 5
Rule 84 : Rule 85 :
Rule 86 : Rule 87 :1=
3=
7
7
7
7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
1
1 1sgn 2 43 32[ ]t t t t
i i i iu u u u
+− +− + −−= −+( ) []| || || || || || || || | 1
1 13 2 2 1sgn[ ]t t t t
i i i iu u u u
+− +− − − + −= [| || || || |
1
1 1sgn 2 1[ ]t t t t
i i i iu u u u
+− += − + + [] 1
1sgn 1[ ]t t t
i i iu u u
++= +− + []
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
-1
1
1
1
-1
1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
-1
1
1
1
-1
1
vertex
n
0
2
3
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6
7
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-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
1
1
-1
1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
1
-1
1
1
1
-1
1
vertex
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
vertex
vertex
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A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part II 2437
Table 5. (Continued)
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
3 5
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
3 5
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
0
3 5
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
0
3 5
Rule 88 : Rule 89 :
Rule 90 : Rule 9 :
2= 2=
2= 1=
7
7
7
7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
1
1 1 13 2 2sgn[ ]t t t t
i i i iu u u u
+− ++ − −−= []| || || || | 1
1 12sgn[ ]t t t t
i i i iu u u u
+− ++ −−= []| || || || |
1
1 121 2 1sgn[ ]t t t t
i i i iu u u u
+− +− + +− + += []| || || || | 1
1 1sgn 2[ ]t t t t
i i i iu u u u
+− += − +− + []
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
1
1
1
1
-1
1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
1
1
1
1
-1
1
vertex
n
0
2
3
4
6
7
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-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
1
1
1
1
-1
1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
1
1
1
1
1
-1
1
vertex
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
vertex
vertex
![Page 62: A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM’S NEW KIND …nonlinear.eecs.berkeley.edu/CellularAutomata/PartII.pdf · in A New Kind of Science [Wolfram, 2002] for the one-dimensionalbinaryCellular](https://reader034.fdocuments.us/reader034/viewer/2022051604/5ff5c0f683eff57983406834/html5/thumbnails/62.jpg)
2438 L. O. Chua et al.
Table 5. (Continued)
1=Rule 92 :
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
0
6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
0
6
Rule 93 :
Rule 94 : Rule 95 :
2=
2= 2=
7
7
7
7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
1
1sgn 1[ ]t t t
i i iu u u
+−= + − [] 1
1 13 1sgn 4 2[ ]t t t t
i i i iu u u u
+− +− − −−= []| || || || |
1
1 13 2 4 1sgn[ ]t t t t
i i i iu u u u
+− +− + +− −= []| || || || | 1
11sgn[ ]t t t
i i iu u u
+−−= − + []| || || || |
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
-1
-1
-1
-1
1
1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
-1
-1
-1
-1
1
1
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
-1
-1
1
1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
1
-1
-1
-1
-1
1
1
vertex
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
vertex
vertex
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A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part II 2439
Table 5. (Continued)
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
0
6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
0
6
1=Rule 96 : Rule 97 :
2=Rule 98 : Rule 99 :
3=
2=
7
7
7
7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
1
1 1sgn 2 1[ ]t t t t
i i i iu u u u
+− += + − − [] 1
1 1sgn 2 3 3 4 2 1[ ]t t t t
i i i iu u u u
+− +− −= − + + −( ) [| || || || || || || || |
1
1 122 1sgn[ ]t t t t
i i i iu u u u
+− + +−− −= []| || || || | 1
1 13 2 2 1sgn[ ]t t t t
i i i iu u u u
+− +− − +−= []| || || || |
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
1
-1
-1
-1
1
1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
1
-1
-1
-1
1
1
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
1
-1
-1
-1
1
1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
1
1
-1
-1
-1
1
1
vertex
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
vertex
vertex
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2440 L. O. Chua et al.
Table 5. (Continued)
2=
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
3 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
3 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
0
3 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
0
3 6
Rule 200 : Rule 20 :
Rule 202 : Rule 203 : 2=
1=
3=
7
7
7
7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
1
1 1sgn 2 1[ ]t t t t
i i i iu u u u
+− += + + − [] 1
1 122 1sgn[ ]t t t t
i i i iu u u u
+− +− + +−= []| || || || |
1
1 1sgn 2 3 3 4 2 1[ ]t t t t
i i i iu u u u
+− +− + +−= −( ) []| || || || || || || || | 1
1 13 2 2 1sgn[ ]t t t t
i i i iu u u u
+− +− + −− −= []| || || || |
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
-1
1
-1
-1
1
1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
-1
1
-1
-1
1
1
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
1
1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
1
-1
1
-1
-1
1
1
vertex
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
vertex
vertex
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A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part II 2441
Table 5. (Continued)
1=
1=
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
3 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
3 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
0
3 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
0
3 6
1=Rule 204 : Rule 205 :
Rule 206 : Rule 207 :1=
7
7
7
7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
1 sgn[ ]t t
i iu u
+ = [] 1
1 1sgn 2 1[ ]t t t t
i i i iu u u u
+− += +− + − []
1
1 1sgn 2 1[ ]t t t t
i i i iu u u u
+− += +− + + [] 1
1sgn 1[ ]t t t
i i iu u u
+−= +− + []
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
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1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
1
1
-1
-1
1
1
n
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2
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-1
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-1
1
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1
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-1
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1
1
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1
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-1
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-1
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vertex
n
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1
-1
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-1
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n
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1
-1
1
-1
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1
1
1
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-1
-1
1
1
vertex
1
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iu −
t
iu 1
t
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1t
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+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
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iu 1
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iu +
1t
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+1
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iu 1
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1t
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+
vertex
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2442 L. O. Chua et al.
Table 5. (Continued)
1=Rule 208 :
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
0
6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
0
6
Rule 209 :
Rule 2 0 : Rule 2 :2=
3=
2=
7
7
7
7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
1
1 1sgn 2 1[ ]t t t t
i i i iu u u u
+− += + − − [] 1
1 1sgn 2 33 42[ ]t t t t
i i i iu u u u
+− +−= + − −−( ) [| || || || || || || || |
1
1 122 1sgn[ ]t t t t
i i i iu u u u
+− +− + −− += []| || || || | 1
1 13 2 2 1sgn[ ]t t t t
i i i iu u u u
+− +− + −− += [| || || || |
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
-1
-1
1
-1
1
1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
-1
-1
1
-1
1
1
vertex
n
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2
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-1
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-1
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1
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1
1
-1
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-1
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-1
1
-1
1
-1
1
-1
-1
1
-1
1
1
n
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2
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-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
1
-1
-1
1
-1
1
1
vertex
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
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1t
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+
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vertex
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A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part II 2443
Table 5. (Continued)
1=
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
0
6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
0
6
1=Rule 2 2 : Rule 2 3 :
2=Rule 2 4 : Rule 2 5 : 2=
7
7
7
7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
1
1 1sgn[ ]t t t t
i i i iu u u u
+− += + − [] 1
1 1sgn 3 2[ ]t t t t
i i i iu u u u
+− += −+ + []
1
1 11 1sgn[ ]t t t t
i i i iu u u u
+− ++ −− + += []| || || || | 1
1 121 2 1sgn[ ]t t t t
i i i iu u u u
+− ++ +− + −= []| || || || |
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
1
-1
1
-1
1
1
n
0
2
3
4
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7
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-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
1
-1
1
-1
1
1
vertex
n
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-1
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1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
1
-1
1
-1
1
1
n
0
2
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7
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-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
1
1
-1
1
-1
1
1
vertex
1
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iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
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iu −
t
iu 1
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iu +
1t
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+
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vertex
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2444 L. O. Chua et al.
Table 5. (Continued)
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
3 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
3 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
0
3 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
0
3 6
Rule 2 6 : Rule 2 7 :
Rule 2 8 : Rule 2 9 :
2=
2= 2=
3=
7
7
7
7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
1
1 1sgn 3 4 232[ ]t t t t
i i i iu u u u
+− +− += − − +− +( ) []| || || || || || || || | 1
1 13 2 2 1sgn[ ]t t t t
i i i iu u u u
+− +− − + − −= []| || || || |
1
1 13 2 2 1sgn[ ]t t t t
i i i iu u x u
+− +− + −−= []| || || || | 1
1 12sgn[ ]t t t t
i i i iu u u u
+− +− − + −= []| || || || |
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
-1
1
1
-1
1
1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
-1
1
1
-1
1
1
vertex
n
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2
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-1
-1
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-1
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1
1
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-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
1
-1
1
1
n
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-1
-1
-1
-1
1
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1
1
-1
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1
-1
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-1
1
-1
1
-1
1
1
1
-1
1
1
-1
1
1
vertex
1
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iu −
t
iu 1
t
iu +
1t
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+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
t
iu 1
t
iu +
1t
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+1
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iu −
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iu 1
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+
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A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part II 2445
Table 5. (Continued)
1=
1=
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
3 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
3 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
0
3 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
0
3 6
Rule 220 : Rule 22 :
Rule 222 : Rule 223 :2=
1=
7
7
7
7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
1
1 1sgn 2 1[ ]t t t t
i i i iu u u u
+− += + − + [] 1
1sgn 1[ ]t t t
i i iu u u
++= − + []
1
1 121 2sgn[ ]t t t t
i i i iu u u u
+− +− − +− + −= []| || || || | 1
1 1sgn 2[ ]t t t t
i i i iu u u u
+− += + +− − []
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
1
1
1
-1
1
1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
1
1
1
-1
1
1
vertex
n
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2
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-1
-1
-1
-1
1
1
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1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
1
1
1
-1
1
1
n
0
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-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
1
1
1
1
-1
1
1
vertex
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
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+
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2446 L. O. Chua et al.
