Look at the possible ways to put a single
regular polygon together with more of the same to make a spatial "cap"
about a single vertex. This involves equilateral triangles (3,4, or 5),
squares (3) or pentagons (3).
This shows that there were at most five vertex caps possible. These actually do work to make
Regular polygons around a vertex.
All vertices are "the same"
Activity: Counting vertices, edges, and faces for the platonic solids to become more familiar with them.