Spherical tilings by congruent 4-gons on Archimedean dual ...akama/iamc.pdf · bipyramid, an...

Introduction Main Theorem Allowed and forbidden vertex types Key Lem. Family of sets of 3-val. vertex types T24 T60 R12 and R30 Summary Future work

Spherical tilings by congruent 4-gons onArchimedean dual skeletons

Yohji Akama

Mathematical Institute, Tohoku University, Sendai Miyagi Japan, 980-8578.

June 13, 2016

Yohji Akama

Spherical tilings by congruent 4-gons on Archimedean dual skeletons


We classify all spherical tilings by congruent 4-gons such that the4-gon has only 3 equal sides and the skeleton of the tiling is thatof an Archimedean dual. Such a tiling induces a perfectface-matching of the skeleton. The classification is by checking thesolvability of systems of integer-coefficient linear equations. Such asystem is induced by an assignment of inner angles respecting aperfect face-matching to the skeleton. The number of suchsystems of integer-coefficient linear equations is huge, because thenumber of the tiles and the number of the perfect face-matchingsof the skeleton of the tiling are large. We overcome this difficultyin not only convex 4-gonal tile cases [A.-van Cleemput15] but alsoconcave 4-gonal tile cases, by linear algebraic approach for the3-valent vertex types of the tilings. We scrutinize how [K. Coolsaet,Spherical quadrangles with three equal sides and rational angles,2015] reduces the complexity for the convex 4-gonal tile cases.

Yohji Akama



Motivation.

Theorem (Grunbaum-Shephard81)

The skeleton of a spherical isohedral tiling (i.e., a tiling s.t. thesymmetry group acts transitively on the tiles) is exactly theskeleton of a trapezohedron(i.e., the dual of an antiprism), abipyramid, an Archimedean dual, or a Platonic solid.

Proposition [A.13, Hiroshima Math. J.]

There is a spherical non-isohedral tiling A by 12congruent concave 4-gons on a trapezohedron skeleton.

Problem[A.15, Japan Conf. on Discr. & Comput. Geom. & Graphs]

Is A the only spherical monohedral, non-isohedral tiling with theskeleton listed in Grunbaum-Shephard’s Theorem?

This reverse problem of Grunbaum-Shephard’s theorem comparesthe symmetry group of spherical isohedral tilings to theautomorphism group of the skeleton.

Yohji Akama



Symmetry group vs. automorphism group of the skeleton.Deformation.

Mani’s theorem ([Math. Ann. 1971], a refinement of Steinitz’stheorem): For each polyhedral graph G , there exists a3-dimensional convex polytope P such that the automorphismgroup of G is the symmetry group of P. Mani’s theorem is forpolyhedra, and is generalized on any sphere in [Deza et al.,

“Geometric Structure of Chemistry- Relevant Graphs. Zigzags and

Central Circuits,” Springer, 15]. Other extensions of Steinitz’stheorem is found in [Seok-Hee Hong and H. Nagamochi11].Cauchy’s rigidity theorem for convex polyhedra,For deformation of vertex-transitive polyhedra, see[Robertson-Carter-Morton1970].Angle-assignments to the skeleton (I. Rivin’s solution toSteinitz’s problem [Ann. of Math1996]).

Yohji Akama



Every sph. tiling by congruent convex 4-gons on thetrapezohedron skeleton is isohedral.

Case-by-case analysis by finding forbidden substructures.

Yohji Akama



All spherical monohedral polygonal tilings with Platonic orbipyramidal skeleton are isohedral [A.-Yan16]. Theircombinatorial types are:

F4

G8

C6

D

I

G4n (n ≥ 3)

G4n+2 (n ≥ 1)

abc

Figure: For each of them, the degree of freedom of the continuousdeformation is the maximum # of change of the symmetry type.

Yohji Akama



Any spherical tiling by congruent 3-gons on the skeleton of an Archimedean dual

and any spherical tiling by congr. kites (darts, rhombi) are isohedral [Sakano-A.15].

