See [Baa02b] and references there is an intimate relation to number theory, for example using the triangular planar tiling with
γ
=
[
ξ 3
]
the ring of Eisenstein integers what we quickly describe now. Let l2(m) denote the number of sublattices with index m, then the Dirichlet series generating function F2 can be expressed using Riemann's Zeta function:
F 2
s
m
1
∞
l 2
m
m s
ζ
s
ζ
s-1
Equation 15.1. Dirichlet Series Generating Function F2
Another similarity problem leads to Dedekinds Zeta function, cyclotomic fields, principal ideal domains and Galois groups.
Besides Theta series (how many faces are in the shell at distance n for a given type of piece and graph, with Jacobi and friends) and the whole topic of modular forms are candidates for more inspection as well.