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 = fkite; dartg 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 2 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 = ff1; f2;:::; fN g similarities with scaling ratios rn = s and G p = f (p): f 2F 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]1 = the set of all infinite strings over the alphabet f1; 2; 3;:::; Ng For θ 2 [N]1 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 2 [N]1 (1) For L a leaf of the k-tree: −1 t(θ; k; L) := (f(θ;k) ◦ fL)(p) (2) T (θ; k) := ft(θ; a(θ; k); L): L is a leaf of the a(θ; k)-tree g (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.