 # Tilings

On these pages I will show some of my research into tiling patterns.

## Tiling Viewer Applet

I have written a Tiling Viewer that allows you to see most of the tilings that I found in my research. I used it to generate all the pictures on this page. Click the link for more information and to download it.

## Terminology

Tiling or tessellation
A dissection of the infinite flat plane into shapes of a finite area.
Tile
One of the shapes that forms a tiling.
Isometry
A distance-preserving mapping of the plane. There are four types: translations, rotations, reflections, and glide reflections.
Symmetry of a tiling
An isometry that maps the tile boundaries onto tile boundaries. In other words this is some transformation that leaves the tiling looking the same as before.
Periodic tiling
A tiling that has two independent translation symmetries, i.e. a tiling that repeats itself along two different axes like a wallpaper pattern.
Primitive unit or Unit Parallelogram
A section of the tiling (usually a parallelogram or a set of neighbouring tiles) that generates the whole tiling using only translations, and which is as small as possible.
Fundamental unit
A section of the tiling (usually a set of neighbouring tiles) that generates the whole tiling using the tiling symmetries (not just the translations), and which is as small as possible.
Monohedral tiling
A tiling where all the tiles are congruent to each other, i.e. all have the same size and shape (though they are allowed to be mirror images).
Isohedral tiling
A monohedral tiling where for any two tiles there is a symmetry of the tiling that maps one tile to the other.
k-Isohedral tiling (k is a positive integer)
A monohedral tiling where the tiles form k classes such that for any two in the same class there is a symmetry of the tiling that maps one tile to the other, and for any pair of tiles in different classes no such symmetry exists. Note that 1-isohedral is the same as isohedral. In the applet, each class of tile has its own colour.
Edge-to-edge tiling
A tiling of polygons such that no corner of one tile touches the side of another.

## Tilings with a convex pentagonal tile

It is well known that any triangle can tile the plane, as can any quadrangle, convex or not.

The situation with pentagons is much more complicated. A regular pentagon does not tile the plane, but various non-regular convex pentagons do. In 1975 Martin Gardner wrote an article in Scientific American, reporting the results of Richard B. Kershner about which types of convex pentagons can tile the plane [MG]. Richard had attempted to enumerate them, and thought that his list of 8 types was complete until in a reaction to Gardner's article Richard E. James III wrote in with another tilable convex pentagon (now known as type 10). Marjorie Rice [MR] found four others which she also sent to Gardner (types 9, 11-13), bringing the total to 13. A further type was found by Rolf Stein in 1985 (type 14).

That is how the situation stayed for a long time. The uncertainty about whether the list of 14 pentagon types is complete inspired me to do my own research into tilings. In 2010 I wrote a program that generates all possible tilings that use k copies of a single polygon tile as its fundamental unit. With this program I tried to find new types of tilable convex pentagons, but that quest was not successful. It nevertheless generated many fascinating tilings, some that I think have not been seen before, most of which I have incorporated into the Tiling Viewer applet. My programs were too slow and generated too many duplicates and false solutions to make much headway when k is 3 or more.

In 2015 a 15th convex pentagon type was found by Casey Mann, Jennifer McLoud-Mann and David Von Derau [MMD]. It turned out that they had used approximately the same methods as I had, except more efficiently and with more proofs. In May 2017, Michaël Rao [MR2] announced a proof that the list of 15 types is complete. The proof relies on a computer program to first enumerate a finite number of cases which cover all possible tilings, and then eliminate any that are not one of the 15 known types. The program's proof has now been verified by Thomas Hales so it is likely to be correct.

## Convex Tiles

### Isohedral tilings

Isohedral tilings with convex polygons are well-studied and can be found in [G+S] and [TM]. There are:
14 isohedral tilings with a triangle tile
56 isohedral tilings with a quadrangle tile
24 isohedral tilings with a convex pentagon tile
13 isohedral tilings with a convex hexagon tile

Convex n-gons with n>6 cannot tile the plane. This is because for n>6, the average of the tile's angles is more than 120 degrees, but if at every vertex in the tiling at least three or more tiles meet then the average angle must be 120 degrees or less. Therefore there are vertices in the tiling where only two tiles meet, and either one of the angles is more than 180, or they are both equal to 180. So either the tile is not convex, or it has redundant vertices. This is also fairly easy to prove by using Euler's characteristic.

All the isohedral tilings with a convex tile can be found in the Tiling Viewer applet. Some of them are special cases of others, where the settings are such that the tiling has extra symmetries, and those have not all been implemented separately.

### 2-Isohedral tilings

Most 2-isohedral tilings have a fundamental unit consisting of two tiles, one from each isohedrality class. In the Tiling Viewer applet the tiling will then have an equal amount of the two colours. There are also some rare cases of 2-isohedral tilings where the two types of tile occur in a 2:1 ratio. In these cases the fundamental unit consists of 3 tiles, and has an extra mirror symmetry that maps two of the tiles to each other. My programs only find these when searching for 3-isohedral tilings, sometimes as a special case of a more general 3-isohedral tiling.

My search program has exhaustively generated all of the tilings for which the fundamental unit consists of two identical convex polygons, and assuming no mistakes or oversights, the Tiling Viewer applet contains a complete set of these types of tiling.

#### Convex Pentagons

No new types of tilable convex pentagon were found. Most of the tilings used type 1 pentagons, seven of type 2, one of type 4, and there is one alternative way to tile the type 6 pentagon tile.

### 3-Isohedral tilings

I have not yet searched for all 3-isohedral tilings, as that will take a lot of time. I have therefore limited myself to the edge-to-edge 3-isohedral tilings for triangles, quadrangles and pentagons, all of which have been implemented in the Tiling Viewer applet. Nevertheless, some tilings that are not edge-to-edge have been implemented too.

### k-Isohedral tilings, k>3

Searching for k-isohedral tilings with k>3 is probably infeasible with my current search program. I did start it searching for edge-to-edge 4-isohedral convex pentagon tilings, and it found the following neat one using a type 8 pentagon tile.

## Non-convex Tiles

I have made a separate page for the tilings I've found that have non-convex tiles.

Here are the two articles I wrote as exchange gifts for the Gathering for Gardner conference.