There are many relations among the uniform polyhedra. This List of uniform polyhedra by spherical triangle groups them by the Wythoff symbol.
Key
Image Name Bowers pet name V Number of vertices,E Number of edges,F Number of faces=Face configuration ?=Euler characteristic, group=Symmetry group Wythoff symbol - Vertex figure W - Wenninger number, U - Uniform number, K- Kalido number, C -Coxeter number alternative name second alternative name
The vertex figure can be discovered by considering the Wythoff symbol: p|q r - 2p edges, alternating q-gons and r-gons. Vertex figure (q.r)p. p|q 2 - p edges, q-gons (here r=2 so the r-gons are degenerate lines). 2|q r - 4 edges, alternating q-gons and r-gons q r|p - 4 edges, 2p-gons, q-gons, 2p-gons r-gons, Vertex figure 2p.q.2p.r. q 2|p - 3 edges, 2p-gons, q-gons, 2p-gons, Vertex figure 2p.q.2p. p q r|- 3 edges, 2p-gons, 2q-gons, 2r-gons, vertex figure 2p.2q.2r
Convex Spherical trianglepq rqp rrp qq rpp rqp qrp q rp q rOctahedronCuboctahedronTruncated octahedronIcosahedron
Non-convex
a b 2 3 3 2 Group Spherical trianglepq rqp rrp qq rpp rqp qrp q rp q r 4 3 2 Group Spherical trianglepq rqp rrp qq rpp rqp qrp q rp q roctahedroncube
5 3 2 Group Spherical trianglepq rqp rrp qq rpp rqp qrp q rp q rp q rp q r
5 5 2 Group Spherical trianglepq rqp rrp qq rpp rqp qrp q rp q rp q r
a b 3 3 3 3 Group Spherical trianglepq rqp rrp qq rpp rqp qrp q rp q r 4 3 3 Group Spherical trianglepq rqp rrp qq rpp rqp qrp q rp q r 5 3 3 Group Spherical trianglepq rqp rrp qq rpp rqp qrp q rp q rp q r
4 4 3 Group Spherical trianglepq rqp rrp qq rpp rqp qrp q rp q r
5 5 3 Group Spherical trianglepq rqp rrp qq rpp rqp qrp q rp q r
a b 5 5 5 5 Group Spherical trianglepq rqp rrp qq rpp rqp qrp q rp q r
References