The three polyhedral examples (and also for example L27) arise as well in the context of Hurwitz inertia groups (automorphism groups of compact Riemannian surfaces, factor groups, torsion-free Fuchsian groups, Galois coverings, see [Bre00]).
The Riemann surface S of the function field K is the set of nontrivial discrete valuations on K. The set S corresponds to the ideals of the ring A of integers of K over
The theory of functions having values for every face (or directed edge or point) on those lattice - boards (grids) might be investigated. The complex imaginary unit 'i' rotates x into y, and y into -x in standard complex analysis, and this bivector behavior carries over to general polytopes (instead of poly=4 some lcm then), replacing the complex numbers with some algebraic extension of the Rationals
We already have constructed some holonomy groups, and monodromy (extending local to global results) is cool as well. Is there a connection to meromorphic functions (that is every singularity is a pole)? Does that mean every hole (invalid face or bigger areas) can be looked as partitioned into regular faces, so that we can say we made some faces invalid?
Other things like for example Cauchy-Riemann equations, holomorphic functions, Residue Theorem, meromorph functions, Riemannian Existence Theorem, Riemann-Roch-Hirzebruch-Grothendieck Theorem etc. look interesting. And harmonic analysis, integral transformations (e.g. s.th. like Fourier, Laplace, Mellin; the inverse is done using Residue Theorem) and families of polynomials (Lejendre, Gegenbauer etc.), FFT on groups should be treated as well.
Families of functions which are complete sets/orthonormal bases in separable Hilbert spaces and their dual spaces are certainly of interest here as well. And let's investigate the adjoint of an operator, Hermitian (self-adjoint) operators for observables as well as unitarian (where adjoint is identical to the inverse).