OEIS/Tilings

New format

On May 02 2020 Brian Galbach wrote:

Here's a new file with a very compact and simple code for each tiling. It has several sections separated by semicolons, which I will explain below.

Each code starts with a single capital letter representing the uniformity of the tiling. A is 1-uniform, B is 2-uniform, etc.

Next, we have a letter representing the vertex type for each vertex class in the tiling. These are as follows:

A: 3.3.3.3.3.3
B: 3.3.3.3.6
C: 3.3.3.4.4
D: 3.3.4.3.4
E: 3.3.4.12
F: 3.3.6.6
G: 3.4.3.12
H: 3.4.4.6
I: 3.4.6.4
J: 3.6.3.6
K: 3.12.12
L: 4.4.4.4
M: 4.6.12
N: 4.8.8
O: 6.6.6

Following the vertex type designations, we have alternating sections:

The first section describes the classes connected to class A. The number of connections must of course be the same as the number of polygons in class A. And each connected class must be from A to the nth letter of the alphabet, where our tiling is n-uniform. (Important: Do not confuse the vertex type, which is A to O, with the vertex class, which is A, B, or C in a 3-uniform tiling, for example.)

The second section describes the orientation of each of those vertices connected to class A. The letter, whether it is uppercase or lowercase, gives us the edge of the connected vertex that is connected to the current edge. A or a is the first edge, B or b is the second edge, etc. And uppercase means that the direction of rotation of the connected vertex is the same as the current vertex, while lowercase means that the direction of rotation is reversed.

Then we repeat the above two sections for class B, class C, etc.

That's it. Let's do an example just so this is clear. I'll pick a line pretty much at random. Okay, this one:

C;FJO;BABC;BBAA;AABB;CADC;ACC;DCB

So we have a 3-uniform tiling (C). The three vertex classes have the vertex types 3.3.6.6, 3.6.3.6, and 6.6.6 (FJO).

Now let's look at vertex class A. It has connected to it vertex classes B, A, B, and C. And the edges connected from each of those vertices are edge 2, edge 2, edge 1, and edge 1. And since these are all uppercase letters, there is no reversing of the direction of rotation relative to the direction of our central vertex.

Class B has connected to it vertex classes A, A, B, and B. And the connected edges are edge 3, edge 1, edge 4, and edge 3, and no direction reversal.

Finally, class C has connected to it vertex classes A, C, and C. And the connected edges from each of those vertices are edge 4, edge 3, and edge 2.