T60

R30

T24

R12

Figure: The 4 4-gon-faced ADs R12,T24,R30,T60 among the 13ADs (left). The right upper (middle, bottom, resp.) is a spherical tilingby congruent kites (darts, rhombi, resp.) on the trapezohedron skel.

Yohji Akama



Any spherical tiling by congruent type-2 4-gons on an AD skeleton is isohedral.

Definition

A 4-gon is said to be type-2, if the edge-lengths are aaab (a 6= b).

Lemma (Ueno-Agaoka01)

A tile of a sph. tiling by congr. 4-gons is a rhomb., a kite, a dart,a type-2, or has cyclic list of edge-lengths aabc (a, b, c distinct).

We will prove:

Theorem (Main)

There is no sph. tiling by congr. type-2 4-gons on R12-, R30-, orT60-skel, but only one sp. ti. by congr. type-2 4-gons on T24-skel.,

It is obtained from the central projection of T24, by twist-ing each Y -figure in common irrational multiple of π rad.

We will prove this below.Yohji Akama



A spherical tiling by congruent 4-gons of edge-lengthsaaab (a 6= b) induces a solvable angle-assignment of theskeleton S respecting a perfect face-matching of S .

T60 has 6066688 perfect face-matchings, i.e., 1-regular spanningsubgraphs of the dual graph. So T60 has ≥ 6066688 · 260

angle-assgnmnts to the skel.

Figure: The left is the 8 perfect face matchings of the cube skel. Theright are 4-gons of type 2. A type-2 4-gon has, by definition, 3 equilateraledges (blue) and 1 face-matching edge (red) of different length. α, β, γ, δstands for inner angles. Each type-2 4-gon has 2 a.-assignments.Yohji Akama



The ten 3-valent vertex types in spherical tilings by congr.type-2 4-gons.

Definition

A vertex consisting of i inner angles α, j inner angles β, k innerangles γ, and l inner angles δ (i , j , k, l ≥ 0) is said to have typeiα + jβ + kγ + lδ.

A spherical tiling by congruent 4-gons of type 2 has at most 103-valent vertex types:

V3 := {3β, 2β + γ, α + β + δ, 2α + γ, 2α + β,

3γ, 2γ + β, α + γ + δ, 2δ + β, 2δ + γ}.

See [A.-van Cleemput, Ars Mathematica Contemporanea, 15].

Yohji Akama



Allowed vertex types

Lemma

∀n ∈ {12, 24, 30, 60}.∃ a family Cn of nonempty sets of vtx typs∀sph. tiling T by n congr. type-2 4-gons on AD skel.∃P ∈ Cn.P ⊆ T

Yohji Akama



Prf – Count face-match’ edges on the AD bipartites.

C24 := AC (V4,>0). 6-set C12 := AC (V3,0, V3,1, V4,>0). (# = 156)

C30 := AC (V3,0, V3,1, V5,1, V5,2). (# = 1200)

C60 := AC (V3,1, V4,>0, V5,>0). (# = 660)

AC (T1,T2, . . . ,Tn) := {{t1, . . . , tn} | ti ∈ Ti} .

Vd ,i := {types of d-val. vertices incident to i face-match’edges}V3,0 := {3β, 2β + γ, β + 2γ, 3γ}. (4-set)

V3,1 := {2α + β, 2α + γ, α + β + δ, α + γ + δ, β + 2δ, γ + 2δ}. (6-set)

V4,>0. 11-set V5,i . (10-set) (i = 1, 2).

(α, β, γ, δ) ↔ (δ, γ, β, α).Yohji Akama



Forbidden vertex types

Lemma

∃ family F12,F24,F30,F60 ⊂ [V3]<4 s.t.

∀sph. tiling T by n > 6 congr. type-2 4-gons.

1 ∀F ∈ F24.F 6⊆ T .

2 n = 12, 30, 60 =⇒ ∀F ∈ Fn.F 6⊆ T .

Yohji Akama



Prf

F24 := {{3β, α + γ + δ}, {α + β + δ, 3γ}}∪{{2α + γ, 3γ}, {2α + β, 2γ + β}, {3β, β + 2δ}, {2β + γ, 2δ + γ}}.20-set F12 := F30 := F24 ∪R62-setF60 := F24 ∪ {R ∪ {S} | R ∈ R,S ∈ V3,0}.

