Building 4d Polytopes

That is a group where the generators have relations of the form: \((ab)^{m_{ab}}\), as above: $$(RG)^4 = (GB)^3 = (RB)^2$$ The generators must also be their own inverse: $$R^2 = B^2 = G^2$$ Since the generators are their own inverses, they can be associated with reflections. Here is a demo (use the ‘step’ button to fill the table):

Let us look at the various cosets you get when you use Todd-Coxeter on the symmetry groups for the regular polytopes, using different subsets of the generators to specify a subgroup:

An interesting structure emerges here: looking at the cosets for the subgroups generated by leaving out one of the generators, the number of cosets matches the number of vertices, edges, or faces! Given the generator relations, we know the angles between the reflector planes: If there is a relation on the form: \((RB)^{N}\), the angle between the reflection planes R and B must be \( \pi / N \) – which again corresponds to a rotation of \( 2\pi / N \) degrees.

Source: syntopia.github.io