Existence of 2D Quasicrystal with vertex having only even (or odd) coordination number.

Question

The coordination number for a vertex is the number of links connected to it. In the Penrose lattice, it takes range from 3 to 7. However, the mean coordination number is 4, the same as a square lattice.

For Ammann-Beenker quasicrystal, coordination number takes value in range 3-8, and interestingly it's mean is the same as the coordination number of the square and Penrose lattice.

The same story also applies to generalized Rauzy tiling. The coordination number change between 3-5 with a mean equal to 4!

We can see all of them have a continuous distribution of coordination numbers. I am curious to know is there any quasicrystal in 2D with, for example, only even (or odd) coordination number?

Answer 1

I think this can be an answer regarding existence. Yes, there is a systematic way to make a tiling with only odd or even coordination numbers. The strategy uses the center model.

For even coordinated tilings:

1. First find the tiling with parallelogram tiles.
2. Put a vertex inside each tile
3. Connect these vertices if their corresponding tiles have common edges.

For odd coordinated tiling

1. triangulate (connect vertices in a way to all tiles become a triangles without overlap) the tiling by respecting quasiperiodicity (for instance use inflation rules).
2. Put a vertex inside each tile
3. Connect these vertices if their corresponding tiles have common edges.