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| {{more footnotes|date=October 2011}}
| | 42 year-old Aircraft Preservation Engineer (Avionics) Carmouche from Montebello, enjoys to spend some time kites, [http://www.vansuthong.net/new-launch-property-singapore/ property launch singapore] developers in singapore and soccer. Is a travel maniac and these days made a vacation in Sichuan Giant Panda Sanctuaries. |
| [[File:Wythoffian construction diagram.png|480px|right]]
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| [[File:Polyhedron truncation example3.png|480px|right|Example forms from the [[cube]] and [[octahedron]]]]
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| A '''uniform polyhedron''' is a [[polyhedron]] which has [[regular polygon]]s as [[Face (geometry)|faces]] and is [[vertex-transitive]]
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| ([[Transitive group action|transitive]] on its [[vertex (geometry)|vertices]], isogonal, i.e. there is an [[isometry]] mapping any vertex onto any other). It follows that all vertices are [[Congruence (geometry)|congruent]], and the polyhedron has a high degree of [[reflectional symmetry|reflectional]] and [[rotational symmetry]]. | |
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| [[Uniform polytope|Uniform]] polyhedra may be [[Regular polyhedron|regular]] (if also face and edge transitive), [[Quasiregular polyhedron|quasi-regular]] (if edge transitive but not face transitive) or [[Semiregular polyhedron|semi-regular]] (if neither edge nor face transitive). The faces and vertices need not be [[Convex polyhedron|convex]], so many of the uniform polyhedra are also [[Star polyhedron|star polyhedra]].
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| Excluding the [[infinite set]]s, there are 75 uniform polyhedra (or 76 if edges are allowed to coincide).
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| * Convex
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| ** 5 [[Platonic solid]]s – regular convex polyhedra
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| ** 13 [[Archimedean solid]]s – 2 [[Quasiregular polyhedron|quasiregular]] and 11 [[Semiregular polyhedron|semiregular]] convex polyhedra
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| * Star
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| ** 4 [[Kepler–Poinsot polyhedra]] – regular nonconvex polyhedra
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| ** 53 [[uniform star polyhedra]] – 5 [[Quasiregular polyhedron#Nonconvex examples|quasiregular]] and 48 semiregular
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| ** 1 star polyhedron found by John Skilling with pairs of edges that coincide, called the [[great disnub dirhombidodecahedron]] (Skilling's figure).
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| There are also two infinite sets of uniform [[Prismatic uniform polyhedron|prisms and antiprisms]], including convex and star forms.
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| [[Dual polyhedra]] to uniform polyhedra are [[face-transitive]] (isohedral) and have regular [[vertex figure]]s, 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]].
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| 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.
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| ==History==
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| * The [[Platonic solid]]s date back to the classical Greeks and were studied by [[Plato]], [[Theaetetus (mathematician)|Theaetetus]] and [[Euclid]].
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| *[[Johannes Kepler]] (1571–1630) was the first to publish the complete list of [[Archimedean solid]]s after the original work of [[Archimedes]] was lost.
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| '''Regular star polyhedra:'''
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| * [[Johannes Kepler|Kepler]] (1619) discovered two of the regular [[Kepler–Poinsot polyhedra]] and [[Louis Poinsot]] (1809) discovered the other two.
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| '''Other 53 nonregular star polyhedra:'''
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| * Of the remaining 53, Albert Badoureau (1881) discovered 36. [[Edmund Hess]] (1878) discovered 2 more and Pitsch (1881) independently discovered 18, of which 15 had not previously been discovered.
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| * The geometer [[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]] discovered the remaining twelve in collaboration with [[J. C. P. Miller]] (1930–1932) but did not publish. M.S. and H.C. Longuet-Higgins and independently discovered 11 of these.
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| * {{harvtxt|Coxeter|Longuet-Higgins|Miller|1954}} published the list of uniform polyhedra.
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| * {{harvtxt|Sopov|1970}} proved their conjecture that the list was complete.
