How can one define vertexes, edges and faces from the Coxeter group? For example, for all platonic solids? I would like to create a general function that takes the Coxeter diagram as input, and gives vertexes, edges and faces as output :-)
I have seen a related question here. However, this discussion is only about how to define the vertexes. I think I now have understood this part fairly well by reading this discussion. But I still don't understand how to define the edges and faces using the same group. I assume it should have something to do with the subgroups. I would like to solve this the elegant way, not the brute force way :-)