Table 5. (Continued)
Rule 224 :
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
0
5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
0
5 6
Rule 225 :
Rule 226 : Rule 227 :
2=
2=3=
1=
7
7
7
7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
1
1 1sgn 2 1[ ]t t t t
i i i iu u u u
+− += + + − [] 1
1 122 1sgn[ ]t t t t
i i i iu u u u
+− +− + +− += [| || || || |
1
1 132sgn 2 4 3[ ]t t t t
i i i iu u u u
+− += − − − +− −( ) []| || || || || || || || | 1
1 13 1sgn 2 2[ ]t t t t
i i i iu u u u
+− +− −−= − []| || || || |
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
-1
-1
-1
1
1
1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
-1
-1
-1
1
1
1
vertex
n
0
2
3
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-1
-1
-1
-1
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-1
-1
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1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
-1
1
1
1
n
0
2
3
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7
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-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
1
-1
-1
-1
1
1
1
vertex
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
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vertex
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A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part II 2447
Table 5. (Continued)
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
0
5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
0
5 6
Rule 228 : Rule 229 :
Rule 230 : Rule 23 :
2=
2= 2=
3=
7
7
7
7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
1
1 1sgn 2 33 42[ ]t t t t
i i i iu u u u
+− += + +−− −( ) []| || || || || || || || | 1
1 13 2 2 1sgn[ ]t t t t
i i i iu u u u
+− +− − + + += []| || || || |
1
1 13 2 2 1sgn[ ]t t t t
i i i iu u u u
+− +− − + + −= []| || || || | 1
1 12sgn[ ]t t t t
i i i iu u u u
+− +−−= − []| || || || |
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
1
-1
-1
1
1
1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
1
-1
-1
1
1
1
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
1
-1
-1
1
1
1
n
0
2
3
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-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
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1
1
-1
-1
1
1
1
vertex
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+1
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1t
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+
1
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iu −
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1t
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+1
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iu −
t
iu 1
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1t
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+
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2448 L. O. Chua et al.
Table 5. (Continued)
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
3 5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
3 5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
0
3 5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
0
3 5 6
Rule 232 : Rule 233 :
Rule 234 : Rule 235 :
2=
2=
1=
1=
7
7
7
7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
1
1 1sgn[ ]t t t t
i i i iu u u u
+− += + + [] 1
1 11 1sgn[ ]t t t t
i i i iu u u u
+− ++ +− + += []| || || || |
1
1 1sgn 2 1[ ]t t t t
i i i iu u u u
+− += + + + [] 1
1 121 2 1sgn[ ]t t t t
i i i iu u u u
+− ++ +− + += []| || || || |
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
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-1
-1
-1
-1
1
1
1
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-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
-1
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-1
1
1
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-1
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-1
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1
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vertex
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iu −
t
iu 1
t
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1t
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+1
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t
iu 1
t
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1t
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+
1
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iu −
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+1
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A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part II 2449
Table 5. (Continued)
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
3 5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
3 5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
0
3 5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2
0
3 5 6
Rule 236 : Rule 237 :
Rule 238 : Rule 239 :
2=1=
1= 1=
7
7
7
7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
1
1 1sgn 2 1[ ]t t t t
i i i iu u u u
+− += + + + [] 1
1 121 2 1sgn[ ]t t t t
i i i iu u u u
+− +− −+ + −= []| || || || |
1
1sgn 1[ ]t t t
i i iu u u
++= + + [] 1
1 1sgn 2[ ]t t t t
i i i iu u u u
+− += + +− + []
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
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-1
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-1
-1
1
1
-1
1
-1
1
-1
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-1
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-1
-1
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-1
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1
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+1
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iu 1
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1t
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+
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2450 L. O. Chua et al.
Table 5. (Continued)
1=Rule 240 :
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
0
5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
0
5 6
Rule 24 :
1=Rule 242 : Rule 243 :
1=
1=
7
7
7
7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
1
1sgn[ ]t t
i iu u
+−= [] 1
1 1sgn 2 1[ ]t t t t
i i i iu u u u
+− += − − + []
1
1 1sgn 2 1[ ]t t t t
i i i iu u u u
+− += − + + [] 1
1sgn 1[ ]t t t
i i iu u u
+−= − + []
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
-1
-1
1
1
1
1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
-1
-1
1
1
1
1
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1
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1
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1
-1
-1
1
1
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iu 1
t
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1t
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+1
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t
iu 1
t
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1t
iu
+
1
t
iu −
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iu 1
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iu +
1t
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+1
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iu 1
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+
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A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part II 2451
Table 5. (Continued)
1=
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
0
5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
0
5 6
Rule 244 : Rule 245 :
2=Rule 246 : Rule 247 :
1=1=
7
7
7
7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
1
1 1sgn 2 1[ ]t t t t
i i i iu u u u
+− += + − + [] 1
1 1sgn 1[ ]t t t
i i iu u u
+− += − + []
1
1 12 21 1sgn[ ]t t t t
i i i iu u u u
+− +− − + −+ −= []| || || || | 1
1 1sgn 2[ ]t t t t
i i i iu u u u
+− += − − + []
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
1
-1
1
1
1
1
n
0
2
3
4
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7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
1
-1
1
1
1
1
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1
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1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
1
-1
1
1
1
1
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0
2
3
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7
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-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
1
1
-1
1
1
1
1
vertex
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
t
iu 1
t
iu +
1t
iu
+1
t
iu −
t
iu 1
t
iu +
1t
iu
+
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2452 L. O. Chua et al.
Table 5. (Continued)
2=
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
3 5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
3 5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
0
3 5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
4
0
3 5 6
Rule 248 : Rule 249 :
Rule 250 : Rule 25 : 1=
1=
1=
7
7
7
7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
1
1 1sgn 2 1[ ]t t t t
i i i iu u u u
+− += + + + [] 1
1 121 2sgn[ ]t t t t
i i i iu u u u
+− +− −− + += []| || || || |
1
1 1sgn 1[ ]t t t
i i iu u u
+− += + + [] 1
1 1sgn 2[ ]t t t t
i i i iu u u u
+− += − + + []
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
-1
1
1
1
1
1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
-1
1
1
1
1
1
vertex
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-1
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1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
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-1
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1
1
1
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-1
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1
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-1
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-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
1
-1
1
1
1
1
1
vertex
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iu −
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iu 1
t
iu +
1t
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+1
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iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
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iu 1
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iu +
1t
iu
+1
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iu −
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iu 1
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+
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A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part II 2453
Table 5. (Continued)
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
3 5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
3 5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
0
3 5 6
Firing Patterns
0 red pixel :
red pixel :
2 red pixels :
3 red pixels :
2 4
0
3 5 6
Rule 252 : Rule 253 :
Rule 254 : Rule 255 :1=
1=
1=
1=
7
7
7
7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
2 3
5
0
4
6 7
1
1sgn 1[ ]t t t
i i iu u u
+−= + + [] 1
1 1sgn 2[ ]t t t t
i i i iu u u u
+− += −+ + []
1
1 1sgn 2[ ]t t t t
i i i iu u u u
+− += + + + [] 1 sgn 1[ ]t
iu
+ = []
Difference Equation Difference Equation
Difference Equation Difference Equation
vertex
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
1
1
1
1
1
1
n
0
2
3
4
6
7
5
-1
-1
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
1
-1
1
1
1
1
1
1
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-1
-1
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1
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-1
1
-1
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-1
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1
1
1
1
1
1
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1
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-1
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-1
-1
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1
-1
1
-1
1
-1
1
-1
1
1
1
1
1
1
1
1
1
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1
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iu −
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iu 1
t
iu +
1t
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+1
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iu −
t
iu 1
t
iu +
1t
iu
+
1
t
iu −
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iu 1
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iu +
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+1
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2454 L. O. Chua et al.
(κ = 1, 2, or 3) defined in [Chua et al., 2002] isprinted in the upper right-hand corner.
2. Top upper-half portion: Explicit equation used togenerate the sample pattern given in the lower-half portion.
3. Middle-half portion: The firing patterns associ-ated with the red vertices of the Boolean cubeare given on the left. Each pattern is identifiedbelow by the red vertex number it is associatedwith. The truth table encoded by the Booleancube is shown on the right.
4. Lower-half portion: The first 30 iterations fromthe same “red center-pixel” initial pattern usedin [Wolfram, 2002]. The array size is 30 × 61.
The nonlinear expression on the right-hand sideof the difference equation given in Table 5 for eachof the 256 local rules assume one of the followingthree forms of increasing complexity:
1. Complexity index κ = 1. This case pertains toall 104 Boolean cubes in Table 1 with a redID number. In this case, all red vertices of thecube can be separated from the blue vertices(linearly separable) by a single plane. Hence noabsolute-value function is needed and we cansimply redefine
F(uti−1, u
ti, u
ti+1)
= sgn{z0 + b1u
ti−1 + b2u
ti + b3u
ti+1
}(9)
In this case, only four parameters {z0, b1, b2, b3}are needed.
