Semiregular polyhedra: Encyclopedia II - Semiregular polyhedra - Existence

Semiregular polyhedra - Existence

Every enumerated vertex configuration potentially uniquely defines a semiregular polyhedron. However not all configurations are possible.

Topological requirements limit existence. Specifically p.q.r implies that a p-gon is surrounded by alternating a q-gons and r-gons, so either p is even or q=r. Similarly q is even or p=r. Therefore potentially possible triples are 3.3.3, 3.4.4, 3.6.6, 3.8.8, 3.10.10, 3.12.12, 4.4.n (for any n>2), 4.6.6, 4.6.8, 4.6.10, 4.6.12, 4.8.8, 5.5.5, 5.6.6, 6.6.6.

In fact, all these configurations with three faces meeting at each vertex turn out to exist (in parentheses the number of vertices, as can be computed from the configuration, see above):

• Platonic solids 3.3.3 (4), 4.4.4 (8), 5.5.5 (20)
• prisms 3.4.4 (6), 4.4.4 (8; also listed above), 4.4.n (2n)
• Archimedean solids 3.6.6 (12), 3.8.8 (24), 3.10.10 (60), 4.6.6 (24), 4.6.8 (48), 4.6.10 (120), 5.6.6 (60).
• regular tessellation 6.6.6
• semiregular tessellations 3.12.12, 4.6.12, 4.8.8

Similarly configurations with four faces meeting at each vertex, p.q.r.s, require that if one number is odd, two of the other three are the same:

• Platonic solid 3.3.3.3 (6)
• antiprisms 3.3.3.3 (6; also listed above), 3.3.3.n (2n)
• Archimedean solids 3.4.3.4 (12), 3.5.3.5 (30), 3.4.4.4 (24), 3.4.5.4 (60)
• regular tessellation 4.4.4.4
• semiregular tessellations 3.6.3.6, 3.4.6.4

Finally configurations with five and six faces meeting at each vertex:

• Platonic solid 3.3.3.3.3 (12)
• Archimedean solids 3.3.3.3.4 (24), 3.3.3.3.5 (60) (both chiral)
• semiregular tessellations 3.3.3.3.6 (chiral), 3.3.3.4.4, 3.3.4.3.4 (note that the two different orders of the same numbers give two different patterns)
• regular tessellation 3.3.3.3.3.3

Adapted from the Wikipedia article "Existence", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki