Dual polyhedra to uniform polyhedra are face-transitive (isohedral) and have regular vertex figures, and are generally classified in parallel with their dual (uniform) polyhedron. The dual of a regular polyhedron is regular, while the dual of an Archimedean solid is a Catalan solid.
The concept of uniform polyhedron is a special case of the concept of uniform polytope, which also applies to shapes in higher-dimensional (or lower-dimensional) space.
Definition
The Original Sin in the theory of polyhedra goes back to Euclid, and through Kepler, Poinsot, Cauchy and many others continues to afflict all the work on this topic (including that of the present author). It arises from the fact that the traditional usage of the term "regular polyhedra" was, and is, contrary to syntax and to logic: the words seem to imply that we are dealing, among the objects we call "polyhedra", with those special ones that deserve to be called "regular". But at each stage— Euclid, Kepler, Poinsot, Hess, Brückner,... —the writers failed to define what are the "polyhedra" among which they are finding the "regular" ones.
Coxeter, Longuet-Higgins & Miller (1954) define uniform polyhedra to be vertex-transitive polyhedra with regular faces. They define a polyhedron to be a finite set of polygons such that each side of a polygon is a side of just one other polygon, such that no non-empty proper subset of the polygons has the same property. By a polygon they implicitly mean a polygon in 3-dimensional Euclidean space; these are allowed to be non-convex and intersecting each other.[2]
There are some generalizations of the concept of a uniform polyhedron. If the connectedness assumption is dropped, then we get uniform compounds, which can be split as a union of polyhedra, such as the compound of 5 cubes. If we drop the condition that the realization of the polyhedron is non-degenerate, then we get the so-called degenerate uniform polyhedra. These require a more general definition of polyhedra. Grünbaum (1994) gave a rather complicated definition of a polyhedron, while
McMullen & Schulte (2002) gave a simpler and more general definition of a polyhedron: in their terminology, a polyhedron is a 2-dimensional abstract polytope with a non-degenerate 3-dimensional realization. Here an abstract polytope is a poset of its "faces" satisfying various condition, a realization is a function from its vertices to some space, and the realization is called non-degenerate if any two distinct faces of the abstract polytope have distinct realizations.
Some of the ways they can be degenerate are as follows:
Hidden faces. Some polyhedra have faces that are hidden, in the sense that no points of their interior can be seen from the outside. These are usually not counted as uniform polyhedra.
Degenerate compounds. Some polyhedra have multiple edges and their faces are the faces of two or more polyhedra, though these are not compounds in the previous sense since the polyhedra share edges.
Double covers. Some non-orientable polyhedra have double covers satisfying the definition of a uniform polyhedron. There double covers have doubled faces, edges and vertices. They are usually not counted as uniform polyhedra.
Double faces. There are several polyhedra with doubled faces produced by Wythoff's construction. Most authors do not allow doubled faces and remove them as part of the construction.
Double edges. Skilling's figure has the property that it has double edges (as in the degenerate uniform polyhedra) but its faces cannot be written as a union of two uniform polyhedra.
Archimedes (287 BC – 212 BC) discovered all of the 13 Archimedean solids. His original book on the subject was lost, but Pappus of Alexandria (c. 290 – c. 350 AD) mentioned Archimedes listed 13 polyhedra.
Piero della Francesca (1415 – 1492) rediscovered the five truncations of the Platonic solids—truncated tetrahedron, truncated octahedron, truncated cube, truncated dodecahedron, and truncated icosahedron—and included illustrations and calculations of their metric properties in his book De quinque corporibus regularibus. He also discussed the cuboctahedron in a different book.[4]
Of the remaining 53, Edmund Hess (1878) discovered 2, Albert Badoureau (1881) discovered 36 more, and Pitsch (1881) independently discovered 18, of which 3 had not previously been discovered. Together these gave 41 polyhedra.
Skilling (1975) independently proved the completeness and showed that if the definition of uniform polyhedron is relaxed to allow edges to coincide then there is just one extra possibility (the great disnub dirhombidodecahedron).
In 1987, Edmond Bonan drew all the uniform polyhedra and their duals in 3D with a Turbo Pascal program called Polyca. Most of them were shown during the International Stereoscopic Union Congress held in 1993, at the Congress Theatre, Eastbourne, England; and again in 2005 at the Kursaal of Besançon, France.[5]
In 1993, Zvi Har'El (1949–2008)[6] produced a complete kaleidoscopic construction of the uniform polyhedra and duals with a computer program called Kaleido and summarized it in a paper Uniform Solution for Uniform Polyhedra, counting figures 1-80.[7]
Also in 1993, R. Mäder ported this Kaleido solution to Mathematica with a slightly different indexing system.[8]
In 2002 Peter W. Messer discovered a minimal set of closed-form expressions for determining the main combinatorial and metrical quantities of any uniform polyhedron (and its dual) given only its Wythoff symbol.[9]
The convex uniform polyhedra can be named by Wythoff construction operations on the regular form.
In more detail the convex uniform polyhedron are given below by their Wythoff construction within each symmetry group.
Within the Wythoff construction, there are repetitions created by lower symmetry forms. The cube is a regular polyhedron, and a square prism. The octahedron is a regular polyhedron, and a triangular antiprism. The octahedron is also a rectified tetrahedron. Many polyhedra are repeated from different construction sources, and are colored differently.
The Wythoff construction applies equally to uniform polyhedra and uniform tilings on the surface of a sphere, so images of both are given. The spherical tilings include the set of hosohedra and dihedra which are degenerate polyhedra.