2. Complexity index κ = 2. This case includes allBoolean cubes whose red vertices can be sep-arated from the blue vertices by two parallelplanes. In this case, only one absolute-value func-tion is needed, and we must redefine
F(uti−1, u
ti, u
ti+1)
= sgn{z1 + c1
∣∣(z0 + b1uti−1 + b2u
ti
+ b3uti+1
)∣∣} (10)
We remark that Eq. (10) can also be usedto generate any local rule of complexity indexκ = 1. However, in this case, six parameters{z1, c1, z0, b1, b2, b3} are needed in general.
3. Complexity index κ = 3. This is the most com-plicated case where three parallel planes arenecessary to separate all red vertices from theblue vertices of the Boolean cube. Two absolute-value functions are needed in this case, and
we have to resort to the two-nested universalCA map F defined in Eq. (7). Observe thatEq. (7) can also be used to generate any lo-cal rule of complexity index κ = 1, or κ =2. However, in this case all eight parameters{z2, c2, z1, c1, z0, b1, b2, b3} are needed in general.
Finally we remark that the synaptic weightschosen in this table are different from those givenin Table 4. The reason is that for all local ruleswith a complexity index κ ≤ 2, the theory pre-sented in [Chua et al., 2002] shows that either one orno absolute value function is needed in the discretemap. In this case, we have opted for the simplerdiscrete map (9) or (10). Even for those local rulesrequiring two absolute-value functions, we havechosen a different set of parameters (extracted fromTable 2 of [Chua et al., 2002]) to emphasize our as-sertion that many synaptic weights can be chosento realize a local rule.
4. Genealogic Classification of LocalRules
The patterns shown in Table 5 are all generatedfrom a “single red center-pixel” initial pattern,i.e. for the 1D CNN ring shown in Fig. 1, only cell“0” is red (which codes for 1) at t = 0. All othercells are blue (which codes for −1) at t = 0. A totalof 29 generations (i.e. iterations) of evolved patternsare shown for each local rule for a ring of N = 61cells. Exactly the same initial and evolved patternsare given in [Wolfram, 2002].
A cursory glance of the 256 patterns (30 rows ×61 columns) in Tables 5 reveals many duplicationsand qualitatively similar dynamics. Our goal in thissection is to discover a unifying principle which willallow us to reshuffle the 256 local rules into 16 dis-tinct families, henceforth called CA gene families,which have different primary firing patterns to bedefined below. Each family has 16 members, hence-forth called CA gene siblings, all sharing the sameprimary firing patterns, but every member is char-acterized by a distinct secondary firing pattern to bedefined below. We will show that all gene siblingsbelonging to the same family exhibit some commonqualitative properties.
4.1. Primary and secondary firingpatterns
Recall that all of the 256 evolution patterns given inTable 5 start from the same initial pattern, namely,
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2
0
3
6
1
4
7
5
3
6
1
4
(a)
(b)
w2=4
w0=1
w4=16
w1=2
w6=64w7=128
w3=8
w5=32
Fig. 2. Subdivision of the Boolean cube into (a) a primary trihedron spanned by the four primary firing vertices ©0 , ©1 , ©2and ©4 and (b) a secondary trihedron spanned by the four secondary firing vertices ©3 , ©5 , ©6 and ©7 .
N = 0 ⇒ red and N �= 0 ⇒ blue at t = 0. We willcontinue to assume this “single red center-pixel” ini-tial pattern throughout this paper.
The output of each pixel Ci after each iterationdepends only on the color pattern of three pixels;namely, Ci−1, Ci and Ci+1 (see Fig. 1). If we scanthis initial pattern using a black mask with only
a three-pixel wide window, we will see that onlyfour firing patterns (out of the eight neighborhoodpatterns shown on the bottom-left of Table 1) arepossible, namely, P0, P1, P2 and P4 correspondingto the four vertices ©0 , ©1 , ©2 and ©4 . If any oneof these four patterns appears at the above “scan-ning window”, and if the corresponding vertex of
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2456 L. O. Chua et al.
the Boolean cube is painted in red, then the cell atthe center of the window will be painted red in thenext generation. We will henceforth refer to P0, P1,P2 and P4 as the primary firing patterns. Since the“single red center pixel” initial pattern cannot elicitany response from the remaining four neighborhoodpatterns P3, P5, P6 and P7, we will henceforth referto them as secondary firing patterns. They are sec-ondary because they can influence the outcome ofthe evolution only after the first generation, whoseoutput pattern may contain one or more of thesesecondary firing patterns.
To derive an intuitive feeling on the evolutionof patterns from one generation to the next one, itis useful to partition the Boolean cube into the twoTrihedrons shown in Fig. 2.
The upper trihedron is called the primary tri-hedron because it contains the four primary patternfiring vertices ©0 , ©1 , ©2 and ©4 at the intersectionof its three faces. The lower trihedron is called thesecondary trihedron for corresponding reasons.
From a group-theoretic perspective, the primarytrihedron and the secondary trihedron exhibit an S2
inversion symmetry in the sense that each pair ofthe vertices
{©0 ,©7 } , {©1 ,©6 } , {©2 ,©5 } , {©4 ,©3 } ,
lie on a straight line through the origin of theBoolean cube. Observe that each red vertex withineach pair maps onto a blue vertex of the same pairvia a 180◦-rotation about the vertical axis throughthe origin (center of Boolean cube), and followed bya reflection above the horizontal plane through theorigin.
4.2. Partitioning 256 local rulesinto 16 gene families
Since only four firing patterns can elicit a responsefrom the single red center-pixel initial pattern, thereare a total of 16 possible combinations, as shownin Tables 6 and 7. Each combination is coded bya four-bit binary number, where a bit is “1” if andonly if the corresponding vertex of the Boolean cubeis painted red. Since each vertex ©k is associatedwith a vertex weight wk = 2k, the sum WP of thevertex weights wk of all “red” (coded by 1) primaryfiring vertices can be used to identify the combina-tion of primary firing patterns characterizing eachof the 256 Boolean cubes in Table 1. These 16 com-bined vertex weights are equal to 0, 1, 2, 3, 4, 5, 6,7, 16, 17, 18, 19, 20, 21, 22 and 23. They define 16
distinct groups of Boolean cubes, henceforth calledCA gene families, and denoted by the correspond-ing symbols G0, G1, G2, G3, G4, G5, G6, G7, G16, G17,G18, G19, G20, G21, G22 and G23. Note the subscript ofeach gene family is equal to its corresponding totalvertex weight WP .
Observe that there is an abrupt change (from7 to 16) in the subscript code of the 2 consecutivegene families G7 and G16 even though their corre-sponding four-bit binary words are in consecutiveorder (7 and 8). This comes from the discontinuityin the subscript of P4 and that of its neighbor P2.We will see shortly that this discontinuity induces ared ↔ blue vertex transformation between each genefamily in Table 6 and a corresponding gene familyin Table 7.
Since the secondary firing patterns do not elicita response from the “single red center-pixel” initialpattern, the outcomes after one iteration (first gen-eration) of all local rules N belonging to a particulargene family are identical regardless of the color ofthe vertices ©3 , ©5 , ©6 and ©7 , and are given in therightmost column in Tables 6 and 7.
4.3. Each gene family has 16 genesiblings
The four secondary firing patterns P3, P5, P6 and P7
can evoke a response when the output pattern fromthe first iteration, or subsequent iterations, containthese patterns, and if the corresponding vertices arepainted in red. In such cases, each combination ofthese secondary firing patterns will give rise to a dif-ferent evolved output pattern. The 16 distinct com-binations of these patterns are shown in Tables 8–10for the three gene families G2, G6 and G22, usingthe same binary format as Table 7. The sum WS
of the vertex weights wk of all “red” (coded by 1)secondary firing vertices is listed in the adjacent col-umn. Adding the total secondary vertex weight WS
to the corresponding total firing vertex weight WP
from Table 7 (which we reproduce in Tables 8–10for gene families G2, G6 and G22) for each gene fam-ily gives the number N of the unique Boolean cubewhose vertices bear the eight binary bits (not in thesame order as the eight-bit word in Table 1) listedin the corresponding row in Tables 8–10, for genefamilies G2, G6 and G22, and from the correspond-ing row in Table 7. Note that WS is invariant inTables 8–10. This is why the 16 local rules listedin the rightmost column of Tables 8–10 are calledgene siblings. We pick only three gene families for
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Table 6. Definition of CA Gene Family G0, G1, G2, G3, G4, G5, G6 and G7. The symbol Pk denotes the firing pattern at redvertex ©k whose vertex weight is equal to wk = 2k and where WP = sum of wk of all firing patterns (coded by 1).
w0= 1
PPPP0
w1 = 2w2 = 4w4 = 16
PPPP1PPPP2PPPP4
1110w2+ w1
+ w0
= 7
G7
0110w2+ w1
= 6G6
1010w2+ w0
= 5G5
0010w2
= 4G4
1100w1+ w0
= 3G3
0100w1
= 2G2
1000w0
= 1G1
00000G0
Primary Firing PatternTotal
Primary
Vertex
Weight
WP
CA
Gene
Family
Code
Initial Pattern:
1 Red pixel at center
Evolved Pattern:
1st Generation
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Table 7. Definition of CA Gene Family G16, G17, G18, G19, G20, G21, G22 and G23. The symbol Pk denotes the firing patternat red vertex ©k whose vertex weight is equal to wk = 2k and where WP = sum of wk of all firing patterns (coded by 1).
w0= 1
PPPP0
w1 = 2w2 = 4w4 = 16
PPPP1PPPP2PPPP4
1111w4+ w2
+ w1 + w0
= 23
G23
0111w4+ w2
+ w1
= 22
G22
1011w4+ w2
+ w0
= 21
G21
0011w4+ w2
= 20G20
1101w4+ w1
+ w0
= 19
G19
0101w4+ w1
= 18G18
1001w4+ w0
= 17G17
0001w4
= 16G16
Primary Firing PatternTotal
Primary
Vertex
Weight
WP
CA
Gene
Family
Code
Initial Pattern:
1 Red pixel at center
Evolved Pattern:
1st Generation
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Table 8. CA Gene Family G2 and its siblings.
