Uniform polyhedron
A uniform polyhedron or semiregular polyhedron is a finite planar polyhedron that is isogonal (vertex-transitive) and has only regular polygons as faces. (Additionally, as with all uniform polytopes, they are not allowed to be compounds and no elements may be doubled.) The set of uniform polyhedra is known to be complete, and is classified like so:
- the infinite families of prisms and antiprisms, including those based on star polygons
- 75 special cases:
- the 5 convex regular Platonic solids
- the 4 nonconvex regular Kepler–Poinsot solids
- the 13 convex nonregular Archimedean solids
- 53 nonconvex nonregular uniform star polyhedra
| Dodecahedron | Small rhombicuboctahedron | Great dodecahedron | Enneagonal antiprism | Small icosicosidodecahedron |
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| Platonic solid | Archimedean solid | Kepler-Poinsot polyhedron | antiprism | uniform star polyhedron |
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History[edit | edit source]
The platonic solids were known in ancient Greece, although Plato was probably not their discoverer (he may have known about the cuboctahedron). The Kepler–Poinsot polyhedra were known as geometrical designs before Kepler, but Kepler first discovered the stellated dodecahedra as regular polyhedra, and Poinsot rediscovered them and discovered their duals. The Archimedean solids were discovered by Archimedes, although his book on them is lost. The uniform star polyhedra were discovered by faceting by Edmund Hess, Albert Badoureau, Johann Pitsch, H. S. M. Coxeter, and J. C. P. Miller, the latter of whom created the complete list. S. P. Sopov proved the list complete.
Related concepts[edit | edit source]
Allowing compounds leads to the uniform compound polyhedra, which have also been completely enumerated.
Relaxing dyadicity to allow any number of even faces to meet at an edge results in exactly six exotic polyhedroids. Five of them can be seen as compounds, but the sixth, the great disnub dirhombidodecahedron, cannot. It was discovered by J. Skilling in 1975.
The concept of uniform polytopes has been generalized to higher dimensions. The convex uniform polytopes in four dimensions and higher have received some professional study, while the nonconvex uniforms are largely the domain of the enthusiast community.