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| * In 1974, [[Magnus Wenninger]] published his book [[List of Wenninger polyhedron models|''Polyhedron models'']], which lists all 75 nonprismatic uniform polyhedra, with many previously unpublished names given to them by [[Norman Johnson (mathematician)|Norman Johnson]].
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| * {{harvtxt|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.
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| * In 1993, Zvi Har'El produced a complete kaleidoscopic construction of the uniform polyhedra and duals with a computer program called '''Kaleido''', and summarized in a paper ''Uniform Solution for Uniform Polyhedra'', counting figures 1-80.
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| * Also in 1993, R. Mäder ported this Kaleido solution to [[Mathematica]] with a slightly different indexing system.
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| * 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]].<ref>[http://www.springerlink.com/content/me48wm7823jhdcpe/fulltext.pdf?page=1 Closed-Form Expressions for Uniform Polyhedra and Their Duals, Peter W. Messer, Discrete Comput Geom 27:353–375 (2002)]</ref>
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| == Uniform star polyhedra ==
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| The 57 nonprismatic nonconvex forms are compiled by Wythoff constructions within [[Schwarz triangle]]s.
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| {{main|uniform star polyhedron}}
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| == Convex forms by Wythoff construction ==
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| The convex uniform polyhedra can be named by [[Wythoff construction]] operations and can be named in relation to the regular form.
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| In more detail the convex uniform polyhedron are given below by their [[Wythoff construction]] within each symmetry group.
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| 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.
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| The Wythoff construction applies equally to uniform polyhedra and [[Spherical polyhedron|uniform tilings on the surface of a sphere]], so images of both are given. The spherical tilings including the set of [[hosohedron]]s and [[dihedron]]s which are degenerate polyhedra.
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| These symmetry groups are formed from the reflectional [[point groups in three dimensions]], each represented by a fundamental triangle (p q r), where p>1, q>1, r>1 and 1/p+1/q+1/r<1.
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| * [[Tetrahedral symmetry]] (3 3 2) - order 24
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| * [[Octahedral symmetry]] (4 3 2) - order 48
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| * [[Icosahedral symmetry]] (5 3 2) - order 120
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| * [[Dihedral symmetry]] (n 2 2), for all n=3,4,5,... - order ''4n''
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| The remaining nonreflective forms are constructed by [[Alternation (geometry)|alternation]] operations applied to the polyhedra with an even number of sides.
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| Along with the prisms and their [[dihedral symmetry]], the spherical Wythoff construction process adds two ''regular'' classes which become degenerate as polyhedra - the ''[[Dihedron|dihedra]]'' and [[Hosohedron|hosohedra]], the first having only two faces, and the second only two vertices. The truncation of the regular ''hosohedra'' creates the prisms.
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| Below the convex uniform polyhedra are indexed 1-18 for the nonprismatic forms as they are presented in the tables by symmetry form. Repeated forms are in brackets.