Total
Secondary
Vertex
Weight
WS
Local
Rule
Number
N=WP+WS
Total
Primary
Vertex
Weight
WP
2342W7+ w6+ w5 + w3
= 2321111
2022W7+ w6+ w3
= 2001011
1702W7+ w5+ w3
= 1681101
1382W7+w3
= 1361001
1062w6+ w5 + w3
= 1041110
742w6+ w3
= 721010
422w5+ w3
= 401100
102w3
= 81000
2
2
2
2
2
2
2
2
Gene Family GGGG2
W7+ w6+ w5
= 224
W7+ w6
= 192
W7+ w5
= 160
W7
= 128
w6+ w5
= 96
w6
= 64
w5
= 32
0
w3= 8
PPPP3
w5 = 32w6 = 64w7 = 128
PPPP5PPPP6PPPP7
2260111
1940011
1620101
1300001
980110
660010
340100
20000
Secondary Firing Pattern
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Table 9. CA Gene Family G6 and its siblings.
Total
Secondary
Vertex
Weight
WS
Local
Rule
Number
N=WP+WS
Total
Primary
Vertex
Weight
WP
2386W7+ w6+ w5 + w3
= 2321111
2066W7+ w6+ w3
= 2001011
1746W7+ w5+ w3
= 1681101
1426W7+w3
= 1361001
1106w6+ w5 + w3
= 1041110
786w6+ w3
= 721010
466w5+ w3
= 401100
146w3
= 81000
6
6
6
6
6
6
6
6
Gene Family GGGG6
W7+ w6+ w5
= 224
W7+ w6
= 192
W7+ w5
= 160
W7
= 128
w6+ w5
= 96
w6
= 64
w5
= 32
0
w3= 8
PPPP3
w5 = 32w6 = 64w7 = 128
PPPP5PPPP6PPPP7
2300111
1980011
1660101
1340001
1020110
700010
380100
60000
Secondary Firing Pattern
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Table 10. CA Gene Family G22 and its siblings.
Total
Secondary
Vertex
Weight
WS
Local
Rule
Number
N=WP+WS
Total
Primary
Vertex
Weight
WP
25422W7+ w6+ w5 + w3
= 2321111
22222W7+ w6+ w3
= 2001011
19022W7+ w5+ w3
= 1681101
15822W7+w3
= 1361001
12622w6+ w5 + w3
= 1041110
9422w6+ w3
= 721010
6222w5+ w3
= 401100
3022w3
= 81000
22
22
22
22
22
22
22
22
Gene Family GGGG22
W7+ w6+ w5
= 224
W7+ w6
= 192
W7+ w5
= 160
W7
= 128
w6+ w5
= 96
w6
= 64
w5
= 32
0
w3= 8
PPPP3
w5 = 32w6 = 64w7 = 128
PPPP5PPPP6PPPP7
2460111
2140011
1820101
1500001
1180110
860010
540100
220000
Secondary Firing Pattern
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2462 L. O. Chua et al.
illustrative purposes. Needless to say, each of the 13remaining gene families will generate a different setof 16 siblings.
The 16 gene siblings associated with each of theeight gene families G0, G1, G2, G3, G4, G5, G6 and G7
(from Table 6) are listed in Table 11. Altogether,Table 11 contains 128 of the 256 Boolean cubes listedin Table 1. For reasons that will be obvious in thefollowing section, we will henceforth refer to thiscollection as the blue gene family group B.
The 16 gene siblings associated with each ofthe remaining eight gene families G16, G17, G18, G19,G20, G21, G22 and G23 (from Table 7) are listed in Ta-ble 12. Altogether, Table 11 contains the remaining128 of the 256 Boolean cubes listed in Table 1. Forreasons that will be obvious in the following section,we will henceforth refer to this collection as the redgene family group R.
5. The Double-Helix Torus
Our objective in this section is to introduce avery compact and enlightening representation of the256 local rules from Table 1, henceforth called thedouble-helix torus, which can be used to encode allof the data from Tables 6 and 7, as well as otherrelevant new information and relationships to beintroduced below.
5.1. Algorithm for generating all 16local rules belonging to eachgene family
The 16 local rules belonging to each gene familyGm ∈ {G0, G1, G2, G3, G4, G5, G6, G7, G16, G17, G18,G19, G20, G21, G22, G23} can be generated by adding,alternatingly, the number “8” and “24” to the ger-minating local rule number “m”, where m is thesubscript associated with the first member of thegene families listed in Tables 11 and 12.
Example. Generate all members of G2.
Here, m = 2, and the germinating local rule is there-fore identified by the number N = 2, which is thefirst member of G2. Adding “8” to N = 2 gives thesecond member N = 2 + 8 = 10. Adding “24” toN = 10 gives the third member N = 10 + 24 = 34.Adding “8” to N = 34 gives the fourth memberN = 34 + 8 = 42. Adding “24” to 42 gives the fifthmember N = 42 + 24 = 66. Continuing this “8/24”“alternating” algorithm, we generate the remaining
members 74, 98, 106, 130, 138, 162, 170, 194, 202,226 and 234. The readers should verify that these16 numbers are identical to those listed under G2
(column 3) in Table 11.To verify that the above algorithm can be used
to generate all of the 16 local rules listed under eachgene family (i.e. column) in Tables 11 and 12, ob-serve that if we add the number “8” to all numbersin any odd-numbered row, we would obtain the cor-responding number listed in the next row (which isnow an even-numbered row) of Tables 11 and 12.Similarly, if we add the number “24” to all num-bers in any even-numbered row, we would obtainthe number listed in the next row (which reverts toan odd-numbered row) of Tables 11 and 12.
To understand why this simple algorithm worksfor all gene families, let us examine the first fourcolumns in Table 8. Note that the bit “1” appearsin the column under the secondary firing patternP3 of every even-numbered row, thereby contribut-ing its associated vertex weight number w3 = 23 = 8to the total secondary vertex weight WS .
Moreover, observe that whenever the bit “1”appears (at even-numbered rows) under the right-most column P3, its three associated binary bits onits left (at the same row) are identical to the corre-sponding bits in the preceding row. Consequently,only the number “8” needs to be added to WP toobtain the local rule number N in the last column.
Now examine the odd-numbered rows in Table 8and note that if we add all of the vertex weightsalgebraically (adding if it flips from 0 to 1, and sub-tracting if it flips from 1 to 0) where a bit hadflipped from the preceding row, we would alwaysobtain a net contribution equal to 24. This prop-erty holds also for the gene family G6 in Table 9,and G22 in Table 10, and in fact for all gene fami-lies because the first four columns used in the aboveanalysis are identical in all cases.
5.2. “8/24” Distribution pattern ingene siblings
In order to visualize the distribution of gene siblingsin each gene family listed in Tables 11 and 12, letus print the entire list of 256 Boolean cubes (localrules) from Table 1 on a uniformly labeled loop asshown in Fig. 3. The two colors here have no sig-nificance other than to help visualization. A closedloop instead of an open-ended string is chosen herein view of the “periodicity” property of the genefamilies to be discussed next.
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Table 11. Members of the blue gene family group B.
Gene Family
23923823723623523423323216
23123022922822722622522415
20720620520420320220120014
19919819719619519419319213
17517417317217117016916812
16716616516416316216116011
14314214114013913813713610
1351341331321311301291289
1111101091081071061051048
103102101100999897967
79787776757473726
71706968676665645
47464544434241404
39383736353433323
151413121110982
765432101
GGGG7GGGG6GGGG5GGGG4GGGG3GGGG2GGGG1GGGG0
Gen
e S
ibli
ngs
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Table 12. Members of the red gene family group R.