These symmetry groups are formed from the reflectional point groups in three dimensions, each represented by a fundamental triangle (pqr), where p > 1, q > 1, r > 1 and 1/p + 1/q + 1/r < 1.
The remaining nonreflective forms are constructed by alternation operations applied to the polyhedra with an even number of sides.
Along with the prisms and their dihedral symmetry, the spherical Wythoff construction process adds two regular classes which become degenerate as polyhedra : the dihedra and the hosohedra, the first having only two faces, and the second only two vertices. The truncation of the regular hosohedra creates the prisms.
Below the convex uniform polyhedra are indexed 1–18 for the nonprismatic forms as they are presented in the tables by symmetry form.
For the infinite set of prismatic forms, they are indexed in four families:
(The sphere is not cut, only the tiling is cut.) (On a sphere, an edge is the arc of the great circle, the shortest way, between its two vertices. Hence, a digon whose vertices are not polar-opposite is flat: it looks like an edge.)
The tetrahedral symmetry of the sphere generates 5 uniform polyhedra, and a 6th form by a snub operation.
The tetrahedral symmetry is represented by a fundamental triangle with one vertex with two mirrors, and two vertices with three mirrors, represented by the symbol (3 3 2). It can also be represented by the Coxeter group A2 or [3,3], as well as a Coxeter diagram: .
There are 24 triangles, visible in the faces of the tetrakis hexahedron, and in the alternately colored triangles on a sphere:
The octahedral symmetry of the sphere generates 7 uniform polyhedra, and a 7 more by alternation. Six of these forms are repeated from the tetrahedral symmetry table above.
The octahedral symmetry is represented by a fundamental triangle (4 3 2) counting the mirrors at each vertex. It can also be represented by the Coxeter group B2 or [4,3], as well as a Coxeter diagram: .
There are 48 triangles, visible in the faces of the disdyakis dodecahedron, and in the alternately colored triangles on a sphere:
The icosahedral symmetry of the sphere generates 7 uniform polyhedra, and a 1 more by alternation. Only one is repeated from the tetrahedral and octahedral symmetry table above.
The icosahedral symmetry is represented by a fundamental triangle (5 3 2) counting the mirrors at each vertex. It can also be represented by the Coxeter group G2 or [5,3], as well as a Coxeter diagram: .
There are 120 triangles, visible in the faces of the disdyakis triacontahedron, and in the alternately colored triangles on a sphere:
The dihedral symmetry of the sphere generates two infinite sets of uniform polyhedra, prisms and antiprisms, and two more infinite set of degenerate polyhedra, the hosohedra and dihedra which exist as tilings on the sphere.
The dihedral symmetry is represented by a fundamental triangle (p 2 2) counting the mirrors at each vertex. It can also be represented by the Coxeter group I2(p) or [n,2], as well as a prismatic Coxeter diagram: .
Below are the first five dihedral symmetries: D2 ... D6. The dihedral symmetry Dp has order 4n, represented the faces of a bipyramid, and on the sphere as an equator line on the longitude, and n equally-spaced lines of longitude.
(2 2 2) Dihedral symmetry
There are 8 fundamental triangles, visible in the faces of the square bipyramid (Octahedron) and alternately colored triangles on a sphere:
The edges are fully truncated into single points. The polyhedron now has the combined faces of the parent and dual. Polyhedra are named by the number of sides of the two regular forms: {p,q} and {q,p}, like cuboctahedron for r{4,3} between a cube and octahedron.
The birectified (dual) is a further truncation so that the original faces are reduced to points. New faces are formed under each parent vertex. The number of edges is unchanged and are rotated 90 degrees. A birectification can be seen as the dual.
Each original vertex is cut off, with a new face filling the gap. Truncation has a degree of freedom, which has one solution that creates a uniform truncated polyhedron. The polyhedron has its original faces doubled in sides, and contains the faces of the dual.
Bitruncated (2t) (also truncated dual)
2t{p,q} t1,2{p,q}
A bitruncation can be seen as the truncation of the dual. A bitruncated cube is a truncated octahedron.
In addition to vertex truncation, each original edge is beveled with new rectangular faces appearing in their place. A uniform cantellation is halfway between both the parent and dual forms. A cantellated polyhedron is named as a rhombi-r{p,q}, like rhombicuboctahedron for rr{4,3}.
The truncation and cantellation operations are applied together to create an omnitruncated form which has the parent's faces doubled in sides, the dual's faces doubled in sides, and squares where the original edges existed.
The alternated cantitruncated. All the original faces end up with half as many sides, and the squares degenerate into edges. Since the omnitruncated forms have 3 faces/vertex, new triangles are formed. Usually these alternated faceting forms are slightly deformed thereafter in order to end again as uniform polyhedra. The possibility of the latter variation depends on the degree of freedom.
Snub (s)
s{p,2q}
Alternated truncation
Cantic snub (s2)
s2{p,2q}
Alternated cantellation (hrr)
hrr{2p,2q}
Only possible in uniform tilings (infinite polyhedra), alternation of For example,
Grünbaum, B. (1994), "Polyhedra with Hollow Faces", in Tibor Bisztriczky; Peter McMullen; Rolf Schneider; et al. (eds.), Proceedings of the NATO Advanced Study Institute on Polytopes: Abstract, Convex and Computational, Springer, pp. 43–70, doi:10.1007/978-94-011-0924-6_3, ISBN978-94-010-4398-4
McMullen, Peter; Schulte, Egon (2002), Abstract Regular Polytopes, Cambridge University Press
Sopov, S. P. (1970). "A proof of the completeness on the list of elementary homogeneous polyhedra". Ukrainskiui Geometricheskiui Sbornik (8): 139–156. MR0326550.
