Uniform polyhedron

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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.

Excluding the infinite sets there are 75 uniform polyhedra (or 76 if edges are allowed to coincide).

Categories include:

• Infinite set of uniform prisms (including star prisms)
• Infinite set of uniform antiprisms (including star antiprisms)
• 5 Platonic solid regular convex polyhedra
• 4 Kepler-Poinsot solid regular nonconvex polyhedra
• 13 Archimedean solid semiregular convex polyhedra
• 14 nonconvex polyhedra with convex faces
• 39 nonconvex polyhedra with nonconvex faces
• 1 polyhedron found by Skilling with pairs of edges that coincide.

They can also be grouped by their symmetry group, which is done below.

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

In 1954 H.S.M. Coxeter, M.S. Longuet-Higgins, J.C.P. Miller published the list of uniform polyhedra. In 1970 S. P. Sopov proved their conjecture that the list was complete. In 1975, J. Skilling 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.

• H.S.M. Coxeter, M.S. Longuet-Higgins, J.C.P. Miller, Uniform polyhedra, Phil. Trans. 1954, 246 A, 401-50.
• S. P. Sopov A proof of the completeness on the list of elementary homogeneous polyhedra. (Russian) Ukrain. Geometr. Sb. No. 8, (1970), 139-156.
• J. Skilling The complete set of uniform polyhedra. Philos. Trans. Roy. Soc. London Ser. A 278 (1975), 111-135.

Listed by symmetry groups and vertex arrangements

All the uniform polyhedra are listed below by their symmetry groups and subgrouped by their vertex arrangments.

Regular polyhedra are labeled by their Schläfli symbol. Other nonregular uniform polyhedra are listed by their vertex configuration or their Uniform polyhedron index U(1-80).

Convex forms and fundamental vertex arrangments

The dihedral, tetrahedral, octahedral, and icosahedral symmetry polyhedra can be named by construction operations upon a parent form.

Note: Dihedra are an infinite set of two-sided polyhedra (2 identical polygons) which generates the prisms as truncated forms.

Each of these convex forms define a vertex arrangement that can be identified for the nonconvex forms in the next section.

	Parent	Truncated	Rectified	Truncated dual	Dual	Runcinated	Omnitruncated	Snub
Extended Schläfli symbol	[\begin p , q \end]	[t\begin p , q \end]	[\begin p \ q \end]	[t\begin q , p \end]	[\begin q , p \end]	[r\begin p \ q \end]	[t\begin p \ q \end]	[s\begin p \ q \end]
Wythoff symbol p-q-2	q | p 2	2 q | p	2 | p q	2 p | q	p | q 2	p q | 2	p q 2 |	| p q 2
Dynkin symbols	xPoQo	xPxQo	oPxQo	oPxQx	oPoQx	xPoQx	xPxQx	sPsQs
Vertex figure	pq	(q.2p.2p)	(p.q.p.q)	(p.2q.2q)	qp	(p.4.q.4)	(4.2p.2q)	(3.3.p.3.q)
Tetrahedral 3-3-2	3 | 3 2	2 3 | 3 (3.6.6)	2 | 3 3 (3.3.3.3)	2 3 | 3 (3.6.6)	3 | 3 2	3 3 | 2 (3.4.3.4)	3 3 2 | (4.6.6)	| 3 3 2 (3.3.3.3.3)
Octahedral 4-3-2	3 | 4 2	2 3 | 4 (3.8.8)	2 | 4 3 (3.4.3.4)	2 4 | 3 (4.6.6)	4 | 3 2	4 3 | 2 (3.4.4.4)	4 3 2 | (4.6.8)	| 4 3 2 (3.3.3.3.4)
Icosahedral 5-3-2	3 | 5 2	2 3 | 5 (3.10.10)	2 | 5 3 (3.5.3.5)	2 5 | 3 (5.6.6)	5 | 3 2	5 3 | 2 (3.4.5.4)	5 3 2 | (4.6.10)	| 5 3 2 (3.3.3.3.5)
Dihedral p-2-2 Example p=5		2.10.10	2.5.2.5	4.4.5		2.4.5.4	4.4.10	3.3.3.5

Definition of operations

Operation	Extended Schläfli symbol	Description
Parent	[\begin p , q \end]	Any regular polyhedron or tiling
Dual	[\begin q , p \end]	The dual is created by creating vertices on the parent face centers, and new faces are formed under each parent vertex. The edges are rotated 90 degrees. The dual of the regular polyhedron is .
Truncated	[t\begin p , q \end]	Each original vertex is cut off, with new faces 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.
Rectified	[\begin p \ q \end]	This is the limit to the truncation process with the original edges truncated to a single point. The polyhedron now has the combined faces of the parent and dual.
Runcinated	[r\begin p \ q \end]	The parent's faces are reduced and the created gaps are filled by squares where the edges were, and the dual's faces where the vertices were. The runcination has a degree of freedom and one solution that creates a uniform polyhedron.
Omnitruncated	[t\begin p \ q \end]	Truncation and rectification applied together create an omnitruncated form which has the parent's faces doubled in sides, the duals faces doubled in sides, and squares where the original edges existed.
Snub	[s\begin p \ q \end]	The snub takes the omnitruncated form and rectifies alternate vertices. (This operation is only possible for polyhedra with all even-sided faces.) All the original faces end up with half as many sides, and the square degenerate into edges. Since the omnitruncated forms have 3 faces/vertex, new triangles are formed.

Note: For nonconvex forms below an additional descriptor Nonuniform is used when the convex hull of the vertex arrangement has same topology as one of these, but has nonregular faces. For example an nonuniform runcinated form may have rectangles created in place of the edges rather than squares.