Gene Family
25525425325225125024924816
24724624524424324224124015
22322222122021921821721614
21521421321221121020920813
19119018918818718618518412
18318218118017917817717611
15915815715615515415315210
1511501491481471461451449
1271261251241231221211208
1191181171161151141131127
95949392919089886
87868584838281805
63626160595857564
55545352515049483
31302928272625242
23222120191817161
GGGG23GGGG22GGGG21GGGG20GGGG19GGGG18GGGG17GGGG16
Gen
e S
ibli
ngs
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27
N=0 2254252 4 6 8 10 12
14
16
18
202224262830323436384042
7068666462605856545250
44
767880828486889092949698
128
124122120118116114112110108106104
46
48
72
74
100
102
126
130
132134136138140142144146148150152154
156
158
180178176174172170168166164162160 182
184
186
188192194196198200202204206208210 190
212
214
216
218
220
222
224
226
228
230
232
234
236
238
240
242
244
246 248 250
1 3255253 5 7 9 11 13247 249 251
15
17
1921232529313335373941
43
45
476967656361595755535149
71
73
757779818385878991939597
99
101
103 125123121119117115113111109107105
127
129
133135137139141143145147149151153 131
155
157
159
215
181179177175173171169167165163161183
185
187
189191193195197199201203205207209211
213
217
219
221
223
225
227
229
231
233
235
237
239
241
243
245
Fig. 3. A loop partitioned into 256 equal units, labeled consecutively from N = 0 to N = 255. Each number corresponds toa local rule.
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2466 L. O. Chua et al.
02
46
22 18
16
20
Fig. 4. Partitioning of all even gene families G0, G2, G4, G6, G16, G18, G20 and G22 into 16 gene siblings for each family.
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13
57
23 19
17
21
Fig. 5. Partitioning of all odd gene families G1, G3, G5, G7, G17, G19, G21 and G23 into 16 gene siblings for each family.
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2468 L. O. Chua et al.
Recall from the “local rule generation algo-rithm” from Sec. 5.1 that if we add the number “24”to any even-numbered rule of any gene family Gm,we would obtain the next local rule number (whichis necessarily an odd-numbered row by construction)listed in Gm. If we apply this algorithm to each fam-ily in the last tow (16) in Tables 11 and 12, which is
an even-numbered row, we would exceed the maxi-mum number 256 in Fig. 3. However, if we performthe addition in modulo 256 arithmetic, we would ob-tain the numbers listed in row 1 of Gm. Hence, eachgene family is periodic modulo 256. This means thatif we track the sequence of 16 local rule numbers ineach gene family Gm in Fig. 3, we would eventually
8
32
40
64
0232
224
200
19
2
168
160
136
128
10496
72
0
1
2
3
4
5
6
7
Fig. 6. Concentric circular representation of the blue gene family group B. Every local rule number in Table 11 is mapped ina one-to-one manner to an intersection between a circle and a half line.
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A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part II 2469
8
32
40
0232
224
200
168
160
136
128
104
96
72
16
17
18
19
20
21
22
23
64
19
2
Fig. 7. Concentric circular representation of the red gene family group R. Every local rule number in Table 11 is mapped ina one-to-one manner to an intersection between a circle and a half line.
return to the starting point. We have plotted thissequence of points for each gene family on the sameloop in Fig. 3, but subdivided them into 16 differentadjacent tracks to avoid clutter. They are shown inFig. 4 for the even-numbered families, and in Fig. 5for the odd-numbered families. Observe that exceptfor a translation in space, the distribution patterns
are identical for all 16 gene families, namely, succes-sive points are separated alternatingly by a distanceof “8” and “24” units, respectively.
In view of this 8/24 alternating periodicityproperty, we can represent the eight gene familiesfrom Tables 11 and 12 by eight concentric circles,as shown in Figs. 6 and 7, respectively. Each set
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2470 L. O. Chua et al.
of concentric circles is superimposed on top of 16uniformly-spaced radial half lines, labeled by weightfactors which increases from 0 via the alternatingincrements of 8 and 24, respectively. Although notshown in Figs. 6 and 7 to avoid clutter, each cir-cle has 16 intersections with each radial line. Eachintersection defines the gene sibling number ob-tained by adding the “circle number” to the “weightfactor” of the corresponding radial line. For exam-ple, the intersection between the “orange” circleno. 4 with the third radial line (with weight factor32) gives the rule no. 4+32 = 36, which is the thirdgene sibling belonging to gene family G4. Every rulelisted in Tables 11 and 12 is mapped in a one-to-onemanner onto one of these intersections.
5.3. Coding local rules on a doublehelix
Each of the two gene family groups B and R listedin Tables 11 and 12 contains 128 distinct Booleancubes extracted from Table 1, and arranged in adifferent order. If we take their complements bychanging the color of the vertices of each Booleancube from blue to red, or red to blue, respectively,we would generate the remaining 128 Boolean cubesfrom Table 1, again arranged in a different order. Itfollows that for each local rule N1 belonging to B,there is a “complementary” rule N2 belonging toR. Their ID numbers are related by the “Red–Bluecomplementary transformation” defined in [Chuaet al., 2002], namely, N1 + N2 = 255.
A careful examination of the entries in Ta-bles 11 and 12 shows that if we rotate Table 12 by180◦, then each Boolean cube in B and its comple-ment in R would occupy the same position. For fu-ture reference, each local rule belonging to the bluegene family B is shown in Table 13 along with itsassociated Boolean cube. The complementary localrule and Boolean cube are redrawn from Table 12(following a 180◦ rotation) and shown in Table 14.They are the members from the red gene familygroup R.
Each gene family Gm listed in Table 13 andits complement in Table 14 can be encoded on twooppositely-directed ribbons, each containing 16 en-tries, where each pair of complementary rules arecoupled together in a “key-and-lock” fit, and de-picted by a pointed stick (attached to a thin blueribbon), mated to a dented stick (attached to a redribbon), as shown in the center of Fig. 8(a) for thegene family G0. To avoid clutter, we have shown only
the first seven local rules (0, 8, 32, 40, 64, 72, 96) andthe last local rule (232) using an expanded scale forclarity. The corresponding “complementary” rules(255, 247, 223, 215, 191, 183, 159) from the lastseven entries in G23 in Table 12, and the comple-mentary local rule (23) from the first entry in G23
in Table 12 are also shown in this figure. If we sim-ply join the ends of these two ribbons together, wewould obtain the 16 local rules represented by theinnermost circle 0 in Fig. 6, and the 16 “comple-mentary” rules represented by the outermost circle23 in Fig. 7. We can repeat this construction pro-cess and obtain eight disconnected double-strandedrings, which together would account for all 256 localrules.
However, since each gene family is a subset ofthe single 256-unit loop presented earlier in Fig. 3, itis more enlightening to join all of these eight disjointdouble-stranded rings together (in the same orderas Table 12) into a single loop, which would occupyhalf as many locations as in the 256-unit loop inFig. 3, but still contains all 256 local rules since twocomplementary rules are encoded at each location.The resulting structure, shown only for the threegene families G7, G0, G1 in Fig. 8(a), is reminiscentof the classic double helix for encoding the DNA[Watson, 2003], which also contains 2 oppositely-directed complementary strands. The main differ-ence is that unlike a DNA, which can be of anylength and is therefore wound on a cylinder, ourdouble-strand is a loop containing exactly 128 localrules on each strand, and hence should be wound ona torus, as depicted in Fig. 8(b), henceforth calledthe double-helix torus.
The double-helix torus gives a compact andcomplete representation of all 256 local rules fromTable 1. However, instead of the ordering schemeused in Table 1 which did not exploit the primary“firing patterns” of local rules, the Boolean cubesencoded into the double-helix torus in Fig. 8(b) arein one-to-one correspondence with those belongingto the 16 gene families listed in Tables 11 and 12,where each family is characterized by a primary fir-ing pattern. The compactness in this representationarises from using the blue strand to encode all localrules from the blue gene family B, and to use thered strand to encode all local rules from the red genefamily R, thereby doubling the bit density.
Let us imagine that the 16 “local rule” ID num-bers in each column of Table 11 are printed on a thinblue ribbon 16 units longs. Similarly, print the eight
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A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part II 2471
Table 13. Members of the blue gene family group B and their Boolean cubes.
23923823723623523423323216
23123022922822722622522415
20720620520420320220120014
19919819719619519419319213
17517417317217117016916812
16716616516416316216116011
14314214114013913813713610
1351341331321311301291289
1111101091081071061051048
103102101100999897967
79787776757473726
71706968676665645
47464544434241404
39383736353433323
151413121110982
765432101
GGGG7GGGG6GGGG5GGGG4GGGG3GGGG2GGGG1GGGG0
Gen
e S
ibli
ng
s
Gene Family
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2472 L. O. Chua et al.
Table 14. Members of the red gene family group R and their Boolean cubes, redrawn after rotating Table 12 by 180◦.
16171819202122231
24252627282930312
48495051525354553
56575859606162634
80818283848586875
88899091929394956
1121131141151161171181197
1201211221231241251261278
1441451461471481491501519
15215315415515615715815910
17617717817918018118218311
18418518618718818919019112
20820921021121221321421513
21621721821922022122222314
24024124224324424524624715
24824925025125225325425516
GGGG16GGGG17GGGG18GGGG19GGGG20GGGG21GGGG22GGGG23
Gen
e S
ibli
ng
s
Gene Family
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32
222
G GGG0
G GGG23
G GGG7
G GGG16
G GGG22G GGG1
191
41
0 255
2478
223
40
64
72
232
1
9
65190
254
246
183
96
176
184
103
79
71
15
248
240
216
152
7
238
102
78
70
177
185
239
153
47
159
(b)
(a)
Fig
.8.