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| For the infinite set of prismatic forms, they are indexed in four families:
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| # [[Hosohedron]]s H<sub>2...</sub> (Only as spherical tilings)
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| # [[Dihedron]]s D<sub>2...</sub> (Only as spherical tilings)
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| # [[Prism (geometry)|Prisms]] P<sub>3...</sub> (Truncated hosohedrons)
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| # [[Antiprism]]s A<sub>3...</sub> (Snub prisms)
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| | |
| === Summary tables ===
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| | |
| {| class="wikitable" width=640
| |
| |- valign=top
| |
| ![[Norman Johnson (mathematician)|Johnson]] name
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| !Parent
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| !Truncated
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| !Rectified
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| !Bitruncated<BR>(tr. dual)
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| !Birectified<BR>(dual)
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| !Cantellated
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| !Omnitruncated<BR>(<small>Cantitruncated</small>)
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| !Snub
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| |-
| |
| !rowspan=3|Extended<BR>[[Schläfli symbol]]
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| !<math>\begin{Bmatrix} p , q \end{Bmatrix}</math>
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| !<math>t\begin{Bmatrix} p , q \end{Bmatrix}</math>
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| !<math>\begin{Bmatrix} p \\ q \end{Bmatrix}</math>
| |
| !<math>t\begin{Bmatrix} q , p \end{Bmatrix}</math>
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| !<math>\begin{Bmatrix} q , p \end{Bmatrix}</math>
| |
| !<math>r\begin{Bmatrix} p \\ q \end{Bmatrix}</math>
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| !<math>t\begin{Bmatrix} p \\ q \end{Bmatrix}</math>
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| !<math>s\begin{Bmatrix} p \\ q \end{Bmatrix}</math>
| |
| |-
| |
| !{p,q}
| |
| !t{p,q}
| |
| !r{p,q}
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| !2t{p,q}
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| !2r{p,q}
| |
| !rr{p,q}
| |
| !tr{p,q}
| |
| !sr{p,q}
| |
| |-
| |
| !t<sub>0</sub>{p,q}
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| !t<sub>0,1</sub>{p,q}
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| !t<sub>1</sub>{p,q}
| |
| !t<sub>1,2</sub>{p,q}
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| !t<sub>2</sub>{p,q}
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| !t<sub>0,2</sub>{p,q}
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| !t<sub>0,1,2</sub>{p,q}
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| !ht<sub>0,1,2</sub>{p,q}
| |
| |-
| |
| ![[Wythoff symbol]]<BR>(p q 2)
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| ! q | p 2
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| ! 2 q | p
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| ! 2 | p q
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| ! 2 p | q
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| ! p | q 2
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| ! p q | 2
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| ! p q 2 |
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| ! | p q 2
| |
| |- valign=top
| |
| ![[Coxeter diagram]]
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| !{{CDD|node_1|p|node|q|node}}
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| !{{CDD|node_1|p|node_1|q|node}}
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| !{{CDD|node|p|node_1|q|node}}<BR>{{CDD|node_1|split1-pq||nodes}}
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| !{{CDD|node|p|node_1|q|node_1}}
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| !{{CDD|node|p|node|q|node_1}}
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| !{{CDD|node_1|p|node|q|node_1}}<BR>{{CDD|node|split1-pq||nodes_11}}
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| !{{CDD|node_1|p|node_1|q|node_1}}<BR>{{CDD|node_1|split1-pq||nodes_11}}
| |
| !{{CDD|node_h|p|node_h|q|node_h}}<BR>{{CDD|node_h|split1-pq||nodes_hh}}
| |
| |-
| |
| ![[Vertex configuration|Vertex figure]]
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| !p<sup>q</sup>
