Triangular spiral tilings and origami (Presentation sheet)

Spiral phyllotaxisQuadrilateral spiral multiple tilings

Triangular spiral multiple tilingsMain theorems for triangular spiral multiple tilings

Triangular spiral tilings and origami

Takamichi SUSHIDA!

Akio HIZUME† and Yoshikazu YAMAGISHI†

!Graduate course of applied mathematics and informatics,†Department of applied mathematics and informatics,

Ryukoku University

Mathematical Software And Free Documents XVIKYOTO UNIVERSITY

March 19, 2013

Takamichi SUSHIDA! Akio HIZUME† and Yoshikazu YAMAGISHI† Triangular spiral tilings and origami



Table of contents

(i) Spiral phyllotaxis

(ii) Quadrilateral spiral multiple tilings

(iii) Triangular spiral multiple tilings

(iv) Main theorems for triangular spiral multiple tilings




Spiral phyllotaxis

Phyllotaxis is the arrangement of leaves and other organs of plants.

Spiral phyllotaxis: ”Sunflower” and ”Pine cone” and so on.

The golden section:

! =1 +

"5

2= 1 +

1

1 +1

1 + · · ·

.

The fibonacci sequence:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 · · · .

The lucas sequence:

2, 1, 3, 4, 7, 11, 18, 29, 47, 76, · · · .

Figure: Capitulum of sunflower

counterclockwise: 89 spiralsclockwise: 55 spirals




Mathematical studies of phyllotaxis

Mathematical studies of phyllotaxis were started by people such as

Bravais brothers in the first half of 19th century.

! Comprehensive text of phyllotaxis:

1994: Roger.V.Jean, Phyllotaxis, A Systemic Study in PlantMorphogenesis, Cambrige University Press.

Figure: Bravais’s sketch ? (http://www.math.smith.edu/phyllo: PHYLLOTAXIS)




! Experiment:

1996: S.Douady and Y.Couder, Phyllotaxis as a DynamicalSelf Organizing Process, J. Theor. Biol, 178, 255-274, 275-294,295-312.

! Bifurcation structure of a dynamical system:

2002: P.Atela, C.Gole and S.Hotton, A dynamical systemfor plant pattern formation: A rigorous analysis, J.Nonlinear Sci.Vol. 12, pp. 641-676.

! Mathematical models:

2006: R.S.Smith at.el, A plausible model of phyllotaxis, Proc.Nat. Acad. Sci. 103 (5) 1301-1306.

2012: Y.Tanaka and M.Mimura, Reaction-di!usion modelfor inflorescence of sunflower, The 3rd Taiwan-Japan jointworkshop for young scholars in applied mathematics, NationalTaiwan University.




Mathematical studies of spiral tilings

2008: A.Hizume, Fibonacci Tornado, in: Proceedings of the11th Bridges Conference, 485-486.

2009: A.Hizume and Y.Yamagishi, Real Tornado, in:Proceedings of the 12th Bridges Conference, 239-242.

2012: T.Sushida, A.Hizume and Y.Yamagishi, Triangularspiral tilings, J.Phys.A: Math.Theor. 45, 23, 235203.

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Figure: Triangular spiral tilings with phyllotactic patterns




Quadrilateral spiral multiple tilings

Let " = re!"1! # D\R. Let m,n > 0 be relatively prime integers.

We denote a point with the complex coordinate "j, j # Z by Aj.

We consider a spiral lattice S = {Aj}j#Z.

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Figure: Quadrilateral spiral multiple tilings:Each j denotes a position of a point Aj , j ! Z.

r = 0.94, ! = 2"# , # = 1+!

52 , (a) (m, n) = (5, 8), (b) (m, n) = (13, 3)




Definition 1

A tiling of a two dimensional manifold X is a family T = {Tj}j of

topological disks Tj $ X which satisfies the following conditions:

X =!

j

Tj , int(Tj) % int(Tk) = & (j '= k).

Each Tj is called a tile.

Let v '= 0 be an integer. Let Cv := C/2#v"(1Z be a cylinder.

We denote C which punctures at the origin O by C$ := C\ {0}.

By the exponential map exp : Cv ) C$, w + 2#v"(1Z ) z = ew,

Cv is a covering space of C$ of a degree |v|.