The
double
-hel
ixto
rus.
2473
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2474 L. O. Chua et al.
columns of Table 12 onto eight thin red ribbons.If we join the bottom edge of each blue ribbon tothe top edge of its right adjacent ribbon (i.e. con-nect the pairs (232, 1), (233, 2), (234, 3), (235, 4),(236, 5), (237, 6), (238, 7) together), we would ob-tain a 128-unit long blue ribbon with all 128 localrule numbers from the gene family B (Table 11)printed on it. The blue ribbon shown in Fig. 8(a)shows only three sections of this strand. If we nowjoin the top edge of the blue ribbon (rule 0 of G0) toits bottom edge (rule 239 of G7), we would obtainthe blue ribbon loop shown encircling the torus inFig. 8(b). Applying exactly the same procedure tothe red gene family R in Table 12, we obtain thecomplementary red ribbon loop in Fig. 8(b).
We will discover in the next section how thedouble-helix torus organizes the dynamic (evolu-tion) patterns in Table 5 in a unified manner,thereby allowing us to explain if not predict thecharacteristic features of the patterns in Table 5without carrying out any brute force computersimulations.
6. Explaining and PredictingPattern Features
Our goal in this final section is to identify some uni-fying principles which will allow us to either predictor explain the garden variety of patterns presentedin Table 5. All of the 30 × 61 patterns in this tablewere generated from the same “single red center-pixel” initial pattern, and are identical to those pre-sented in [Wolfram 2002] and [Chua et al., 2002].
6.1. Gallery of gene family patterns
The 256 patterns scattered throughout Table 5 donot display any logical relationships because theordering scheme, though quite elegant, did not makeuse of any unifying dynamic organization principles.We will now show that the primary firing patternswhich we used in Sec. 4.2 to derive the 16 genefamilies provide us with one such unifying princi-ple. In particular, since each of the 16 gene siblingsbelonging to a gene family has the same primaryfiring patterns, we can expect certain common fea-tures. For example, the evolved patterns at the firstgeneration (i.e. after one iteration) are identical forall siblings since they all have the same “initial”
and “firing” patterns. The outcomes after the firstiterations, however will in general differ becauseone or more secondary firing patterns may “kickin”. To uncover the characteristic features of eachgene family, the entries in Table 5 were reorganizedaccording to the gene family they belong, and pre-sented in Tables 15.1–15.16, along with their as-sociated Boolean cubes, complexity index, and theprimary (left side) and secondary (right side) firingpatterns. All red vertices belonging to the primarytrihedron are painted with a light-red interior tohighlight the primary firing vertices.
A cursory examination of Table 15 shows thatall 16 siblings from each of four gene families G0,G2, G4 and G16 display identical output patterns.To explain why the 16 patterns within each fam-ily are identical, observe that each family has onlyone primary firing pattern. This implies that one,and only one, pixel in generation 1 will be paintedin red, at the center location for family G0 andG4, since the primary pattern is symmetric, or atthe first left (resp. first right)-of-center location forfamily G2 (resp., G16), since the primary firing pat-tern is asymmetrical and “fires” if, and only if, thecenter of the window neighborhood pattern is an-chored at the first left (resp., first right)-of-centerposition.
Since the remaining 12 gene families have atleast two primary firing patterns, we can expectdifferent siblings will in general evolve differentpatterns, even though they belong to the same genefamily. In order to explain the genesis of these pat-tern variations, let us turn to the next subsectionwhere more subtle organizational principles will beuncovered.
6.2. Predicting the background
Observe that in all of the patterns exhibited inTable 15 there are vast subregions characterized byeither homogeneous or regularly repeating patternswhich convey no new information as the iterationcontinues. We will henceforth refer to such subre-gions as the background. Assuming that there is nonoise in our iterations,4 the following typical pat-terns will emerge from a “single red center-pixel”initial pattern under the following conditions:
(a) Blue background. This case will emerge when-ever the vertex ©0 is painted in blue. It includes
4A randomly-generated pixel noise can, in certain cases, give rise to uncontrolled dynamics which may eventually obliteratethe previous background.
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Table
15.1
.B
oole
an
Cube
Fam
ilyG 0
ger
min
ate
dfr
om
Rule
0via
asi
ngl
ere
dce
nte
r-pix
el(.
..0,0
,1,0
,0,.
..)
init
ialco
ndit
ion.
Fir
ing
Pa
ttern
s
0 r
ed
pix
el
1 r
ed
pix
el
2 r
ed
pix
els
3 r
ed
pix
els
23 1
5
0
467
Ru
le0
Fir
ing
Pa
ttern
s
0 r
ed
pix
el
1 r
ed
pix
el
2 r
ed
pix
els
3 r
ed
pix
els
23 1
5
0
467
Ru
le8
Fir
ing
Pa
ttern
s
0 r
ed
pix
el
1 r
ed
pix
el
2 r
ed
pix
els
3 r
ed
pix
els
23 1
5
0
467
Ru
le3
2
Fir
ing
Pa
ttern
s
0 r
ed
pix
el
1 r
ed
pix
el
2 r
ed
pix
els
3 r
ed
pix
els
23 1
5
0
467
Ru
le4
0
Fir
ing
Pa
ttern
s
0 r
ed
pix
el
1 r
ed
pix
el
2 r
ed
pix
els
3 r
ed
pix
els
23 1
5
0
467
Ru
le 6
4
Fir
ing
Pa
ttern
s
0 r
ed
pix
el
1 r
ed
pix
el
2 r
ed
pix
els
3 r
ed
pix
els
23 1
5
0
467
Ru
le7
2
Fir
ing
Pa
ttern
s
0 r
ed
pix
el
1 r
ed
pix
el
2 r
ed
pix
els
3 r
ed
pix
els
23 1
5
0
467
Ru
le9
6
Fir
ing
Pa
ttern
s
0 r
ed
pix
el
1 r
ed
pix
el
2 r
ed
pix
els
3 r
ed
pix
els
23 1
5
0
467
Ru
le1
04
Fir
ing
Pa
ttern
s
0 r
ed
pix
el
1 r
ed
pix
el
2 r
ed
pix
els
3 r
ed
pix
els
23 1
5
0
467
Ru
le1
28
Fir
ing
Pa
ttern
s
0 r
ed
pix
el
1 r
ed
pix
el
2 r
ed
pix
els
3 r
ed
pix
els
23 1
5
0
467
Ru
le1
36
Fir
ing
Pa
ttern
s
0 r
ed
pix
el
1 r
ed
pix
el
2 r
ed
pix
els
3 r
ed
pix
els
23 1
5
0
467
Ru
le1
60
Fir
ing
Pa
ttern
s
0 r
ed
pix
el
1 r
ed
pix
el
2 r
ed
pix
els
3 r
ed
pix
els
23 1
5
0
467
Ru
le1
68
Fir
ing
Pa
ttern
s
0 r
ed
pix
el
1 r
ed
pix
el
2 r
ed
pix
els
3 r
ed
pix
els
23 1
5
0
467
Ru
le1
92
Fir
ing
Pa
ttern
s
0 r
ed
pix
el
1 r
ed
pix
el
2 r
ed
pix
els
3 r
ed
pix
els
23 1
5
0
467
Ru
le2
00
Fir
ing
Pa
ttern
s
0 r
ed
pix
el
1 r
ed
pix
el
2 r
ed
pix
els
3 r
ed
pix
els
23 1
5
0
467
Ru
le2
24
Fir
ing
Pa
ttern
s
0 r
ed
pix
el
1 r
ed
pix
el
2 r
ed
pix
els
3 r
ed
pix
els
23 1
5
0
467
Ru
le2
32
35
735
6
77
773
5
77
7
35
6
1=
1=
2=
1=
2=
2=
2=
1=
1=
1=
1=
1=
1=
1=
1=
1=
53
63
65
6
35
63
65
6
2475
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Table
15.2
.B
oole
an
Cube
Fam
ilyG 1
ger
min
ate
dfr
om
Rule
1via
asi
ngl
ere
dce
nte
r-pix
el(.
..0,0
,1,0
,0,.
..)
init
ialco
ndit
ion.