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| !(q.2p.2p)
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| !(p.q)<sup>2</sup>
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| !(p.2q.2q)
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| !q<sup>p</sup>
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| !(p.4.q.4)
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| !(4.2p.2q)
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| !(3.3.p.3.q)
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| |- align=center
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| |[[Tetrahedral symmetry|Tetrahedral]]<BR>(3 3 2)
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| |[[File:Uniform polyhedron-33-t0.png|64px]]<BR>[[Tetrahedron|{3,3}]]
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| |[[File:Uniform polyhedron-33-t01.png|64px]]<BR>[[Truncated tetrahedron|(3.6.6)]]
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| |[[File:Uniform polyhedron-33-t1.png|64px]]<BR>[[Octahedron|(3.3.3.3)]]
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| |[[File:Uniform polyhedron-33-t12.png|64px]]<BR>[[Truncated tetrahedron|(3.6.6)]]
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| |[[File:Uniform polyhedron-33-t2.png|64px]]<BR>[[Tetrahedron|{3,3}]]
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| | [[File:Uniform polyhedron-33-t02.png|64px]]<BR>[[Cuboctahedron|(3.4.3.4)]]
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| |[[File:Uniform polyhedron-33-t012.png|64px]]<BR>[[Truncated octahedron|(4.6.6)]]
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| |[[File:Uniform polyhedron-33-s012.svg|64px]]<BR>[[Icosahedron|(3.3.3.3.3)]]
| |
| |- align=center
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| |[[Octahedral symmetry|Octahedral]]<BR>(4 3 2)
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| |[[File:Uniform polyhedron-43-t0.svg|64px]]<BR>[[Cube|{4,3}]]
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| |[[File:Uniform polyhedron-43-t01.svg|64px]]<BR>[[Truncated cube|(3.8.8)]]
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| |[[File:Uniform polyhedron-43-t1.svg|64px]]<BR>[[Cuboctahedron|(3.4.3.4)]]
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| |[[File:Uniform polyhedron-43-t12.svg|64px]]<BR>[[Truncated octahedron|(4.6.6)]]
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| |[[File:Uniform polyhedron-43-t2.svg|64px]]<BR>[[Octahedron|{3,4}]]
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| |[[File:Uniform polyhedron-43-t02.png|64px]]<BR>[[Small rhombicuboctahedron|(3.4.4.4)]]
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| |[[File:Uniform polyhedron-43-t012.png|64px]]<BR>[[Truncated cuboctahedron|(4.6.8)]]
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| |[[File:Uniform polyhedron-43-s012.png|64px]]<BR>[[Snub cube|(3.3.3.3.4)]]
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| |- align=center
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| |[[Icosahedral symmetry|Icosahedral]]<BR>(5 3 2)
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| |[[File:Uniform polyhedron-53-t0.png|64px]]<BR>[[Dodecahedron|{5,3}]]
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| |[[File:Uniform polyhedron-53-t01.png|64px]]<BR>[[Truncated dodecahedron|(3.10.10)]]
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| |[[File:Uniform polyhedron-53-t1.png|64px]]<BR>[[Icosidodecahedron|(3.5.3.5)]]
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| |[[File:Uniform polyhedron-53-t12.png|64px]]<BR>[[Truncated icosahedron|(5.6.6)]]
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| |[[File:Uniform polyhedron-53-t2.png|64px]]<BR>[[Icosahedron|{3,5}]]
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| |[[File:Uniform polyhedron-53-t02.png|64px]]<BR>[[Rhombicosidodecahedron|(3.4.5.4)]]
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| |[[File:Uniform polyhedron-53-t012.png|64px]]<BR>[[Truncated icosidodecahedron|(4.6.10)]]
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| |[[File:Uniform polyhedron-53-s012.png|64px]]<BR>[[Snub dodecahedron|(3.3.3.3.5)]]
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| |}
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| | |
| And a sampling of Dihedral symmetries:
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| {| class="wikitable"
| |
| |-
| |
| !(p 2 2)
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| !Parent
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| !Truncated
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| !Rectified
