Uniform polyhedron

A uniform polyhedron is a polyhedron with regular polygons as faces and identical vertices. Furthermore, for every two vertices there is an isometry mapping one into the other, so there is a high degree of reflectional and rotational symmetry. Uniform polyhedra are regular or semi-regular but the faces and vertices need not be convex.

Exclusive of the infinite sets of uniform prisms and antiprisms, there are 75 uniform polyhedra.

The 75 nonprismatic uniform polyhedra include:

• 5 Platonic solid regular convex polyhedra
• 4 Kepler-Poinsot solid regular non-convex polyhedra
• 13 Archimedean solid semiregular convex polyhedra
• 53 Remaining non-convex polyhedra are listed here:
  • 14 quasi-regular non-convex polyhedra
  • 17 stellated Archimedean solids
  • 22 other semi-regular non-convex solids

Non-convex quasi-regular polyhedra

Quasi-regular means vertex- and edge-uniform but not face-uniform, and every face is a regular polygon; this implies that there are two "kinds" of faces, and that at every edge one of each meet; also that at every vertex four faces meet: alternating two of each kind.

The quasi-regular polyhedra include the two convex polyhedra

• Cuboctahedron
• Icosidodecahedron

and 14 non-convex polyhedra (Hart):

• 2 with a rhombic vertex figure (convex vertices with 4 adjacent faces):
• Dodecadodecahedron
• Great icosidodecahedron
• 9 Hemihedra: ditrigonal vertex figures, (non-convex verticies with 4 adjacent faces):
  • Tetrahemihexahedron
  • Octahemioctahedron
  • Cubohemioctahedron
  • Small icosihemidodecahedron
  • Small dodecahemidodecahedron
  • Great dodecahemiicosahedron
  • Small dodecahemiicosahedron
  • Great dodecahemidodecahedron
  • Great icosihemidodecahedron
• 3 triambic polyhedra (each vertex has 6 adjacent faces):
  • Small triambic icosidodecahedron (small ditrogonal icosidodecahedron)
  • Triambic dodecadodecahedron (ditrogonal dodecahedron)
  • Great triambic icosidodecahedron (great ditrogonal icosidodecahedron)

The Small dodecicosahedron has a ditrogonal vertex figure but is not edge uniform.

Non-convex semi-regular polyhedra

Main article Semiregular polyhedra.

The remaining uniform polyhedra are all semi-regular non-convex polyhedra and include the 17 nonconvex Archimedean solids:

• 3 have cubic symmetry:
  • Great cubicuboctahedron
  • Cuboctatruncated cuboctahedron or cubitruncated cuboctahedron
  • Quasitruncated hexahedron or stellated truncated cube
• 11 have icosahedral symmetry:
  • Quasitruncated small stellated dodecahedron or small stellated truncated dodecahedron
  • Quasitruncated great stellated dodecahedron or great stellated truncated dodecahedron
  • Great truncated dodecahedron
  • Great truncated icosahedron
  • Rhombidodecadodecahedron
  • Icosidodecatruncated icosidodecahedron **
  • Small ditrigonal icosidodecahedron **
  • Great ditrigonal icosidodecahedron **
  • Great quasitruncated icosidodecahedron or great truncated icosidodecahedron
  • Great dodecicosidodecahedron
  • Great ditrigonal dodecicosidodecahedron **
• 3 have the symmetry of the snub dodecahedron:
  • Snub dodecadodecahedron
  • Great inverted snub icosidodecahedron or great vertisnub icosidodecahedron
  • Great inverted retrosnub icosidodecahedron or great retrosnub icosidodecahedron

There are 23 more semi-regular non-convex polyhedra:

• ? have octahedral symmetry:
  • Small cubicuboctahedron
  • Small rhombicube
  • Great rhombicube
  • Great truncated cuboctahedron
• ? have icosahedral symmetry:
  • Rhombicosahedron
  • Small rhombidodecahedron
  • Great rhombidodecahedron
  • Small dodecicosahedron
  • Great dodecicosahedron
  • Small dodecicosidodecahedron
  • Icositruncated dodecadodecahedron
  • Great truncated dodecadodecahedron
  • Vertisnub dodecadodecahedron
  • Icosidodecadodecahedron
  • Snub icosidodecadodecahedron
  • Small dodekic icosidodecahedron ??
  • Great dodekic icosidodecahedron ??
  • Small icosic icosidodecahedron ??
  • Great icosic icosidodecahedron ??
  • Great snub icosidodecahedron
  • Snub disicosidodecahedron
  • Retrosnub disicosidodecahedron
  • Great snub icosidisdodecahedron
  • Great snub disicosidisdodecahedron

TODO: Check this list for duplicates/alternate names

Given two polyhedra of equal volume, one may ask whether it is then always possible to cut the first into polyhedral pieces which can be reassembled to yield the second polyhedron. This is a version of Hilbert's third problem; the answer is "no", as was shown by Dehn in 1900.

History

The Platonic solids date back to the classical Greeks and were studied by Plato, Theaetetus and Euclid. Johannes Kepler (1571-1630) was the first to publish the complete list of Archimedean solids after the original work of Archimedes was lost.

Kepler (1619) discovered two of the regular Kepler-Poinsot solids and Louis Poinsot (1809) discovered the other two.

Of the remaining 37 were discoved by Badoureau (1881). Hess (1878) discovered 2 more and Pitsch (1881) indepentantly discovered 18, not all previously discovered.

The famous group theorist Donald Coxeter discovered the remaining twelve in colaboration with Miller (1930-1932) but did not publish. M.S. and H.C. Longuet-Higgins and independently discovered 11 of these.

In a seminal paper:

H.M.S. Coxeter, M.S. Longuet-Higgins, J.C.P Millar, Uniform polyhedra, Phil. Trans. 1954, 246 A, 401-50.

the full list of uniform polyhedra was published, and it was conjectured that the list was complete. J. Skilling later confirmed this result.

Mathematics

Euclid was the first to show that for a convex polyhedron the vertex angles of the polygons at each vertex must add up to less than 360°. For example the angles at each vertex of a cube are 90°+90°+90°=270°<360°.

The angle defect at each vertex 360° less the angles of the adjacent polygons, for a cube this is 90°. Descartes proved that for convex polyhedron the total angular deficit for all the vertices is 720°. For a cube this is 8 × 90° = 720°.

Euler's theorem shows that for convex polyhedron V-E+F=2.

See also

• Polyhedron
  • Semiregular polyhedra
• List of uniform polyhedra
• List of Wenninger polyhedron models
• Polyhedron model
• List of uniform polyhedra by vertex figure

External links

• Stella: Polyhedron Navigator - Software for generating and printing nets for all uniform polyhedra
• Paper models
• Uniform Solution for Uniform Polyhedra
• The Uniform Polyhedra
• Virtual Polyhedra Uniform Polyhedra
• Eric W. Weisstein. "Uniform Polyhedron." From MathWorld--A Wolfram Web Resource.