Some Interesting Examples of Wang Tilings · 2013-04-04 · Aperiodic Tile Sets Summary Some...

Aperiodic Tile SetsSummary

Some Interesting Examples of Wang Tilings

M. Zeltzer D. Molho

Department of MathematicsVassar College

April 6th / Hudson River Undergraduate MathematicsConference

Zeltzer, Molho Wang Tilings (cont.)

Wang TilingsPenrose Mappings

Wang Tilings

As previously discussed, a Wang tile is a square tile withcolored edges.In any Wang tiling, the colored edges can be representedby rational numbers:

Aperiodic tile set: a tile set with a valid infinite tiling and novalid periodic tiling.

Wang Tilings

Jarkko Kari, 1995, 14 Tile Aperiodic Tile Set

Tiling Rules for this Set

Define the property that a tile multiplies by q ifaq + b = c + d for any Wang tile:

The numbers along the left and right edge are calledcarries.For example:

1(q)+−1 = 1+0⇒ q = 2 1(q)+0 = 1+−13⇒ q =

Tiling Rules for this Set

Define the property that a tile multiplies by q ifaq + b = c + d for any Wang tile:

The numbers along the left and right edge are calledcarries.For example:

1(q)+−1 = 1+0⇒ q = 2 1(q)+0 = 1+−13⇒ q =

Aperiodicity

Let f : Z2 → T be a tiling from Kari’s tiles. Suppose it isperiodic with horizontal period a and vertical period b.Let i ∈ Z, Generate the set of tilesFi = {f (1, i), f (2, i) · · · , f (a, i)}:

Define ni ∈ R to be the sum of the numbers representingthe top edges of the tiles in Fi .

Aperiodicity

Aperiodicity (part 2)

Fi+1:qini + left carry(f (1, i)) = ni+1 + right carry(f (a, i)),ni+1 = qini .For example:

(1(q) +−1) = (1 + 0), (1(q) + 0) = (2 + 0),(0 + 0) = (−1 + 1), (1(q) +−1) = (2− 1)⇒ niq = 3q,ni+1 = 1 + 2 + 1 + 2 = 6,⇒ q = 2

f is also vertically periodic, F1 will have the same tiles asFb+1:

n1 = nb+1 = q1q2 · · · qb · n1

q1q2 · · · qb = 1qi = 2 or 2

3 , and any product of 2’s and 23 ’s can never be 1.

A contradiction.Therefore the 14 tile set is aperiodic.

n1 = nb+1 = q1q2 · · · qb · n1

q1q2 · · · qb = 1qi = 2 or 2

n1 = nb+1 = q1q2 · · · qb · n1

q1q2 · · · qb = 1qi = 2 or 2

n1 = nb+1 = q1q2 · · · qb · n1

q1q2 · · · qb = 1qi = 2 or 2

Penrose Mappings

Robert Penrose suggested that a Wang tiling could beconstructed from a Penrose tiling, and Raphael M.Robinson refined a technique to do so.

Penrose P2 Tiling

The Penrose P2 tiling is an aperiodic tiling consisting oftwo tiles,a kite shaped tile:

and a dart shaped tile:

Penrose P2 Tiling

The Penrose P2 tiling is an aperiodic tiling consisting oftwo tiles,a kite shaped tile:

and a dart shaped tile:

Penrose P2 TIling (Cont.)

Quadrilateral/Pentagonal Partitioning

Use the P2 tiling on the previous page to partition the set intoquadrilateral and non-regular pentagonal sections as shown

above.

Quadrilateral/Pentagonal Partitioning (cont.)

The Penrose P2 matching rule:

The partition on the previous page yields the following 8shapes:

However, 32 Wang tiles are needed to account for rotationand reflection.

Color Assignment

Using 16 colors, each horizontal and vertical edge of thepolygons are assigned colors as follows:

Assign Wang tiles

These 4 colors and orientations for each polygon can beassigned to a Wang tile as follows.Using one example polygon:

Assign Wang Tiles (Cont.)

The Full Mapping

The 32 Tile Wang Tiling

Summary

An aperiodic Wang tiling can be generated from thePenrose P2 Tiling.Since the P2 tiles are aperiodic, the Wang tile setgenerated from it is also aperiodic.An tiling of the plane consisting of 14 Wang tiles wasdemonstrated to be aperiodic.

Further ResearchFurther research can include generating a Wang Tiling,using a similar method, from the Penrose P3 tiling, anaperiodic tiling consisting of two rhombs.

Bibliography I

J. Kari.A Small Aperiodic Set of Wang Tiles.Discrete Mathematics, 160: 259-264, 1995.

G. Shawcross.Wang Tiles and Aperiodic Tiling.(October 12, 2012).In order, rhythm and pattern.Retrieved March 27, 2013, fromhttp://grahamshawcross.com/2012/10/12/wang-tiles-and-aperiodic-tiling/.

Some Interesting Examples of Wang Tilings · 2013-04-04 · Aperiodic Tile Sets Summary Some...

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