Here R := {{t1, t2} ⊂ V3 | t1 = t2 =⇒ α = δ or β = γ}.Proof.

Prop.[A.13] ∀s. ti. T by cngr. type-2 4gns, (1) α 6= γ and β 6= δ,(2) V3,0 ∩ T 6= ∅ 6= T ∩ V3,1 =⇒ α 6= δ and β 6= γ.

n = 12, 30: V3,0 ∩ T 6= ∅ 6= T ∩ V3,1.n = 60: By T ∩ V3,1 6= ∅, if T ∩ V3,0, then Prop (2) implies α 6= δand β 6= γ.

Yohji Akama



The characterization of the inner angles(α, β, γ, δ) ∈ (Qπ)4 of convex type-2 sph. 4-gons (1/2).

Theorem (Coolsaet15)

If the inner angles (α, β, γ, δ) of a conv. type-2 sph. 4-gons are in(Qπ)4, then (α, β, γ, δ) or (δ, γ, β, α) satisfy one of the following:

1 α = δ , β = γ. (isosceles trapezoid. Does not tile the unitsphere [A.-van Cleemput15])

2 α = γ2 , δ = β

2 , α + δ < π.

3 α = 3γ2 , β = π

3 , δ = 2π3 − γ

2 . , π2 < γ < 2π

3 .

4 α = π6 + γ

2 , β = 2γ, δ = π2 + γ

2 , π3 < γ < π

2 .

5 α = π6 + γ

2 , β = 2γ, δ = π2 + 3γ

2 , 4π15 < γ < π

3 .

6 α = π6 + γ

2 , β = 2π − 2γ, δ = 3π2 − 3γ

2 , π2 < γ < 5π

6 .

Yohji Akama



The characterization of the rational inn. angles of conv.type-2 sph. 4-gons (2/2) – sporadic.

(cnt’d) (4π/n is the area of the 4-gon.)

1 π(

56 , 8

21 , 57 , 17

42

), π

(1115 , 7

15 , 35 , 8

15

), π

(1730 , 8

15 , 1315 , 11

30

),

π(

2330 , 1

3 , 1415 , 3

10

), π

(3160 , 3

5 , 56 , 23

60

), π

(4960 , 3

10 , 1415 , 17

60

). (n = 12).

2 π(

1960 , 7

10 , 1415 , 13

60

). (n = 24).

3 π(

5360 , 4

15 , 710 , 17

60

). (n = 30).

4 π(

710 , 4

15 , 1315 , 7

30

), π

(4960 , 4

15 , 710 , 17

60

). (n = 60).

5 1 tuple for n = 6, 2 tuples for n = 42, 1 tuple for n = 84, 1tuple for n = 120, 12 tuples for fractional n.

(∵) [G. Myerson. “Rational products of sines of rational angles.”Aequationes mathematicae, 1993].

Yohji Akama



For ∀tiling T , ∀set of vertex types of T induces solvablelinear system

∀n ∈ {12, 24, 30, 60}, ∀sp. ti. T by n congr. x type-2 4-gons onAD skel., ∀ set T of 3-val. vtx typs of T ,

(ϕx(n, Cn,Fn,T )): ∀F ∈ Fn.F 6⊆ T & ∃P ∈ Cn s.t. a system

{α + β + γ + δ − 2 =4π

n, t = c = 2π | c ∈ P, t ∈ T}

has indefinite solution, or definite sol. (α, β, γ, δ) in(0, 2π)4 \ (0, π)4 (x=cnc) and listed in Coolsaet’s Theorerem(x=cnv).

