Infinite Triangular Tessellation

In a previous post I showed that a finite triangular tessellation cannot have average degree higher than 6. It is difficult to define average degree for an infinite graph, but I found that for any n greater than 5, there is an infinite triangular tessellation where every vertex has degree n. Here is the tessellation for the case when n=7:

The graph is built on an infinite sequence of concentric rings. The first three are shown. There is a vertex at the center, and seven vertices on the first ring. To have a degree of 7, each vertex on the first ring must connect to 4 vertices on the 2nd ring.

The 2nd ring has two types of vertices: those connected with one vertex on the 1st ring, and those connected with two. The latter category only need connect with 3 vertices on the 3rd ring to have degree 7.

The number vertices on each ring is governed by this linear recurrence:

It isn’t hard to derive. Observe that the number of vertices on the n-th ring which are connected to two vertices on the (n-1)-th ring is equal to the number of verticles on the (n-1)-th ring.

The solution to the recurrence is:

Here are the first few values:

1. 7
2. 21
3. 56
4. 147
5. 385
6. 1008
7. 2639
8. 6909
9. 18088

Because the radii of the rings are unspecified, we can choose them so the area of the triangles remain constant. Taking the first ring to have radius one, and approximating the area of the tessellation polygon contained in each ring with the area of the ring itself, we get a recursive formula for the ring radius: