Daniel McNeil

April 3, 2007

Math 371

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

Tiling on the Euclidean Plane

Regular Tilings

Are there any others?

(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

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.

Periodic vs Aperiodic

• Periodic tilings display translational symmetry in two non-parallel directions.

• Aperiodic tilings do not display this translational symmetry.

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

Penrose Tilings

Roger Penrose

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.

• In a Penrose tiling, Nkite/Ndart = Φ

• Given a region of diameter d, an identical region can always be found within d(Φ+½).

Other Geometric ApplicationsTopologically 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

Hyperbolic Tilings

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)180o/n

• 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

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

4,5 4,7

4,8 4,10

Semiregular Tilings

• Just like in Euclidean, there are also semiregular tilings in Hyperbolic.

• This example shows a square and 5 triangles at each vertex.

Poincaré Upper Half Plane

• The vertical distance between two points is ln(y2/y1).

• Faces are all of equal non-Euclidean size.

• Image can be transformed from Poincaré Disc to PUHP.

Poincaré Disc vs PUHP

Poincaré Disc vs PUHP

Tilings in Art and Architecture

Tilings in Nature

• 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