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| !Bitruncated<BR>(tr. dual)
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| !Birectified<BR>(dual)
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| !Cantellated
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| !Omnitruncated<BR>(<small>Cantitruncated</small>)
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| !Snub
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| |-
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| !rowspan=3|Extended<BR>[[Schläfli symbol]]
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| !<math>\begin{Bmatrix} p , 2 \end{Bmatrix}</math>
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| !<math>t\begin{Bmatrix} p , 2 \end{Bmatrix}</math>
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| !<math>\begin{Bmatrix} p \\ 2 \end{Bmatrix}</math>
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| !<math>t\begin{Bmatrix} 2 , p \end{Bmatrix}</math>
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| !<math>\begin{Bmatrix} 2 , p \end{Bmatrix}</math>
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| !<math>r\begin{Bmatrix} p \\ 2 \end{Bmatrix}</math>
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| !<math>t\begin{Bmatrix} p \\ 2 \end{Bmatrix}</math>
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| !<math>s\begin{Bmatrix} p \\ 2 \end{Bmatrix}</math>
| |
| |-
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| !{p,2}
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| !t{p,2}
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| !r{p,2}
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| !2t{p,2}
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| !2r{p,2}
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| !rr{p,2}
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| !tr{p,2}
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| !sr{p,2}
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| |-
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| !t<sub>0</sub>{p,2}
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| !t<sub>0,1</sub>{p,2}
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| !t<sub>1</sub>{p,2}
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| !t<sub>1,2</sub>{p,2}
| |
| !t<sub>2</sub>{p,2}
| |
| !t<sub>0,2</sub>{p,2}
| |
| !t<sub>0,1,2</sub>{p,2}
| |
| !ht<sub>0,1,2</sub>{p,2}
| |
| |-
| |
| ![[Wythoff symbol]]
| |
| ! 2 | p 2
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| ! 2 2 | p
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| ! 2 | p 2
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| ! 2 p | 2
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| ! p | 2 2
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| ! p 2 | 2
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| ! p 2 2 |
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| ! | p 2 2
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| |-
| |
| ![[Coxeter-Dynkin diagram]]
| |
| !{{CDD|node_1|p|node|2|node}}
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| !{{CDD|node_1|p|node_1|2|node}}
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| !{{CDD|node|p|node_1|2|node}}
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| !{{CDD|node|p|node_1|2|node_1}}
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| !{{CDD|node|p|node|2|node_1}}
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| !{{CDD|node_1|p|node|2|node_1}}
| |
| !{{CDD|node_1|p|node_1|2|node_1}}
| |
| !{{CDD|node_h|p|node_h|2x|node_h}}
| |
| |-
| |
| ![[Vertex configuration|Vertex figure]]
| |
| !p<sup>2</sup>
| |
| !(2.2p.2p)
| |
| !(p. 2.p. 2)
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| !(p. 4.4)
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| !2<sup>p</sup>
| |
| !(p. 4.2.4)
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| !(4.2p.4)
| |
| !(3.3.p. 3.2)
| |
| |- align=center
| |
| |[[Dihedral symmetry|Dihedral]]<BR>(2 2 2)
| |
| | [[File:Digonal dihedron.png|64px]]<BR>[[Dihedron|{2,2}]]
| |
| | 2.4.4
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| | [[File:Digonal dihedron.png|64px]]<BR>2.2.2.2
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| | <BR>4.4.2
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| | [[File:Digonal dihedron.png|64px]]<BR>[[Hosohedron|{2,2}]]
| |
| | 2.4.2.4
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| | [[File:Spherical square prism2.png|64px]]<BR>4.4.4
| |
| | [[File:Spherical digonal antiprism.png|64px]]<BR>3.3.3.2
| |
| |- align=center
| |
| |[[Dihedral symmetry|Dihedral]]<BR>(3 2 2)
| |
