### Tiling

September 24, 2015

Every

technological age can be identified by its

materials. For the

Industrial Revolution, these were primarily

iron and

coal. The

computer age is linked with

silicon, but my

childhood seemed to be the age of

linoleum.

Linoleum, named for its principal

ingredient,

linseed oil, was

invented in 1855, but its popularity as a

floor tile peaked in the

1950s. The corridors of my

elementary school were a sea of

square linoleum tiles, chosen for that purpose because they were very easy to clean.

Complete coverage of a

surface is called

tiling. There are a multitude of ways to tile

planar surfaces with

geometrical shapes. Although the floor tiles of my school were square,

nature seems to prefer

hexagonal tiling, as shown in the figure.

Humans have

mimicked nature by using hexagons for tiling.

When you constrain yourself to

regular tiling using just one type of regular polygon, you're limited to hexagons, as shown above figure,

triangles, or squares. While you can place three regular hexagons, four squares, or six

equilateral triangles around a

vertex, you can't do that with a

pentagon, or any

polygon with more than six sides.

| *The pattern of blocks for my patio.*
This is a very common tiling for rectangles whose length is twice the width, and it can be visualized as a shading of a square tiling.
(Photo by author) |

As

Johannes Kepler noted in his

Harmonices Mundi, the 1619 work in which he published his

third law of planetary motion, you can complete tiling with regular pentagons by adding three additional shapes (see figure).

Once we allow our tilings to include more than one shape, a vast

panoply of

designs has opened. As an example, the figure below shows how two six-sided shapes can fill a plane with a more

artistic effect than a simple hexagonal tiling.

The previous examples are tilings having

translational symmetry; that is, the

pattern is formed by just stacking a primitive object

horizontally and

vertically in the plane. Even the Keplerian tiling of pentagons and three other shapes has a group of elements that can be repeatedly stamped to fill the plane. However, there are

aperiodic tilings without translational symmetry.

A

Penrose tiling, named after

mathematician,

Roger Penrose, tiles the plane using an

aperiodic set of prototiles. A Penrose tiling does not have translational symmetry, and it's an example of a two-dimensional

quasicrystal with a

diffraction pattern having five-fold

rotational symmetry. An example of Penrose tiling appears below, and a nice summary of Penrose tiling and other tiling appears as ref. 2.[2]

Although regular pentagons will not by themselves tile a plane, relaxing the requirement to any

convex polygon of five sides yields quite a few

pentagonal tilings. Up to the present, fourteen had been found, and these are shown on their

Wikipedia page. Just last month, three mathematicians from the

University of Washington Bothell,

Casey Mann,

Jennifer McLoud, and David Von Derau, discovered a fifteenth with help from a

computer algorithm.[4-6] This follows discovery of the fourteenth by thirty years.

The

German mathematician,

Karl Reinhardt, discovered the first five classes of pentagonal tilings in 1918.[6] R. B. Kershner found three more in 1968, Richard James found an additional one in 1975, and

Marjorie Rice, an

amateur mathematician, discovered four further types.[6] Rolf Stein found the fourteenth in 1985.[6] Casey Mann, part of the team that discovered the new pentagonal tile that covers the plane, is quoted on NPR as saying,

"We discovered the tile using using a computer to exhaustively search through a large but finite set of possibilities... We were of course very excited and a bit surprised to find the new type of pentagon."[6]

Mann likens this discovery with finding a new

elementary particle, and it could have practical application in

biochemistry and

chemistry, where

molecules are constrained by

geometry to form in just certain shapes, and in structural design.[4] Von Derau was already a

professional software developer when he arrived at the university to complete his

undergraduate degree, and he was recruited by the two other

authors, both

professors, to assist in their tiling

research.[4]

Von Derau coded an

algorithm the others had developed, and he ran it on a

cluster of computers. Eventually, the pentagonal tile shown in the figure popped out of the program.[5] This tile is characterized by the

angles shown, and the following side dimensions:

**a=c=e**

**b=2a**

**d = 2a/((√2)((√3)-1))**

Von Derau's

computer program is being ported to some

high performance computers, and the search for additional tilings continues.[4]

### References:

- Johannes Kepler, "Harmonices Mundi," 1619, from the Carnegie Mellon University Posner Collection.
- Craig Kaplan, "The trouble with five," Plus Magazine, December 1, 2007.
- Roger Penrose, "Set of tiles for covering a surface," US Patent No. 4,133,152, January 9, 1979 (via Google Patents).
- Discovery rocks the math world, University of Washington, Bothell, Press Release, August 14, 2015.
- Eyder Peralta, "With Discovery, 3 Scientists Chip Away At An Unsolvable Math Problem," NPR, August 14, 2015.
- Alex Bellos, "Attack on the pentagon results in discovery of new mathematical tile," The Guardian (UK), August 25, 2015.