Let T % ="T %

j

#j

be a tiling of Cv. Then exp (T %) ="exp (T %

j)#

T "j#T "

is a multiple tiling of C$ of multiplicity |v|.

Definition 2

Let V % be an additive subgroup of Cv.We say that T % admits a transitive action by V % if

(i) *T % # T %, *$ # V %, T % + $ # T % and

(ii) *T %1, T

%2 # T %, +$ # V % such that T %

2 = T %1 + $.

If T % admits a transitive action by an additive subgroup %Z which isgenerated by a single element % # Cv,

then T = exp(T %) is called a spiral multiple tiling of multiplicity |v|.





We denote the principal argument of z # C$ by (# < arg (z) , #.

We suppose that T0 := "A0AmAm+nAn is a quadrilateral in C$ inthis order of vertices.

Let

%m = m log (r) +"(1

$m& ( 2#[[

m&

2#]]%

# log ("m),

%n = n log (r) +"(1

$n& ( 2#[[

n&

2#]]%

# log ("n),

where [[x]] # Z is an integer which is the nearest to x # R

such that (12

< -x. := x ( [[x]] , 12.

We obtain a tiling T % := {T %0 + k1%m + k2%n}k1,k2#Z of Cv.

where T %0 is a component of log (T0) which has 0, %m, %m + %n and

%n on its boundary.




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Im

2!

!2!

!!

!

Figure: A quadrilateral spiral tiling and a tiling of Cv, v = 1:r = 0.94, ! = 2"# , # = 1+

!5

2 , (m, n) = (5, 8)

In the right figure, each j ! Z denotes a position of

$j = j log (r) +"#1(j(! + 2"%) + 2"kv), k ! Z, % ! Z




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Figure: A quadrilateral spiral tiling and a tiling of Cv, v = 1:r = 0.94, ! = 2"# , # = 1+

!5

2 , (m, n) = (5, 8)


$j = j log (r) +"#1(j(! + 2"%) + 2"kv), k ! Z, % ! Z




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Figure: A quadrilateral spiral multiple tiling and a tiling of Cv, v = 2:r = 0.94, ! = 2"# , # = 1+

!5

2 , (m, n) = (13, 3)


$j = j log (r) +"#1(j(! + 2"%) + 2"kv), k ! Z, % = 2%" + 1, %" ! Z




Let a, b # Z be integers such that bm ( an = 1.

Then we obtain the followings:

% := b%m ( a%n = log (r)+"(1(&+2#') # log ("), ' = a[[

n&

2#]](b[[

m&

2#]].

v := m[[n&

2#]] ( n[[

m&

2#]] =

12#

(n arg("m) ( m arg("n)).

In Cv, we have [[n!2" ]]%m ( [[m!

2" ]]%n / 0,

m% / %m, n% / %n and %mZ + %nZ / %Z mod 2#v"(1Z.

Let T % = {T %0 + k1%m + k2%n}k1,k2#Z = {T %

0 + k%}k#Z.

Then T % is a tiling of Cv that admits a transitive action by %Zand we have T = exp(T %).





We denote C which punctures at the origin O by C$ := C\ {0}.We denote the principal argument of z # C$ by (# < arg (z) , #.

Proposition 3

Suppose that "m, "n '# R".If T0 := "A0AmAm+nAn is a quadrilateral in C$ in this order ofvertices, then a family of quadrilaterals

T = {Tj := "AjAj+mAj+m+nAj+n}j#Z (1)

is a quadrilateral spiral multiple tiling of multiplicity |v|,where v is given by

v =12#

(n arg ("m) ( m arg ("n)). (2)

In (1), a combinational index (m,n) is called a parastichy pair.




Definition 4

(i) A parastichy pair (m,n) is an opposed parastichy pair

if (arg ("m))(arg ("n))< 0.

(ii) A parastichy pair (m,n) is a non-opposed parastichy pair

if (arg ("m))(arg ("n))> 0.

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Figure: r = 0.94, (m, n) = (5, 8), (a) ! = 2"# , (b) ! = 2"# + &




For x # R, let

x = a0 +1

a1 + 1a2+···

= [a0, a1, a2, · · · ], a0 # Z, ai # Z+, i 0 1

be a continued fraction expansion of x.

