1. Daniel McNeil April 3, 2007 Math 371

2. What is a tiling? A tiling, or tessellation, refers to a collection of figures that cover a plane with no gaps and no overlaps. Tessella is Latin term describing a piece of clay or stone used to make a mosaic

3. Tiling on the Euclidean Plane

4. Regular Tilings Are there any others?

5. (3,12,12)(3,6,3,6) (4,4,3,3,3) (4,6,12)(3,4,6,4) (3,3,3,3,6) (8,8,4) (3,3,4,3,4) Semiregular Tilings

6. Tilings and Patterns Book written in 1986 by Branko Grünbaum and G.C. Shepherd. Remains most extensive collection of work to date. Took particular interest in periodic and aperiodic tilings.

7. Periodic vs Aperiodic Periodic tilings display translational symmetry in two non- parallel directions. Aperiodic tilings do not display this translational symmetry.

8. Is there a polygon that tiles the plane but cannot do so periodically? From Old and New Unsolved Problems in Plane Geometry and Number Theory

9. Penrose Tilings Roger Penrose

10. Penrose Tilings Discovered by Roger Penrose in 1973 Most prevalent form of aperiodic tilings No translational symmetry, so never repeats exactly, but does have identical parts In 1984, Israeli engineer Dany Schectman discovered that aluminum manganese had a penrose crystal structure.

12. In a Penrose tiling, N kite /N dart = Φ Given a region of diameter d, an identical region can always be found within d(Φ+½).

13. Other Geometric Applications Topologically Equivalent Tilings a=average number of sides per polygon F=number of faces b=average number of sides meeting at a vertex V=number of vertices Euler Characteristic

14. Hyperbolic Tilings

15. Regular Tilings In Euclidean we saw that the angle of a regular n-gon depends on n. What about Hyperbolic geometry? In Hyperbolic, the angle depends on both n and the length of each side. 0<θ<(n-2)180 o /n

16. In Euclidean we could construct a regular tiling with 4 squares at each vertex. Now in Hyperbolic we need 5 or more. In general, we have regular hyperbolic tilings of k n-gons whenever 1/n+1/k<1/2 Result: Infinitely many regular hyperbolic tilings Regular Tilings

17. 1/n+1/k = 1/4+1/6 = 10/24 < 1/2

18. 4,54,7 4,84,10

19. Semiregular Tilings Just like in Euclidean, there are also semiregular tilings in Hyperbolic. This example shows a square and 5 triangles at each vertex.

20. Poincaré Upper Half Plane The vertical distance between two points is ln(y 2 /y 1 ). Faces are all of equal non-Euclidean size. Image can be transformed from Poincaré Disc to PUHP.

21. Poincaré Disc vs PUHP

23. Tilings in Art and Architecture

24. Tilings in Nature

25. Abelson, Harold and DiSessa, Andrea. 1981. Turtle Geometry. Cambridge: MIT Press Baragar, Arthur. 2001 A Survey of Classical and Modern Geometries: With Computer Activities. New Jersey: Prentice Hall Klee, Victor and Wagon Stan. 1991. Old and New Unsolved Problems in Plane Geometry and Number Geometry. New York: The Mathematical Association of America Livio, Mario. 2002 The Golden Ratio. New York: Broadway Books Stillwell, John. 2005. The Four Pillars of Geometry. New York: Springer www.wikipedia.org www.mathworld.wolfram.com