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Transcript
Tilings, Finite Groups, and Hyperbolic Geometry at the Rose-Hulman REU S. Allen Broughton Rose-Hulman Institute of Technology 1 Outline A Philosopy of Undergraduate Research Tilings: Geometry and Group Theory Tiling Problems - Student Projects Example Problem: Divisible Tilings Some results & back to group theory Questions 2 A Philosopy of Undergraduate Research doable, interesting problems student - student & student -faculty collaboration computer experimentation (Magma, Maple) student presentations and writing 3 Tilings: Geometry and Group Theory show ball tilings: definition by example tilings: master tile Euclidean and hyperbolic plane examples tilings: the tiling group group relations & Riemann Hurwitz equations Tiling theorem 4 Icosahedral-Dodecahedral Tiling 5 (2,4,4) -tiling of the torus 6 Tiling: Definition Let S be a surface of genus . Tiling: Covering by polygons “without gaps and overlaps” Kaleidoscopic: Symmetric via reflections in edges. Geodesic: Edges in tilings extend to geodesics in both directions 7 Tiling: The Master Tile - 1 8 Tiling: The Master Tile - 2 maily interested in tilings by triangles and quadrilaterals p, q , r reflections in edges: rotations at corners: a , b, c , , angles at corners: l m n terminology: (l,m,n) -triangle, (s,t,u,v) quadrilateral, etc., 9 Tiling: The Master Tile - 3 terminology: (l,m,n) -triangle, (s,t,u,v) quadrilateral, etc. hyperbolic when 2 or 2 l m n or 1 1 1 1 0 l m n 10 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 11 Group Relations (simple geometric and group theoretic proofs) p q r 2 a b n 2 l 2 m 1. c 1, abc 1, ( pqqrrp 1) ( a ) qaq qpqq qp a , 1 1 (b) qbq qqrq rq b . 1 1 12 Riemann Hurwitz equation ( euler characteristic proof) Let S be a surface of genus then: 2 2 1 1 1 1 | G| l m n 13 Tiling Theorem A surface S of genus tiling group has a tiling with G p, q , r * if and only if the group relations hold the Riemann Hurwitz equation holds 14 Tiling Problems - Student Projects Tilings of low genus (Ryan Vinroot) Divisible tilings (Dawn Haney, Lori McKeough) Splitting reflections (Jim Belk) Tilings and Cwatsets (Reva Schweitzer and Patrick Swickard) 15 Divisible Tilings torus - euclidean plane example hyperbolic plane example Dawn & Lori’s results group theoretic surprise 16 Torus example ((2,2,2,2) by (2,4,4)) 17 Euclidean Plane Example ((2,2,2,2) by (2,4,4)) show picture the Euclidean plane is the “unwrapping” of torus “universal cover” 18 Hyperbolic Plane Example show picture can’t draw tiled surfaces so we work in hyperbolic plane, the universal cover 19 Dawn and Lori’s Problem and Results Problem find divisible quadrilaterals restricted search to quadrilaterals with one triangle in each corner show picture used Maple to do – combinatorial search – group theoretic computations in 2x2 complex matrices 20 Dawn & Lori’s Problem and Results cont’d Conjecture: Every divisible tiling (with a single tile in the corner is symmetric 21 A group theoretic surprise we have found divisible tilings in hyperbolic plane Now find surface of smallest genus with the same divisible tiling for (2,3,7) tiling of (3,7,3,7) we have: | G | 2357200374260265501327360000 * 14030954608692056555520001 22 A group theoretic surprise - cont’d | G | 2 22! and * 21 1 Z G 22 1 21 2 * 23 Thank You! Questions??? 24