| | [[File:Trigonal dihedron.png|64px]]<BR>[[Dihedron|{3,2}]]
| |
| | [[File:Hexagonal dihedron.png|64px]]<BR>2.6.6
| |
| | 2.3.2.3
| |
| | [[File:Spherical triangular prism.png|64px]]<BR>4.4.3
| |
| | [[File:Spherical trigonal hosohedron.png|64px]]<BR>[[Hosohedron|{2,3}]]
| |
| | [[File:Spherical triangular prism.png|64px]]<BR>2.4.3.4
| |
| | [[File:Spherical hexagonal prism2.png|64px]]<BR>4.4.6
| |
| | [[File:Spherical trigonal antiprism.png|64px]]<BR>3.3.3.3
| |
| |- align=center
| |
| |[[Dihedral symmetry|Dihedral]]<BR>(4 2 2)
| |
| | [[Dihedron|{4,2}]]
| |
| | 2.8.8
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| | 2.4.2.4
| |
| | [[File:Spherical square prism.png|64px]]<BR>4.4.4
| |
| | [[File:Spherical square hosohedron.png|64px]]<BR>[[Hosohedron|{2,4}]]
| |
| | [[File:Spherical square prism.png|64px]]<BR>2.4.4.4
| |
| | [[File:Spherical octagonal prism2.png|64px]]<BR>4.4.8
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| | [[File:Spherical square antiprism.png|64px]]<BR>3.3.3.4
| |
| |- align=center
| |
| |[[Dihedral symmetry|Dihedral]]<BR>(5 2 2)
| |
| | [[Dihedron|{5,2}]]
| |
| | 2.10.10
| |
| | 2.5.2.5
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| | [[File:Spherical pentagonal prism.png|64px]]<BR>4.4.5
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| | [[File:Spherical pentagonal hosohedron.png|64px]]<BR>[[Hosohedron|{2,5}]]
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| | [[File:Spherical pentagonal prism.png|64px]]<BR>2.4.5.4
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| | [[File:Spherical decagonal prism2.png|64px]]<BR>4.4.10
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| | [[File:Spherical pentagonal antiprism.png|64px]]<BR>3.3.3.5
| |
| |- align=center
| |
| |[[Dihedral symmetry|Dihedral]]<BR>(6 2 2)
| |
| | [[File:Hexagonal dihedron.png|64px]]<BR>[[Dihedron|{6,2}]]
| |
| | [[File:Dodecagonal dihedron.png|64px]]<BR>2.12.12
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| | [[File:Hexagonal dihedron.png|64px]]<BR>2.6.2.6
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| | [[File:Spherical hexagonal prism.png|64px]]<BR>4.4.6
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| | [[File:Spherical hexagonal hosohedron.png|64px]]<BR>[[Hosohedron|{2,6}]]
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| | [[File:Spherical hexagonal prism.png|64px]]<BR>2.4.6.4
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| | [[File:Spherical dodecagonal prism2.png|64px]]<BR>4.4.12
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| | [[File:Spherical hexagonal antiprism.png|64px]]<BR>3.3.3.6
| |
| |}
| |
| | |
| === Wythoff construction operators ===
| |
| {| class="wikitable"
| |
| !Operation
| |
| !Symbol
| |
| ![[Coxeter diagram|Coxeter<BR>diagram]]
| |
| !Description
| |
| |-
| |
| !width=150|Parent
| |
| !width=60|{p,q}<BR>t<sub>0</sub>{p,q}
| |
| |{{CDD|node_1|p|node|q|node}}
| |
| | Any regular polyhedron or tiling
| |
| |-
| |
| ! [[Rectification (geometry)|Rectified]] (r)
| |
| !r{p,q}<BR>t<sub>1</sub>{p,q}
| |
| |{{CDD|node|p|node_1|q|node}}
| |
| |The edges are fully truncated into single points. The polyhedron now has the combined faces of the parent and dual.
| |
| |-
| |
| !Birectified (2r)<BR>(also [[Dual polyhedron|dual]])
| |
| !2r{p,q}<BR>t<sub>2</sub>{p,q}
| |
| |{{CDD|node|p|node|q|node_1}}
| |
| |[[File:Dual Cube-Octahedron.svg|100px|right]]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. The dual of the regular polyhedron {p, q} is also a regular polyhedron {q, p}.
| |
| |-
| |
| ![[Truncation (geometry)|Truncated]] (t)
| |
| !t{p,q}<BR>t<sub>0,1</sub>{p,q}
| |
| |{{CDD|node_1|p|node_1|q|node}}
| |
| |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.<BR>[[File:Cube truncation sequence.svg|400px]]
| |
| |-
| |
| !Bitruncated (2t)<BR>(also truncated dual)
| |
| !2t{p,q}<BR>t<sub>1,2</sub>{p,q}
| |
| |{{CDD|node|p|node_1|q|node_1}}
| |
| |Same as truncated dual.
| |
| |-
| |
| ! [[Cantellation (geometry)|Cantellated]] (rr)<BR>(Also [[Expansion (geometry)|expanded]])
| |
| !rr{p,q}
| |
| |{{CDD|node_1|p|node|q|node_1}}
| |
| |In addition to vertex truncation, each original edge is ''beveled'' with new rectangular faces appearing in their place. A uniform cantellation is half way between both the parent and dual forms.<BR>[[File:Cube cantellation sequence.svg|400px]]
| |
| |-
| |
| !Cantitruncated (tr)<BR>(Also [[Omnitruncated polyhedron|omnitruncated]])
| |
| !tr{p,q}<BR>t<sub>0,1,2</sub>{p,q}
| |
| |{{CDD|node_1|p|node_1|q|node_1}}
| |
| |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.