There are 2 convex uniform polyheda, tetrahedron, and truncated tetrahedron, and one nonconvex form, the tetrahemihexahedron which have tetrahedral symmetry. The tetrahedron is self dual.

In addition the octahedron, truncated octahedron, cuboctahedron, and icosahedron have tetrahedral symmetry as well as higher symmetry. They are added for completeness below, although their nonconvex forms with octahedral symmetry are not included here.

Vertex group	Convex	Nonconvex
(Tetrahedral)
Truncated (*)	(3.6.6)
Rectified (*)		(4.3/2.4.3)
Runcinated (*)	(3.4.3.4)
Omnitruncated (*)	(4.6.6)
Snub (*)

There are 8 convex forms, and 10 nonconvex forms with octahedral symmetry.

Vertex group	Convex	Nonconvex
(Octahedral)
Truncated (*)	(4.6.6)
Rectified (*)	(3.4.3.4)	(6.4/3.6.4)	(6.3/2.6.3)
Truncated dual (*)	(3.8.8)	(4.8/3.4/3.8/5)	(8/3.3.8/3.4)	(4.3/2.4.4)
Dual (*)
Runcinated (*)	(3.4.4.4)	(4.8.4/3.8)	(8.3/2.8.4)	(8/3.8/3.3)
Omnitruncated (*)	(4.6.8)
Nonuniform omnitruncated (*)	(4.6.8)	(8/3.4.6)	(8/3.6.8)
Snub (*)	(3.3.3.3.4)

There are 8 convex forms and 46 nonconvex forms. Some of the nonconvex snub forms have nonuniform chiral symmetry, and some have achiral symmetry.

There are many nonuniform forms of varied degrees of truncation and runcination.

Vertex group	Convex	Nonconvex
(Icosahedral)
Truncated (*)	(5.6.6)
Nonuniform truncated (*)	(5.6.6)	U32	U37	U61	U38	U44	U56	U67	U73
Rectified (*)	(3.5.3.5)	U49	U51	U54	U70	U71	U36	U62	U65
Truncated dual (*)	(3.10.10)	U42	U48	U63
Nonuniform truncated dual (*)	(3.10.10)	U68	U72	U45
Dual (*)			U30	U41	U47
Runcinated (*)	(3.4.5.4)	U33	U39
Nonuniform runcinated (*)	(3.4.5.4)	U31	U43	U50	U55	U58	U75	U64	U66
Omnitruncated (*)	(4.6.10)
Nonuniform omnitruncated (*)	(4.6.10)	U59
Snub (*)	(3.3.3.3.5)
Nonuniform Snub (*)	(3.3.3.3.5)	U40	U46	U57	U69	U60	U74

There are four infinite sets of uniform prisms and antiprisms: (1) convex prisms, (2) star prisms, (3) convex antiprisms, and (4) star antiprims.

They share two sets of vertex arrangements:

• Achiral dihedral: D_nh symmetry forms are named by their generating polygon, where n is p for star forms with p/q.
• Chiral dihedral:D_nd symmetry forms are similarly named by the generating polygon but prefixed by the term gyrated since the two polygons have a half twist applied.

Prisms and star prisms are D_nh. Antiprisms are D_nd. Star antiprisms exist in both form, depending on whether q is odd or even for a given star polygon .

For a given star polygon , if q>p/2, it is considered a crossed form with a center point reflection applied between the two halves. However some do not exist, such as , , and , as well as any others that fail the vertex figure existence constraint:

cos(2π/n) < cos(π/n), where n=p/q

which is equivalent to p/q > 3/2.

Note: The cube and octahedron are listed here with dihedral symmetry (as a tetragonal prism and trigonal antiprism respectively), although if uniformly colored, they also have octahedral symmetry.

Vertex group	Convex	Nonconvex
Trigonal	3.3.4
Gyrated trigonal	3.3.3.3
Tetragonal	4.4.4
Gyrated tetragonal	3.3.3.4
Pentagonal	4.4.5	4.4.5/2	3.3.3.5/2
Gyrated pentagonal	3.3.3.5	3.3.3.5/3
Hexagonal	4.4.6
Gyrated hexagonal	3.3.3.6
Heptagonal	4.4.7	4.4.7/2	4.4.7/3	3.3.3.7/2 3.3.3.7/4
Gyrated heptagonal	3.3.3.7	3.3.3.7/3
Octagonal	4.4.8	4.4.4.8/3
Gyrated octagonal	3.3.3.8	3.3.3.8/3 3.3.3.8/5
Enneagonal	4.4.9	3.3.3.9/2 3.3.3.9/4
Gyrated enneagonal	3.3.3.9	3.3.3.9/5
Decagonal	4.4.10	4.4.10/3
Gyrated decagonal	3.3.3.10	3.3.3.10/3
11-agonal	4.4.11	4.4.11/2 4.4.11/5	3.3.3.11/2 3.3.3.11/4 3.3.3.11/6
Gyrated 11-agonal	3.3.3.11	3.3.3.11/3 3.3.3.11/5 3.3.3.11/7
Dodecagonal	4.4.12	4.4.12/5	3.3.3.12/7
Gyrated dodecagonal	3.3.3.12	3.3.3.12/5
...

Skilling's figure

One further nonconvex uniform polyhedron is the Great disnub dirhombidodecahedron, also known as Skilling's figure, which is vertex-uniform, but has pairs of edges which coincide in space such that four faces meet at some edges. This has I_h symmetry.

See also

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

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.

 

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From Wikipedia, the Free Encyclopedia. Original article here. Support Wikipedia by contributing or donating.
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