In a semi-regular polyhedron, why does a polygon with odd sides (e.g. a triangle) have to be surrounded by polygons of the same type (e.g. squares)?

Question

In a semi-regular polyhedron with 4 faces meeting at each vertex, why does a polygon with an odd number of sides (such as a triangle) have to be surrounded by polygons of the same type (e.g. squares)?

(Each vertex of a semi-regular polyhedron has the same arrangement of polygons around it.)

Answer 1

In a semi-regular polyhedron every vertex ought be equivalent.

If you'd attach to say a triangle the polygons A, A, B, then there would be a vertex (3, A, ..., A) and 2 vertices (3, A, ..., B).

If you'd attach polygons A, B, C, then you'll get the 3 vertices (3, A, ..., B), (3, A, ..., C), (3, B, ..., C).

Thence the only possibility indeed would be to have all 3 adjacent polygons the same, as only then all vertices would look like (3, A, ..., A).

--- rk