Define the sequence

{pi}i&0 , {qi}i&0 , {pi,k}i&0,0<k<ai+1, {qi,k}i&0,0<k<ai+1

by p0 = a0, q0 = 1; p1 = a0a1 + 1, q1 = a1;

pi = pi"2 + aipi"1, qi = qi"2 + aiqi"1, i 0 2; and

pi,k = pi"1 + kpi, qi,k = qi"1 + kqi, i 0 0, 0 < k < ai+1.

pi

qi= [a0, a1, · · · , ai] is called a principal convergent of x and

pi,k

qi,k= [a0, a1, · · · , ai, k] is called an intermediate convergent of x.




Lemma 5

Let 0 < x < 1. Let m,n, a, b be positive integers such that

a

m< x <

b

n, bm ( an = 1.

Thena

m,

b

nare principal or intermediate convergents of x,

at least one of which is principal.





Let a, b be integers such that bm ( an = 1.

Proposition 6

Suppose that "m, "n '# R".

Suppose that T = {Tj := "AjAj+mAj+m+nAj+n}j#Zis a quadrilateral spiral multiple tiling of a multiplicity |v| and(m,n) is an opposed parastichy pair.

Thena

m,

b

nare principal or intermediate convergents of

1v

$&

2#+ '

%, at least one of which is principal,

where ' is given by ' = a[[n&

2#]] ( b[[

m&

2#]].




! Proof of proposition 6:We may suppose that arg("n) < 0 < arg("m) without loss ofgenerality. So we have

-n&2#

. =n&

2#( [[

n&

2#]] < 0 <

m&

2#( [[

m&

2#]] = -m&

2#.

By ' = a[[n&

2#]] ( b[[

m&

2#]] and v = m[[

n&

2#]] ( n[[

m&

2#]], We have

[[m&

2#]] = av ( m', [[

n&

2#]] = bv ( n'.

Thus we obtain

n

$&

2#+ '

%( bv < 0 < m

$&

2#+ '

%( av,

and hencea

m<

1v

$&

2#+ '

%<

b

n, bm ( an = 1.

"Takamichi SUSHIDA! Akio HIZUME† and Yoshikazu YAMAGISHI† Triangular spiral tilings and origami



Triangular spiral multiple tilings

LetT = {Tj := "AjAj+mAj+m+nAj+n}j#Z

be a quadrilateral spiral multiple tiling of multiplicity |v|.

If three vertices of a quadrilateral T0 = "A0AmAm+nAn lie on asame line, then T becomes a triangular spiral multiple tiling.

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Figure: Deformation between quadrilateral tilings through a triangulartiling with an opposed parastichy pair (3, 5)




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Figure: Parastichy transition: ! = 2"# , # = 1+!

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Figure: Deformation between quadrilateral tilings through a triangulartiling with a non-opposed parastichy pair (3, 5)

Questions

(i) For each pair (m,n), which complex numbers " = re!"1! # D\R

produce triangular spiral multiple tilings ?

(ii) Which triangles admit spiral multiple tilings ?




Let

(m,k(z) =zk ( 1zm ( 1

(3)

be a rational function with one complex variable.

Lemma 7

Let " = re!"1! # C\R. Let m,n > 0 be relatively prime integers.

Suppose that "m '= 1. Then the following conditions are mutuallyequivalent.

(i) The three points Am, Am+n and An lie on a same line.

(ii) The four points O, A0, Am and An lie on a same circle.

(iii) rm sinn& ( rn sinm& + sin (m ( n)& = 0.

(iv) (m,m"n(") # R.




Main theorems for triangular spiral multiple tilings

We proved the following theorem about triangular spiral multipletilings with opposed parastichy pairs.

Theorem A

Let m,n > 0 be relatively prime integers.Let v be an integer which satisfies 0 < |v| < max {m,n}

2 .

Let Pm,n,v be the set of generators " = re!"1! # D\R

for triangular spiral multiple tilings of multiplicity |v|with an opposed parastichy pair (m,n).

Then Pm,n,v is a branch of a real algebraic curve parameterized by& = arg (").

Moreover, the union P =&

v>0

&(m,n) Pm,n,v is a dense subset of D.




