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?

 A: 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.
A: With the 3-regularity- and the embedding- constraint, mentioned above, we can narrow down valid transformations, such that $M_x=\pmatrix{A&B&C\\D&E&F\\G&H&J}$ has the following structure:
$$
M_x=\pmatrix{A&B&C\\\frac32(A-1)&\frac32B+1&\frac32C\\\frac12(A-1)&\frac12B&\frac12C+1}
$$
This result is supported

*

*by the identity operation with:
$$
M_c=\pmatrix{1&0&0\\0&1&0\\0&0&1}
$$


*by the chamfer operation with:
$$
M_c=\pmatrix{1&2&0\\0&4&0\\0&1&1}
$$


*and by the whirl operation with:
$$
M_c=\pmatrix{1&4&0\\0&7&0\\0&2&1}
$$
all these leave 3-regularity invariant...
