1 Tilings, Finite Groups, and Hyperbolic Geometry at the Rose-Hulman REU Rose-Hulman REU S. Allen Broughton Rose-Hulman Institute of Technology

2 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

3 A Philosopy of Undergraduate Research  doable, interesting problems  student - student & student -faculty collaboration  computer experimentation (Magma, Maple)  student presentations and writing

4 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

5 Icosahedral-Dodecahedral Tiling

6 (2,4,4) -tiling of the torus

7 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

8 Tiling: The Master Tile - 1

9 Tiling: The Master Tile - 2  maily interested in tilings by triangles and quadrilaterals  reflections in edges:  rotations at corners:  angles at corners:  terminology: (l,m,n) -triangle, (s,t,u,v) - quadrilateral, etc.,

10 Tiling: The Master Tile - 3  terminology: (l,m,n) -triangle, (s,t,u,v) - quadrilateral, etc.  hyperbolic when or

11 The Tiling Group Observe/define: Tiling Group: Orientation Preserving Tiling Group:

12 Group Relations (simple geometric and group theoretic proofs)

13 Riemann Hurwitz equation ( euler characteristic proof) Let S be a surface of genus then:

14 Tiling Theorem A surface S of genus has a tiling with tiling group if and only if  the group relations hold  the Riemann Hurwitz equation holds

15 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)

16 Divisible Tilings  torus - euclidean plane example  hyperbolic plane example  Dawn & Lori’s results  group theoretic surprise

17 Torus example ((2,2,2,2) by (2,4,4))

18 Euclidean Plane Example ((2,2,2,2) by (2,4,4))  show picture  the Euclidean plane is the “unwrapping” of torus “universal cover”

19 Hyperbolic Plane Example  show picture  can’t draw tiled surfaces so we work in hyperbolic plane, the universal cover

20 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

21 Dawn & Lori’s Problem and Results cont’d  Conjecture: Every divisible tiling (with a single tile in the corner is symmetric

22 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:

23 A group theoretic surprise - cont’d

24 Thank You! Questions???