Self-Similar Polygonal Tilings

Andrew Vince

Second Malta Conference on Graph Theory and Combinatorics

Andrew Vince Self-Similar Polygonal Tilings Andrew Vince Self-Similar Polygonal Tilings Tiling in the Alhambra palace, Granada, Spain, build in the 13th centruy

Andrew Vince Self-Similar Polygonal Tilings Escher print, about 1938

Andrew Vince Self-Similar Polygonal Tilings Archimedean tilings

Andrew Vince Self-Similar Polygonal Tilings Kepler “monster” (1619)

Andrew Vince Self-Similar Polygonal Tilings Penrose tiling

Aperiodic prototile set P = {kite, dart}

Andrew Vince Self-Similar Polygonal Tilings A Penrose tiling has ﬁnite order:

The prototile set is ﬁnite.

A Penrose tiling T is repetitive:

For every r > 0 there is R > 0, such that a copy of every r-patch in T is contained in every R-patch in T .

Andrew Vince Self-Similar Polygonal Tilings A perfect tile (Duijvestijn, 1978) and a rep-tile

Andrew Vince Self-Similar Polygonal Tilings A tiling T is self-similar if there is a similarity transformation φ : R2 → R2 such that, for for every t ∈ T , the larger tile φ(t) is in turn tiled by tiles in T .

Andrew Vince Self-Similar Polygonal Tilings Self-Similar Polygonal Tiling

A self-similar polygonal tiling is a tiling of the plane by pairwise similar polygons that is

1 of ﬁnite order

2 self-similar

3 repetitive

Andrew Vince Self-Similar Polygonal Tilings Andrew Vince Self-Similar Polygonal Tilings Golden Bee Tilings

The golden bee is the only polygon, except for right triangles, that can be partitioned into two non-congruent similar copies of itself.

Andrew Vince Self-Similar Polygonal Tilings A self-similar polygonal tiling of order 6.

Andrew Vince Self-Similar Polygonal Tilings The Construction - pair (p, F )

p - a polygon

an F = {f1, f2,..., fN } similarities with scaling ratios rn = s and G p = f (p). f ∈F

N P 2an The number s is determined by a1, a2,..., aN : s = 1. n=1

p(a, b) : where s2a + s2b = 1.

Andrew Vince Self-Similar Polygonal Tilings 2 2 4 r1 = s, r2 = s where s + s = 1

Andrew Vince Self-Similar Polygonal Tilings The k-tree for a1, a2,..., aN

Every non-leaf node has N children, the respective edges labeled a1, a2,..., aN . If L is a leaf, let a(L) = the sum of the labels on the path from the root to L, a−(L) = sum of the labels from the root to the parent of L.

a(L) > k ≥ a−(L)

Andrew Vince Self-Similar Polygonal Tilings notation

For a leaf L

fL = fi1 ◦ fi2 ◦ · · · ◦ fin ,

where ai1 , ai2 ,... ain are the labels on the path from the root to L.

[N]∞ = the set of all inﬁnite strings over the alphabet {1, 2, 3,..., N}

For θ ∈ [N]∞

a(θ, j)= aθ1 + aθ2 + ··· + aθj

f −1 = f −1 ◦ f −1 ◦ · · · ◦ f −1 (θ,j) θ1 θ2 θj

Andrew Vince Self-Similar Polygonal Tilings The construction

Given: pair (p, F ) and θ ∈ [N]∞

(1) For L a leaf of the k-tree:

−1 t(θ, k, L) := (f(θ,k) ◦ fL)(p)

(2)

T (θ, k) := {t(θ, a(θ, k), L): L is a leaf of the a(θ, k)-tree }

(3) [ T (θ) := T (θ, k) k≥1

Andrew Vince Self-Similar Polygonal Tilings Examples

p(a, b) : where s2a + s2b = 1.

Andrew Vince Self-Similar Polygonal Tilings Andrew Vince Self-Similar Polygonal Tilings q(a, b): w = sb−a/2 where sa + sb = 1.

Andrew Vince Self-Similar Polygonal Tilings Figure: A tiling of order 3.

Andrew Vince Self-Similar Polygonal Tilings Andrew Vince Self-Similar Polygonal Tilings Andrew Vince Self-Similar Polygonal Tilings Andrew Vince Self-Similar Polygonal Tilings Andrew Vince Self-Similar Polygonal Tilings Andrew Vince Self-Similar Polygonal Tilings Andrew Vince Self-Similar Polygonal Tilings