Φ(n, i) :={T ∈ [V3]

<i | ϕx(n, Cn,Fn,T )}

Yohji Akama



The sphe. tiling by congr. type-2 4-gons on T24-skel.(1/4)

Proof. A complete of representative of Φcnv (24, 3) is

{{2β + Y } , {2α + Y } , {α + β + δ} | Y = β, γ},

by 2sec computation with MAPLE. There are 1088 perfectface-matchings of the T24-skel, and 8 assignments of α, β, γ, δ tothe skel. such that the set of 3-val. vertex types is in Φcnv (24, 3)and the angle-assignments respect perfect face-matchings. 4assignments among the 8 induce systems of linear equations, whichare unsolvable by the coefficient difference. There are only onesolvable and canonical assignment (cnt’d):

Yohji Akama



0: 1 (α) 2 (α) 3 (α) 4 (α) 16: 8 (β) 23 (δ) 22 (δ) 24 (β)1: 0 (β) 5 (δ) 6 (δ) 7 (β) 17: 8 (δ) 24 (β) 10 (β) 9 (δ)2: 0 (δ) 7 (β) 8 (β) 9 (δ) 18: 10 (δ) 24 (β) 22 (β) 25 (δ)3: 0 (β) 9 (δ) 10 (δ) 11 (β) 19: 10 (β) 25 (δ) 12 (δ) 11 (β)4: 0 (δ) 11 (β) 12 (β) 5 (δ) 20: 12 (β) 25 (δ) 22 (δ) 21 (β)5: 1 (γ) 4 (γ) 13 (γ) 21: 13 (γ) 20 (γ) 14 (γ)6: 1 (α) 13 (α) 14 (α) 15 (α) 22: 14 (α) 20 (α) 18 (α) 16 (α)7: 1 (γ) 15 (γ) 2 (γ) 23: 14 (γ) 16 (γ) 15 (γ)8: 2 (α) 15 (α) 16 (α) 17 (α) 24: 16 (γ) 18 (γ) 17 (γ)9: 2 (γ) 17 (γ) 3 (γ) 25: 18 (γ) 20 (γ) 19 (γ)10: 3 (α) 17 (α) 18 (α) 19 (α)11: 3 (γ) 19 (γ) 4 (γ)12: 4 (α) 19 (α) 20 (α) 13 (α)13: 5 (δ) 12 (δ) 21 (β) 6 (β)14: 6 (δ) 21 (β) 22 (β) 23 (δ)15: 6 (β) 23 (δ) 8 (δ) 7 (β)

Yohji Akama



The sph. tiling by congr. type-2 4-gons on T24-skel. (3/4).

From the angle-assignment A, the spherical tiling by 24 congruenttype-2 quadrangles is constructed from the central projection ofT24 by rotating the central Y-figure of each octant (i.e.,right-angled regular triangle),

by a fixed radian θ in the direction of the parity of theoctant. Here, the octant (+,+,−) has parity − andthe octant (+,−,−) has parity +.

The inner angles of the tiles are α = π/2,β = arcsin

((5− 2

√3)/(12− 6

√3)

), γ = 2π/3, and δ = π − β.

The length of the edge αδ is

b = arccos

(√(4− 2

√3)/(5− 2

√3)

), while the length of the

other three edges is a = π/2− b.Note that β is an irrational multiple of π, because no instance of(α, β, γ, δ) is listed in Coolsaet’s thm.

Yohji Akama



The sph. tiling by congr. type-2 4-gons on T24-skel. (4/4)

¬∃ sph. tiling by congruent concave type-2 4-gons on T24-skel.Proof. By 1min computation with MAPLE,

Φcnc(24, 4) = Φcnv (24, 3) ∪ { [{2 α + γ, 2 β + γ}]∼,

[{2 β + γ, α + γ + δ}]∼, [{α + β + δ, α + γ + δ}]∼}.

9704 assignments of α, β, γ, δ to the skel. such that the set of the3-val. vtx types is in Φcnc(24, 4) and the assgnmnts respect perfectface-matchings. 6564 assgnmnts among the 9704 induce systemsof linear equations, but they are unsolvable by the coefficient diff.3136 assgnmnts among the 9704 induce systems of linearequations, but they are unsolvable by the linear programming. Theother 4 assignment are solvable and correspond to one commoncanonical solution A of the two previous slides.

Yohji Akama



No sph. tiling by cngr convex 4-gons on T60-skel.