Re

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21

Figure: The set of generators for triangular spiral tilings of multiplicityv = 1, with opposed parastichy pairs (m, n): A rational number x on theunit circle denotes e2!

!#1x




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21

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Figure: Parastichy transition: ! = 2"# , # = 1+!

52 , (2, 3) $ (5, 3) $ (5, 8)

! Principal convergents of the Golden section ! = 1+!

52 :

11,32,85,2113

, · · · , !, · · · ,3421

,138

,53,21

! Parastichy transition:

(2, 3) ) (5, 3) ) (5, 8) ) (13, 8) ) (13, 21), · · · .




Next, we proved the following theorem about triangular spiralmultiple tilings with non-opposed parastichy pairs.

Theorem B

Let m,n > 0 be relatively prime integers.Let v be an integer which satisfies 0 < |v| < n

2 .

Let Qm,n,v be the set of generators " = re!"1! # D\R

for triangular spiral multiple tilings of multiplicity |v|with a non-opposed parastichy pair (m,n).

Then Qm,n,v is a branch of a real algebraic curve parameterized byr = |"|.

Moreover, the union Qv =&

(m,n) Qm,n,v is a dense subset of D.




Triangles which admit spiral multiple tilings

Let arg "m = ( arg "n = #/3, and let v = 1 .

n arg "m ( m arg "n = 2# · 1 1 m + n = 6. (m,n) = (1, 5), & = #/3.

rm sinn& ( rn sin m& + sin (m ( n)& = 0 has a unique root

0 < r = 0.7548 · · · < 1.

! !

!

!

!

Re

Im

!m

!n

!m+n 1

O

arg !m =3ù

à arg !n =3ù

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Figure: A spiral tiling by equilateral triangles




Let arg "m = arg "n = #/3, and let v = 1.

n arg "m ( m arg "n = 2#v 1 (m + n = 6. (m,n) = (1, 7), & = #/3.

However, rm sin (m + n)& ( rm+n sin m& ( sin n& = 0 doesn’t have asolution 0 < r < 1.

Hence we can’t obtain a spiral tiling by equilateral triangles, withan non-opposed parastichy pair.

!!

!

!

Re

Im

arg !m =3ù

arg !n =3ù

!m+n

!m

!n

1

O

●

Figure: A equilateral triangle of the case of non-opposed parastichy pair




Let arg "m = #/3, arg "n = (#/6, and let v = 1.

n arg "m ( m arg "n = 2#v 1 m + 2n = 12

(m,n) = (2, 5), & = (5#/6.

The equation rm sinn& ( rn sinm& + sin (m ( n)& = 0

has a unique root 0 < r = 0.9214 · · · < 1.

! !

!

!

!

Re

Im

!m

!n

!m+n

1

O

arg !m =3ù

à arg !n =6ù

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Figure: A spiral tiling by right triangles with angles 30$, 60$, 90$




Let arg "m = #/3, arg "n = #/6, and let v = 1

n arg "m ( m arg "n = 2#v 1 (m + 2n = 12

(m,n) = (2, 7), & = (5#/6.

The equation rm sin (m + n)& ( rm+n sinm& ( sinn& = 0

has two solutions r = 0.883 · · · and 0.754 · · · .

! !

!

!

!

Re

Im

!m

!n

!m+n

1

Oarg !m =

3ù

arg !n =6ù

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Figure: Spiral tilings by right triangles with angles 30$, 60$, 90$




Origami for triangular spiral tilings

Figure: Origami sheets for a Fibonacci tornado:Solid lines are mountain fold and dashed lines are valley fold.




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Figure: Origami sheets for a spiral tiling by regular triangles:Solid lines are mountain fold and dashed lines are valley fold.

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Figure: Top-down view of an origami of a spiral tiling by regular triangles.Takamichi SUSHIDA! Akio HIZUME† and Yoshikazu YAMAGISHI† Triangular spiral tilings and origami



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Figure: An origami sheet for a spiral multiple tiling by right triangles withangles 30$, 60$, 90$

Figure: Top-down view

In the right figure,solid lines are mountain fold anddashed lines are valley fold.

These figures were chosen the coverpage of Journal of Physics A:Mathematical and Theoretical. 45. 23,2012.




Thank you for your attention...


Triangular spiral tilings and origami (Presentation sheet)

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