Fir
ing P
att
ern
s
0 r
ed
pix
el
1 r
ed
pix
el
2 r
ed
pix
els
3 r
ed
pix
els
23 1
5
0
467
Ru
le1
Fir
ing P
att
ern
s
0 r
ed
pix
el
1 r
ed
pix
el
2 r
ed
pix
els
3 r
ed
pix
els
23 1
5
0
467
Ru
le9
Fir
ing P
att
ern
s
0 r
ed
pix
el
1 r
ed
pix
el
2 r
ed
pix
els
3 r
ed
pix
els
23 1
5
0
467
Ru
le3
3
Fir
ing
Pa
ttern
s
0 r
ed
pix
el
1 r
ed
pix
el
2 r
ed
pix
els
3 r
ed
pix
els
23 1
5
0
467
Ru
le4
1
Fir
ing P
att
ern
s
0 r
ed
pix
el
1 r
ed
pix
el
2 r
ed
pix
els
3 r
ed
pix
els
23 1
5
0
467
Ru
le 6
5
Fir
ing P
att
ern
s
0 r
ed
pix
el
1 r
ed
pix
el
2 r
ed
pix
els
3 r
ed
pix
els
23 1
5
0
467
Ru
le7
3
Fir
ing P
att
ern
s
0 r
ed
pix
el
1 r
ed
pix
el
2 r
ed
pix
els
3 r
ed
pix
els
23 1
5
0
467
Ru
le9
7
Fir
ing
Pa
ttern
s
0 r
ed
pix
el
1 r
ed
pix
el
2 r
ed
pix
els
3 r
ed
pix
els
23 1
5
0
467
Ru
le1
05
Fir
ing P
att
ern
s
0 r
ed
pix
el
1 r
ed
pix
el
2 r
ed
pix
els
3 r
ed
pix
els
23 1
5
0
467
Ru
le1
29
Fir
ing P
att
ern
s
0 r
ed
pix
el
1 r
ed
pix
el
2 r
ed
pix
els
3 r
ed
pix
els
23 1
5
0
467
Ru
le1
37
Fir
ing P
att
ern
s
0 r
ed
pix
el
1 r
ed
pix
el
2 r
ed
pix
els
3 r
ed
pix
els
23 1
5
0
467
Ru
le1
61
Fir
ing
Pa
ttern
s
0 r
ed
pix
el
1 r
ed
pix
el
2 r
ed
pix
els
3 r
ed
pix
els
23 1
5
0
467
Ru
le1
69
Fir
ing P
att
ern
s
0 r
ed
pix
el
1 r
ed
pix
el
2 r
ed
pix
els
3 r
ed
pix
els
23 1
5
0
467
Ru
le1
93
Fir
ing P
att
ern
s
0 r
ed
pix
el
1 r
ed
pix
el
2 r
ed
pix
els
3 r
ed
pix
els
23 1
5
0
467
Ru
le2
01
Fir
ing P
att
ern
s
0 r
ed
pix
el
1 r
ed
pix
el
2 r
ed
pix
els
3 r
ed
pix
els
23 1
5
0
467
Ru
le2
25
Fir
ing
Pa
ttern
s
0 r
ed
pix
el
1 r
ed
pix
el
2 r
ed
pix
els
3 r
ed
pix
els
23 1
5
0
467
Ru
le2
33
35
735
6
7
3 77
735
77
7
35
6
1=
2=
2=
2=
2=
2=
2=
3=
2=
2=
2=
2=
2=
2=
2=
2=
00
00
00
00
00
00
00
00
36
56
53
6
5
63
65
6
2476
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Table
15.3
.B
oole
an
Cube
Fam
ilyG 2
ger
min
ate
dfr
om
Rule
2via
asi
ngl
ere
dce
nte
r-pix
el(.
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23 1
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Ru
le2
34
35
735
6
77
773
5
77
7
35
6
1=
1=
2=
2=
2=
1=
1=
1=
1= 1=
53
63
65
6
35
63
65
6
11
11
11
11
11
11
11
11
1=
2=
2=
2=
3=
3=
2477
![Page 102: A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM’S NEW KIND …nonlinear.eecs.berkeley.edu/CellularAutomata/PartII.pdf · in A New Kind of Science [Wolfram, 2002] for the one-dimensionalbinaryCellular](https://reader034.fdocuments.us/reader034/viewer/2022051604/5ff5c0f683eff57983406834/html5/thumbnails/102.jpg)
Table
15.4
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oole
an
Cube
Fam
ilyG 3
ger
min
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om
Rule
3via
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ngl
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ion.
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1
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35
35
735
6
77
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35
6
1=
1=
2=
2=
2=
2=
1=
1=
53
63
65
6
35
63
65
6
00
00
00
00
00
00
00
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11
11
11
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11
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11
11
2=
2=
3=
3=
2=
2=
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2=
2478
![Page 103: A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM’S NEW KIND …nonlinear.eecs.berkeley.edu/CellularAutomata/PartII.pdf · in A New Kind of Science [Wolfram, 2002] for the one-dimensionalbinaryCellular](https://reader034.fdocuments.us/reader034/viewer/2022051604/5ff5c0f683eff57983406834/html5/thumbnails/103.jpg)
Table
15.5
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an
Cube
Fam
ilyG 4
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min
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Rule
4via
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36
35
735
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6
1=
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53
63
65
6
35
63
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0
22
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24
22
22
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2479
![Page 104: A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM’S NEW KIND …nonlinear.eecs.berkeley.edu/CellularAutomata/PartII.pdf · in A New Kind of Science [Wolfram, 2002] for the one-dimensionalbinaryCellular](https://reader034.fdocuments.us/reader034/viewer/2022051604/5ff5c0f683eff57983406834/html5/thumbnails/104.jpg)
Table
15.6
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an
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ilyG 5
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37
35
735
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6
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53
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35
63
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6
00
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22
22
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2480
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Table
15.7
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an
Cube
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ilyG 6
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23 1
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38
35
735
6
77
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5
77
7
35
6
2=
2=
1=
1=
1=
1=
1=
53
63
65
6
35
63
65
6
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
2=
2=
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3=
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2481
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Table
15.8
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oole
an
Cube
Fam
ilyG 7
ger
min
ate
dfr
om
Rule
7via
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53
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6
00
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12
12
12
12
12
12
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12
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3=
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2482
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Table
15.9
.B
oole
an
Cube
Fam
ilyG 1
6ger
min
ate
dfr
om
Rule
16
via
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63
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2483
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Table
15.1
0.
Boole
an
Cube
Fam
ilyG 1
7ger
min
ate
dfr
om
Rule
17
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49
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6
1=
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56
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2484
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Table
15.1
1.
Boole
an
Cube
Fam
ilyG 1
8ger
min
ate
dfr
om
Rule
18
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5
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Ru
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42
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ing
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ed
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el
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ed
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el
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els
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ed
pix
els
23 1
5
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Ru
le2
50
35
735
6
77
773
5
77
7
35
6
1=
2=
2=
1=
1= 1=
53
63
65
6
35
63
65
6
11
11
11
11
11
11
11
11
2=
2=
2=
44
44
44
44
44
44
44
4
2=
2=
3=
3=
2=
2=
1=
4
2485
![Page 110: A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM’S NEW KIND …nonlinear.eecs.berkeley.edu/CellularAutomata/PartII.pdf · in A New Kind of Science [Wolfram, 2002] for the one-dimensionalbinaryCellular](https://reader034.fdocuments.us/reader034/viewer/2022051604/5ff5c0f683eff57983406834/html5/thumbnails/110.jpg)
Table
15.1
2.
Boole
an
Cube
Fam
ilyG 1
9ger
min
ate
dfr
om
Rule
19
via
asi
ngl
ere
dce
nte
r-pix
el(.
..0,0
,1,0
,0,.
..)
init
ialco
ndit
ion.
Fir
ing
Pa
ttern
s
0 r
ed
pix
el
1 r
ed
pix
el
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ed
pix
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ed
pix
els
23 1
5
0
467
Ru
le1
9
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ing
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ttern
s
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5
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467
Ru
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7
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1
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15
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ing
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Ru
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23
Fir
ing
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ed
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els
23 1
5
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Ru
le1
47
Fir
ing
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Ru
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55
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ing
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Ru
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79
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ing
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Ru
le1
87
Fir
ing
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ttern
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ed
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el
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ed
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el
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ed
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els
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ed
pix
els
23 1
5
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Ru
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11
Fir
ing
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ttern
s
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ed
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el
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ed
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el
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ed
pix
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23 1
5
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Ru
le2
19
Fir
ing
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ttern
s
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ed
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Ru
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43
Fir
ing
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Ru
le2
51
35
735
6
77
773
5
77
7
35
6
1=
1=
2=
2=
1=
53
63
65
6
35
63
65
6
00
00
00
00
00
00
00
00
11
11
11
11
11
11
11
11
2=
2=
2=
44
44
44
44
44
44
44
4
3=
1=
3=
1=
2=
1=
1=
1=
4
2486
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Table
15.1
3.
Boole
an
Cube
Fam
ilyG 2
0ger
min
ate
dfr
om
Rule
20
via
asi
ngl
ere
dce
nte
r-pix
el(.
..0,0
,1,0
,0,.
..)
init
ialco
ndit
ion.
Fir
ing
Pa
ttern
s
0 r
ed
pix
el
1 r
ed
pix
el
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ed
pix
els
3 r
ed
pix
els
23 1
5
0
467
Ru
le2
0
Fir
ing
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ttern
s
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ed
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5
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467
Ru
le2
8
Fir
ing
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s
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Ru
le5
2
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ing
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ed
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els
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5
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467
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le6
0
Fir
ing
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s
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ed
pix
els
23 1
5
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467
Ru
le 8
4
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ing
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ttern
s
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ed
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el
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ed
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el
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ed
pix
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ed
pix
els
23 1
5
0
467
Ru
le9
2
Fir
ing
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s
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ed
pix
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ed
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els
23 1
5
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Ru
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16
Fir
ing
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5
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Ru
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24
Fir
ing
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ttern
s
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el
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ed
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el
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ed
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els
23 1
5
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Ru
le1
48
Fir
ing
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ttern
s
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ed
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ed
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el
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ed
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ed
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els
23 1
5
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Ru
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56
Fir
ing
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s
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ed
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Ru
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80
Fir
ing
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5
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Ru
le1
88
Fir
ing
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s
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ed
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el
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ed
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el
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ed
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ed
pix
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23 1
5
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Ru
le2
12
Fir
ing
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ttern
s
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ed
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el
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ed
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ed
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20
Fir
ing
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44
Fir
ing
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Ru
le2
52
35
735
6
77
773
5
77
7
35
62=
1=
2=
1=
1=
1=
53
63
65
6
35
63
65
6
0
22
22
22
42
2
22
22
22
22
2=
2=
2=
44
44
44
4
44
44
44
44
2=
2= 3=
3=
2=
2=
1=
2487
![Page 112: A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM’S NEW KIND …nonlinear.eecs.berkeley.edu/CellularAutomata/PartII.pdf · in A New Kind of Science [Wolfram, 2002] for the one-dimensionalbinaryCellular](https://reader034.fdocuments.us/reader034/viewer/2022051604/5ff5c0f683eff57983406834/html5/thumbnails/112.jpg)
Table
15.1
4.
