This post is a brief introduction to the mathematics of tiling, which I find quite fascinating.

Tiles are a long-standing architectural feature. They have been used in cultures all over the world, but the math underlying how they can cover a space with no gaps is universal. To convey an understanding of this math, I'll start with some terminology and types/classes of tilings, and then some useful notation. Next I'll include some examples. I'll finish this post with some interesting links and further reading.

##### Terminology and Types of Tilings:
• Prototile = a basic shape that is repeated throughout a tiling pattern; there can be one or more prototiles.
• Vertex = the point where tiles meet.
• Regular polygon = a 2-dimensional shape where all the sides are the same length and all of the interior angles are equal.
• Monohedral tiling = a tiling pattern that has only one prototile—every tile is the same shape.
• Regular tiling = a monohedral tiling pattern in which the prototile is a regular polygon; this is only possible for equilateral triangles, squares, and regular hexagons.
• Semi-regular tiling = a tiling pattern that has more than one type of prototile, but all prototiles are regular polygons, and they form the same arrangement at every vertex; all of the prototiles must have the same side length since they meet at the same vertices.
• Periodic tilings = a tiling pattern is periodic if it repeats in such a way that it can be shifted linearly (i.e. translated) and map onto itself; regular and semi-regular tilings are both periodic.
• Aperiodic tilings = tiling patterns that do not repeat themselves with any linear translation (although they may have other types of symmetry).

Note that all regular tilings are monohedral but not all monohedral tilings are regular. Pentagons and enneagons (9-sided shapes) can form interesting tiling patterns, but not in their regular (each side having an equal length) forms. I'm not going to get into k-uniform tilings, which are periodic and have regular polygons as their prototiles, but have more than one vertex configuration.

##### Notation

A type of notation that is useful for describing regular tilings is the Schläfli symbol. It is stated as {# edges per polygon, # polygons meeting at a vertex}. In this notation, the three regular tilings are:

• {3,6} for triangular tiling—each tile has 3 sides and 6 tiles meet at each vertex.
• {4,4} for square tiling—each tile has 4 sides and 4 tiles meet at each vertex.
• {6,3} for hexagonal tiling—each tile has 6 sides and 3 tiles meet at each vertex.

Schläfli notation can also be used for polyhedrons.

For semi-regular tiling, vertex configuration notation is convenient. It comprises a dotted sequence of numbers, where each number represents the number of sides of a regular polygon prototile and the sequence represents their order around each vertex. There are eight possibilities:

4.8.8,   3.4.6.4,   6.3.3.3.3,   3.3.4.3.4,   3.6.3.6,   3.12.12,   4.6.12,   3.3.3.4.4.

The interior angle of a n-sided regular polygon = 180° – (360°/n)

At each vertex, the interior angles must add up to 360°. Using the stated equation, here are the interior angles for relevant regular polygons:

• 3-sided = 60°
• 4-sided = 90°
• 6-sided = 120°
• 8-sided = 135°
• 12-sided = 150°

A quick calculation will show that each of the eight vertex configurations stated adds up to 360°.

The regular tilings can also be represented with vertex configuration notation: 3.3.3.3.3.3 (triangular), 4.4.4.4 (square), and 6.6.6 (hexagonal), although Schläfli notation is more compact.

##### Examples, Links, and Related Topics

As mentioned above, there are interesting monohedral tilings of 5-sided and 9-sided polygons (but irregular). A certain enneagon can be used to make a spiral tiling. Pentagonal tiling is an area of active research. There are currently 15 different irregular pentagons that are known to work in monohedral tilings, with the most recent only discovered this year.

Penrose tiles make nice aperiodic designs. I've mentioned them on this blog before.

Middle Eastern cultures are known for their artistic tilings, like those I saw at the Al-Hambra in Granada, Spain. A neat application is the Persian/Turkish use of "Girih" tiles to produce strapwork patterns; similar patterns can be constructed with a compass and straight-edge.

Closely-related mathematical concepts to tiling are wallpaper groups and circle packing.

The Conway criterion can be used to evaluate potential (monohedral, periodic) prototiles.

If you wanted to get really advanced, you could look at tilings of non-Euclidean (curved) space...

To conclude this post, here is a description of a fictional tiling problem from Anathem, by Neal Stephenson (pp. 551–553):

The plaza where we were enjoying our little picnic—the one where Metekoranes had been buried and cast in stone—was the Teglon made real. It was flat, decagonal, maybe two hundred feet across, paved in smooth slabs of marble. In ancient times the plaza had been plentifully supplied with tiles made of clay baked in molds. There were seven molds, hence seven different shapes of tile. Their shapes were such that it was possible to fit them together in an infinite number of patterns. That's not possible with squares, or equilateral triangles; those fit together in repeating patterns, so there are no choices to make. But as long as you had more copies of the Teglon tiles, you could go on making choices forever. Hundreds of tiles were scattered around the place even now, and from place to place the modern-day Orithenans had been putting them together in little arrangements. I squatted down and looked at one, then looked questioningly at Spry. "Go ahead," she said, "it's a modern reproduction. We found the original molds!"

I picked up a tile for a closer look. This one happened to be four-sided: a rhombus. A groove was molded into its surface, curving from one of its sides to another. I carried it over to the nearest vertex of the Decagon and set it down; its obtuse angle fit perfectly into the corner.

"Ah," Suur Spry teased me, "going straight to the most difficult problem of all, huh?"

She was talking, of course, of the Teglon. She turned away and walked to the opposite vertex and set a tile down there. Meanwhile I scavenged a few other tiles, getting samples of all seven shapes. I chose one at random and set it next to the first. This one also had a groove curving from one side to another—all of the tiles were so made—and I rotated it until its groove mated with, and became a continuation of, the one on the first tile. Into the angle between them, I was able to place a third. That created opportunities to slide in a fourth, a fifth, and so on. I was playing the Teglon. The objective of the game was to build the pattern outward from one vertex and pave the entire Decagon in such a way that the groove formed a continuous, unbroken curve from the first vertex to the last—the one directly opposite, where Suur Spry had put down a tile. Along the way, the curve had to pass across every tile in the entire Decagon. For the first little while, it was easy—it came naturally.

But beyond a certain point, the two objectives—that of tiling the whole surface, and that of keeping the curve going—began to conflict. I had to leave a stretch of groove hanging for a while, then work my way back to it, steering the groove around to make the connection. That was satisfying. But a few minutes later, I found myself with three such segments of marooned groove in different parts of the pattern, and despaired of ever finding an arrangement that would connect them all. On one level, this was all about the shape of the outer boundary, and how it developed. Tiles trapped in the middle were of no further interest to the game—or so you might think. But on the other hand, the way in which an interior tile had been layed down ended up determining the location of every other tile in the whole Decagon.
The Ancient Orithenans suspected, but didn't know how to prove, that the tiles of the Teglon were aperiodic: that no pattern would ever repeat. Again, solving the Teglon would have been easy—it would have been automatic—with square or triangular tiles, or any tile system that was periodic. With aperiodic tiles, it was impossible, or at least very unlikely, unless you had some God-like ability to see the whole pattern in your head at once. Metekoranes had believed that the final pattern existed in the Hylaean Theoric World, and that the Teglon could only be solved by one who had developed the power of seeing into it.