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Kaleidoscopic Tilings on Surfaces, this Time with the Groups S. Allen Broughton Rose-Hulman Institute of Technology 1 Credits • Some of this work has been done and will be done with numerous undergraduates. 2 Outline • • • • Repeated patterns, structure, groups, and semigroups Groups and group actions Talk 1: fundamental domains Talk 2: homology, group algebras, and separability 3 Outline of Talk 1 • • • • • • • • Tilings of surfaces and the plane– examples, definition Kaleidoscopic tilings of the hyperbolic plane Hyperbolic geometry Tiling groups G* and G Riemann-Hurwitz equation Fundamental homomorphism Fundamental domain via a homomorphism Examples via Matlab 4 Patterns • • Many objects have repeated patterns, structures, or other properties due to the underlying presence of a symmetry group or semigroup Semigroup examples • fractal structures in an iterated function system • convolution properties of Laplace, and z-transforms • Group examples • Tilings in 2 or more dimensions • Crystallographic structures 5 Groups and group action Definition for these talks • • • • • • A group G is set of transformations of a set or space X, such that G is closed under composition and taking of inverses. If g,h ε G then gh =g◦h ε G and g-1 ε G For x ε X and g ε G define the action of g on x by gx=g(x). Note that (gh)x =g◦h(x) =g(hx). Typically a group will preserve some property of the space – e.g., distance preserving or rigid motion. Subgroups are subsets of G closed under compositions and taking inverses. 6 Groups and group actions Examples • • • • • X = {1,2,3…,n} and G is a group of permutations of X X is a regular n-gon and G = Dn is the dihedral group of symmetries of X X is euclidian or hyperbolic space and group G is the set of rigid motions of X Any subgroup of the above examples mapping a structure on X to itself G acts on subsets of X, e.g. k-subsets, vertices, edges, lines, or circles 7 Icosahedral-Dodecahedral tiling (2,3,5) – tiling – soccer ball 8 (2,4,4) -tiling of the torus 9 (2,2,2,2) -tiling of the torus 10 (3,3,4) -tiling of the hyperbolic plane 11 (2,3,7) -tiling of the hyperbolic plane 12 Tiling: definition • • • • • Let S be a surface of genus σ . Tiling: Covering by polygons “without gaps and overlaps” Kaleidoscopic: Symmetric via reflections in edges. Geodesic: Edges in tiles extend to geodesics in both directions terminology: (l,m,n)-triangle 13 Hyperbolic geometry • • • Refer to tiling pictures Points, lines and angles Reflections 14 The tiling group - 1 Q c _ π n p r ∆0 _ π R _ π a l m q b P 15 The tiling group - 2 Full Tiling Group for triangle (a finite group) G =< p , q , r > * Group Relations p 2 = q 2 = r ( pq ) = ( qr ) l 2 m = 1. = ( rp ) n = 1. 16 Riemann Hurwitz equation - triangles Let S be a surface of genus number of triangles: σ and |G*| the 1 1 1 4σ − 4 = 1− − − | G* | l m n 17 Tiling theorem - triangles A surface S of genus tiling group: σ has a tiling with G = < p, q , r > * if and only if • the group relations hold, and • the Riemann Hurwitz equation holds. Therefore tiling problems can be solved via group computation Start with the group then make the surface 18 The Tiling Group Observe/define: a = pq,b = qr ,c = rp Tiling Group: G = < p, q , r > * Orientation Preserving Tiling Group: G = < a ,b, c > 19 Group Relations (simple geometric and group theoretic proofs) p 2 = q 2 = r 2 = 1. a = b = c = 1, a b c = 1, ( p q q r r p = 1) l m n θ (a ) = qaq −1 θ (b ) = q b q −1 = qpqq = qp = a = qqrq = rq = b −1 −1 , . 20 Universal Cover and Fundamental Domain • • • • • Euclidean example (on board) The covering group and the fundamental homomorphism The covering group is defined on the universal of the surface A “fundamental domain” comes from the homomorphism Surface constructed by identifying boundaries on fundamental domain 21 Covering group and the homomorphism ~ ~ ~ ~ ~ ~ Λ =< p , q , r > , Λ =< a , b , c > * Γ → Λ → G ~p → p , q~ → q , ~ r → r * * Γ → Λ → G ~ ~ ~ a → a,b → b, c → c 22 Fundamental domain from the homomorphism • • • Pick exactly one tile in the cover for each group element in G*. Matlab examples Problem find a nice way to move in the surface to create nice fundamental domains 23