Lemma

For any sp. ti. T by cngr type-2 4gns on T60-skel, (1)T ∩ V3,0 6= ∅. (2) ∀F ∈ F12.F 6⊆ T .

Proof. (1) Assume otherwise. All 20 3-val. vertices turn out to beincident to match’ edges. Because T60 has bipartital skel., theremaining 10 match’ edges are incident to 5-val. vertices. As # of5-val. vert is 12, some 5-val. vertex is incident to no matchingedge. However,

¬ϕx(60,AC (V3,1, V4,>0, V5,>0, V5,0), ∅, ∅). (x ∈ {cnv , cnc}).

(The total time of computation with MAPLE/Corei7 is 1min.)This is a contradiction.

Yohji Akama



Proof continued

(2) F12 = F24 ∪ {{t1, t2} ⊂ V3 | t1 = t2 =⇒ α = δ or β = γ}.We have the conclusion, by T ∩ V3,1 6= ∅, (1), F24, and

V3,0 ∩ T 6= ∅ 6= T ∩ V3,1 =⇒ α 6= δ&β 6= γ.

Corollary

There is no spherical tiling by congruent convex type-2 4-gons onT60-skeleton.

Proof.

1min computation shows that

¬ϕcnv (60,AC (V3,0, V3,1, V4,>0, V5,>0), ∅, ∅).

By counting matching edges, we have the conclusion.

Yohji Akama



No sph. tiling by cngr. concave type-2 4-gons on T60-skel.

Prf. Any 5-val. vertex is adjacent to a 4-val. one in T60. So,

Lemma

In a sp. ti. by cng type-2 4gns on T60-skel., if every 5-val. vertexdoes δ (α, resp.) 4 times, then some 4-val. vertex contains α (δ,resp.) at least twice .

ϕcnc(60,AC (V3,0, V3,1, V4,>0, V5,>0), ∅, ∅) is witnessed by 8solutions (α, β, γ, δ) =(2π/15, 2π/15, 26π/15, π/15), or(π/30, 28π/15, π/15, π/10) withα ↔ δ and/or β ↔ γ. Note that AC (...) is symmetric w.r.t suchexchange(s) since V’s are defined so.

Yohji Akama



No sph. tiling by cngr. concave type-2 4-gons on T60-skel.(2/2)

If (α, β, γ, δ) = (2π/15, 2π/15, 26π/15, π/15), then a 5-valentvertex type is γ + 4δ and a 4-valent vertex type is α + γ + 2δ orβ + γ + 2δ.If (α, β, γ, δ) = (π/30, 28π/15, π/15, π/10) , then a 5-valentvertex type is 4α + β and a 4-valent vertex type is 2α + β + γ.These contradict against previous Lemma on # of α and # of δ.The other six solution are obtained by α ↔ δ and/or β ↔ γ, andcontradict similarly.

Yohji Akama



No sph. tiling by congr. type-2 4-gons on R12- or R30-skel.

Proof.

Φcnv (12, 3) = 〈{3 β, 2X + Y } , {3 β, α + β + δ} | X = α, δ;Y = β, γ〉⊂

Φcnc(12, 4) = 〈{3 β, 2X + Y } , {β + 2Y , α + β + δ} | X = α, δ;Y = β, γ〉 .

Φcnv (30, 2) = 〈〉.

Φcnc(30, 4) = 〈{3 β, α + β + δ} , {2 X + γ, 2 β + γ} | X = α, δ〉 .

1min, 2min, 10min, 50min computation by (MAPLE/Corei7).

Yohji Akama



No sph. tiling by congr. type-2 4-gons on R12- or R30-skel.– Prf cnt’d

Proof (cnt’d) No member of Φcnc(n, 4) (n = 12, 30) is suitable, by:

Lemma (“3vs3”)

For a spherical tiling by congr. 4-gons on theRe-skeleton (e = 12, 30), if the set of 3-val. vertex types is{t1, t2}, then a homogeneous system of linear equationsit1 + jt2 = kα + `β + `γ + kδ has a non-triv. solution (i , j , k, `)such that i ≥ 0, j ≥ 0.

Proof. The opposite vertex of a 3-val. vertex is 3-val.