Boole
an
Cube
Fam
ilyG 2
1ger
min
ate
dfr
om
Rule
21
via
asi
ngl
ere
dce
nte
r-pix
el(.
..0,0
,1,0
,0,.
..)
init
ialco
ndit
ion.
Fir
ing
Pa
ttern
s
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ed
pix
el
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5
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le2
1
Fir
ing
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s
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5
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467
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le2
9
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3
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ing
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5
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5
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467
Ru
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5
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el
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5
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3
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ing
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17
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5
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Ru
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25
Fir
ing
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ttern
s
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ed
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el
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ed
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els
23 1
5
0
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Ru
le1
49
Fir
ing
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ttern
s
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el
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ed
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ed
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els
23 1
5
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Ru
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57
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ing
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Ru
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81
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ing
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Ru
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89
Fir
ing
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5
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Ru
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13
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ing
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s
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ed
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el
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ed
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el
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ed
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ed
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5
0
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Ru
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21
Fir
ing
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ttern
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45
Fir
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53
35
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6
77
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5
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7
35
6
1=
2=
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2=
1=
53
63
65
6
35
63
65
6
00
00
00
00
00
00
00
00
22
22
22
22
22
22
22
22
1=
2=
2=
2=
44
44
44
44
44
44
44
44
3=
3=
1=
2=
1=
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1=
2488
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Table
15.1
5.
Boole
an
Cube
Fam
ilyG 2
2ger
min
ate
dfr
om
Rule
22
via
asi
ngl
ere
dce
nte
r-pix
el(.
..0,0
,1,0
,0,.
..)
init
ialco
ndit
ion.
Fir
ing
Pa
ttern
s
0 r
ed
pix
el
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ed
pix
el
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ed
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ed
pix
els
23 1
5
0
467
Ru
le2
2
Fir
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ttern
s
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el
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ed
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ed
pix
els
23 1
5
0
467
Ru
le3
0
Fir
ing
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ttern
s
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ed
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el
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el
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pix
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23 1
5
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Ru
le5
4
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ing
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ttern
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ed
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ed
pix
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23 1
5
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467
Ru
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2
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ed
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ed
pix
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23 1
5
0
467
Ru
le 8
6
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s
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ed
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el
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ed
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el
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ed
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els
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ed
pix
els
23 1
5
0
467
Ru
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4
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ing
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ttern
s
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ed
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ed
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ed
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ed
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els
23 1
5
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18
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ed
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Ru
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26
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ing
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s
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ed
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ed
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el
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ed
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ed
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els
23 1
5
0
467
Ru
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50
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ing
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s
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ed
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el
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ed
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el
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ed
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ed
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els
23 1
5
0
467
Ru
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58
Fir
ing
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ttern
s
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ed
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ed
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el
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ed
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els
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ed
pix
els
23 1
5
0
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Ru
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82
Fir
ing
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s
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ed
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el
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ed
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el
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ed
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ed
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23 1
5
0
467
Ru
le1
90
Fir
ing
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ttern
s
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ed
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el
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ed
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el
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ed
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els
3 r
ed
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els
23 1
5
0
467
Ru
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14
Fir
ing
Pa
ttern
s
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ed
pix
el
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ed
pix
el
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ed
pix
els
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ed
pix
els
23 1
5
0
467
Ru
le2
22
Fir
ing
Pa
ttern
s
0 r
ed
pix
el
1 r
ed
pix
el
2 r
ed
pix
els
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ed
pix
els
23 1
5
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Ru
le2
46
Fir
ing
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ttern
s
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ed
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el
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ed
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ed
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ed
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23 1
5
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Ru
le2
54
35
735
6
77
773
5
77
7
35
6
2=
2=
1=
53
63
65
6
35
63
65
6
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
2=
2=
2=
2=
2=
2=
44
44
44
44
44
44
44
44
2=
2=
2=
3=
2=
2=
2=
2489
![Page 114: A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM’S NEW KIND …nonlinear.eecs.berkeley.edu/CellularAutomata/PartII.pdf · in A New Kind of Science [Wolfram, 2002] for the one-dimensionalbinaryCellular](https://reader034.fdocuments.us/reader034/viewer/2022051604/5ff5c0f683eff57983406834/html5/thumbnails/114.jpg)
Table
15.1
6.
Boole
an
Cube
Fam
ilyG 2
3ger
min
ate
dfr
om
Rule
23
via
asi
ngl
ere
dce
nte
r-pix
el(.
..0,0
,1,0
,0,.
..)
init
ialco
ndit
ion.
Fir
ing
Pa
ttern
s
0 r
ed
pix
el
1 r
ed
pix
el
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A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part II 2491
all even-numbered gene families; namely, G0, G2,G4, G6, G16, G18, G20 and G22.
To prove the above assertion, observe thatsince the neighborhood pattern P0 associatedwith vertex ©0 consists of three consecutive bluepixels, it would coincide with all of the bluepixels in the initial pattern, except possibly forthe two pixels on either side of the center lo-cation. Since vertex ©0 is painted in blue, thesecoincident blue pixels must maintain their bluestatus. Hence, except possibly for some regionon either side of the center location, the back-ground will appear in blue.
(b) Red background. This case will emerge when-ever both vertices ©0 and ©7 are painted in red.It includes the lower-half portion of all odd-numbered gene families; namely, G1, G3, G5, G7,G17, G19, G21 and G23.
To prove the above assertion, observe thatexcept for the two pixels on either side of thecenter location, all blue pixels in the initial pat-tern will switch from blue to red in the firstgeneration. All of these newly converted redpixels, except possibly for some near the cen-ter location, must maintain their red status inall subsequent iterations because the neighbor-hood pattern P7 at vertex ©7 consists of threeconsecutive red pixels. Hence, except for someregion on either side of the center location, thebackground will appear in red.
(c) Striped (alternating red-blue) background. Thiscase will emerge whenever vertex ©0 is paintedin red but vertex ©7 is painted in blue. Itincludes the upper-half portion of all oddnumbered gene families; namely, G1, G3, G5, G7,G17, G19, G21 and G23.
To prove this assertion, observe the out-come of the first iteration will be identicalto case (b), i.e. except possibly for some pix-els near the center location, all pixels will be
painted red in the first generation. But sincethe primary firing pattern P0 at vertex ©0 con-sists of only blue pixels, and since vertex ©7 ispainted in blue all of these red pixels will revertto blue in the second generation, except possi-bly for some pixels near the center location. Theabove scenario will simply repeat itself in all fu-ture iterations. Hence, the background will bemade up of alternating red and blue horizontalstripes.
Acknowledgments
This work is supported in part by the DURINTcontract no. N00014-01-1-0741, the ONR con-tract N000-14-03-1-0698, and the NSF grantCHE-0103447.
References
Chua, L. O. [1998] CNN: A Paradigm for Complexity(World Scientific, Singapore).
Chua, L. O. Yoon, S. & Dogaru, R. [2002] “A non-linear dynamics perspective of Wolfram’s new kindof science. Part I: Threshold of complexity,” Int. J.Bifurcation and Chaos 12, 2655–2766.
Dogaru, R., Chitu, C. & Glesner, M. [2003] “A uni-versal CNN neuron for CMOS technology: Modeland functional capabilities,” Proc. SCS 2003, Int.Symp. Circuits and Systems, Iasi, Romania, July2003, pp. 181–184
Sbitnev, V. I., Yang, T. & Chua, L. O. [2001] “The localactivity criteria for difference-equation CNN,” Int. J.Bifurcation and Chaos 11, 311–419.
Sbitnev, V. I. & Chua, L. O. [2002] “Local activitycriteria for discrete map CNN,” Int. J. Bifurcationand Chaos 12, 1227–1272.
Watson, J. [2003] DNA: The Secret of Life (Alfred A.Knopf, NY).
Wolfram, S. [2002] A New Kind of Sciences (WolframMedia, Inc., Champaign Illinois, USA).
![Developing an Interactive Communicative Search Agent · it is still being developed and it has been yet to see whether Wolfram’s compu-2. tational approach[9] will yield the same](https://static.fdocuments.us/doc/165x107/60ff386168c520035d0ae106/developing-an-interactive-communicative-search-agent-it-is-still-being-developed.jpg)
