What are Valid Goldberg Polyhedra Frequencies?

The list of octahedral Goldberg polyhedra $GP_{IV}(n,m)$ looks like the following:

$$ (1,0)\;,(2,0)\;,(3,0)\;,(4,0)\;,(5,0)\;,(7,0)\;,(8,0)\;(9,0)\;,,...\\ (1,1)\;,(2,1)\;,(2,2)\;,(3,3)\;,(4,4)\;... $$

Is there a (simple) way to see, which pairs $(n,m)$ are valid frequencies?

I tried to get them via transformations like they are described in the Construction site, e.g. "a whirl can transform a GP(a,b) into GP(a+3b,2a-b) for a>b", but the number of operations applicable to a polyhedra is huge and I don't expect them to leave the transformed polyhedra in the octahedral subset.

Knowing that the initial $GP^0$ and the transformed polyhedra $GP^1$ represent bicubic planar graphs some transformations like

• Expansion are obviously ruled out. Further since Euler Formulae shall apply to both we get the following contradiction for expansion: Initially we have $3v=2e$. Then by expansion $v\mapsto 2e $ and $ e \mapsto 4e$. therefore we get $$ 3\cdot 2e\neq 2\cdot 4e $$

• Chamfering would be valid since $v\mapsto v+2e$ and $e\mapsto 4e$ leads to $$ 3(v+2e)=2\cdot4e\\ 3v+6e=8e\\ 3v=2e $$

Generalizing this approach to linear transformations we get: $$ v\mapsto\mathfrak V(v,e,f)=Av+Be+Cf\\ e\mapsto\mathfrak E(v,e,f)=Dv+Ee+Ff\\ f\mapsto\mathfrak F(v,e,f)=Gv+He+Jf\\ $$

• $3$-regularity constraints: $$ 3(Av+Be+Cf)=2(Dv+Ee+Ff)\\ \underbrace{(3A-2D)}_3 v+\underbrace{(3B-2E)}_{-2}e+\underbrace{(3C-2F)}_0f=0, $$

• constraints for embedding of graph on orientable surface with $g$ holes: $$ (Av+Be+Cf)+(Gv+He+Jf)=(Dv+Ee+Ff)+2-2g\\ \underbrace{(A+G-D)}_1v+\underbrace{(B+H-E)}_{-1}e+\underbrace{(C+J-F)}_1f=2-2g\\ $$

• I'd also like to point out the relation between $(n,m)$ and $v,e$ and $f$, which is the following for octahedral polyhedra: $$ \begin{array}{rcl} v&=&8T\\ e&=&12T\\ f&=&4T+2\\ T&=&m^2+nm+n^2 \end{array}, $$ where $T$ is the triangulation number.

Maybe it would shed some light on first question if there is an answer to the following one:

Which transformations act invariant on the set of bicubic planar graphs and how do their linear representations looks like?

Solution 1:

Any non-negative integers $(n,m)$, not both zero, are valid.

"Geodesic Grids" on Wikibooks goes into some detail about how to construct a geodesic grid (the dual to a Goldberg polyhedron) for any values of $n$ and $m$.

Goldberg's original article, "A Class of Multi-Symmetric Polyhedra", also describes how to construct an icosahedral Goldberg polyhedron for any $n$ and $m$, and at the end notes that it works also for tetrahedral or cubic symmetry.