Yohji Akama



Summary.

Theorem (Main)

There is no sph. tiling by congr. type-2 4-gons on R12-, R30-, orT60-skel, but only one spherical tiling by congruent type-2 4-gonson T24-skel.

It is obtained from the central projection of T24 bytwisting each Y -figure in common irrational multi-ple of π radian.

Yohji Akama



For Re- or T60-skel.,1 Plan a family C of compulsory sets of vertex types, by counting

argument # of matching edges in view of the skeleton.2 Do generate the family Φ of sets of 3-val. vertex types,

conjuncting with C, avoiding F , and having indefinite solutionsor Coolsaet-rational (concave, resp.) definite ones.

3 Check 3-val. vertex types s.t.: In Re , the opposite of a 3-val.vertex is 3-val. (“3vs3” Lemma). In T60, a 4-val. vertex and a5-val. vertex have types so that they areadjacent (Lemma (T602)). Conclude that no sph. tiling bycongruent type-2 4-gons exists if Φ = {∅}.

4 Act (1-4) for each remaining subcase T ∈ Φ (Lemma(T60)).

A smaller C may save the total computation time.For T24-skel., generate Φ from some C. Then construct“canonical” sph. tilings by cngr. 4-gons from all LP-solvableangle-assignments respecting perfect face-matchings of T24

s.t. the set of 3-val. vertex types is in Φ. (cf. canonicalconstruction path of McKay)

Yohji Akama



Feature of Coolsaet-rationality for convex type-2 4-gons.

Sp. ti. by cng. type-2’s on T24-skel. has angle in (R \Q)π.Efficient. Let Acnv (n) be Φcnv (n, 3) with “listed in Coolsaet’sthm” replaced by “∈ (0, π)4,” that is, {maximalT ∈ [V3]

3 |∀F ∈ Fn.F 6⊆ T & ∃P ∈ Cn (α + β + γ + δ − 2π = 4π

n ,∀t ∈ T∀c ∈ P. t = c = 2π) has indefinite sol., or definite sol.(α, β, γ, δ) ∈ (0, π)4}.

Φcnv (12, 3) 2min 8 singl. & 8 pairsAcnv (12) 2min 10 singl. & 18 pairs

Φcnv (24, 3) 20sec 6 singl.Acnv (24) 1min 10 singl. & 18 pairs

Φcnv (30, 2) 10min 0Acnv (30) 10min 10 singl. & 10 pairs

Φcnv (60, 3) 3min 2 singl.Acnv (60) 15min 10 singl. & 18 pairs

Yohji Akama



The two 5-gon-faced Archi. duals.

Amon the 13 ADs, there are two 5-gon-faced ones (figures below).The cyclic list of 5 edges of a tile is aaabb (a 6= b). The #s offaces are 24 and 60. On each of the two 5-gon-faced AD skels.,there is a spherical tiling T by congruent equilateral5-gons [Akama&M. Yan]. The two tilings are a pentagonalsubdivision of an octahedron and that of an icosahedron. Hence,any spherical tiling by congruent equilateral 5-gons on an AD skel.is isohedral.

Figure: P24 (left) and P60 (right). Their skeletons are non-bipartitaltripartites.

Yohji Akama



Prf of Key Lem.(1/4) – The 4 4-gonal Archi. duals havebipartital skel.

Rhombic e-hedron (Re), tetragonal 2e-hedron (T2e) (e = 12, 30).

By the constructions [Conway et al., The Symmetries of Things, 08], the skel. of Re is(L({3, q}))∗ & that of T2e is (L(L({3, q})))∗. Here L(G ) is the linegraph (i.e., link the midpts of adjacent edges) of a graph G . {3, q}(q = 4, 5) is Platonic solid with e edges s.t. q 3-gons share a vtx.So, the skel. of Re is a bipartite btw the set V1 of f 3-val. vert. &the set of v q-val. vert., where {3, q} has f faces. & v vert.The skel. of T2e is a bipartite btw V1 ∪ V3 and a set V2 of f 4-val.vert. Here V3 is a set of v q-val. vert.

Yohji Akama


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