| |
| |}
| |
| | |
| {| class="wikitable"
| |
| |+ Alternation operations
| |
| |-
| |
| !width=150|Operation
| |
| !width=70|Symbol
| |
| ![[Coxeter diagram|Coxeter<BR>diagram]]
| |
| !Description
| |
| |-
| |
| | |
| !Snub rectified (sr)
| |
| !sr{p,q}
| |
| |{{CDD|node_h|p|node_h|q|node_h}}
| |
| |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.<BR>[[File:Snubcubes in grCO.svg|320px]]
| |
| | |
| |-
| |
| !Snub (s)
| |
| !s{p,2q}
| |
| |{{CDD|node_h|p|node_h|2x|q|node}}
| |
| |Alternated truncation
| |
| |-
| |
| !Cantic snub (s<sub>2</sub>)
| |
| !s<sub>2</sub>{p,2q}
| |
| |{{CDD|node_h|p|node_h|2x|q|node_1}}
| |
| |
| |
| | |
| |-
| |
| !Alternated cantellation (hrr)
| |
| !hrr{2p,2q}
| |
| |{{CDD|node_h|2x|p|node|2x|q|node_h}}
| |
| |Only possible in uniform tilings (infinite polyhedra), alternation of {{CDD|node_1|2x|p|node|2x|q|node_1}}<BR>For example, {{CDD|node_h|4|node|4|node_h}}
| |
| |-
| |
| !Half (h)
| |
| !h{2p,q}
| |
| |{{CDD|node_h1|2x|p|node|q|node}}
| |
| |align=left|[[Alternation (geometry)|Alternation]] of {{CDD|node_1|2x|p|node|q|node}}, same as {{CDD|labelp|branch_10ru|split2-qq|node}}
| |
| |-
| |
| !Cantic (h<sub>2</sub>)
| |
| !h<sub>2</sub>{2p,q}
| |
| |{{CDD|node_h1|2x|p|node|q|node_1}}
| |
| |Same as {{CDD|labelp|branch_10ru|split2-qq|node_1}}
| |
| |-
| |
| !Half rectified (hr)
| |
| !hr{2p,2q}
| |
| |{{CDD|node|2x|p|node_h1|2x|q|node}}
| |
| |align=left|Only possible in uniform tilings (infinite polyhedra), alternation of {{CDD|node|2x|p|node_1|2x|q|node}}, same as {{CDD|labelp|branch_10ru|2a2b-cross|branch_10lu|labelq}} or {{CDD|labelp|branch_10r|iaib|branch_01l|labelq}}<BR>For example, {{CDD|node|4|node_h1|4|node}} = {{CDD|nodes_10ru|2a2b-cross|nodes_10lu}} or {{CDD|nodes_11|iaib|nodes}}
| |
| |-
| |
| !Quarter (q)
| |
| !q{2p,2q}
| |
| |{{CDD|node_h1|2x|p|node|2x|q|node_h1}}
| |
| |Only possible in uniform tilings (infinite polyhedra), same as {{CDD|labelq|branch_11|papb-cross|branch_10l|labelq}}<BR>For example, {{CDD|node_h1|4|node|4|node_h1}} = {{CDD|nodes_11|2a2b-cross|nodes_10lu}} or {{CDD|nodes_11|iaib|nodes_10l}}
| |
| |}
| |
| | |
| === (3 3 2) T<sub>d</sub> Tetrahedral symmetry ===
| |
| | |
| 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]] A<sub>2</sub> or [3,3], as well as a [[Coxeter-Dynkin diagram]]: {{CDD|node|3|node|3|node}}.
| |
| | |
| There are 24 triangles, visible in the faces of the [[tetrakis hexahedron]] and alternately colored triangles on a sphere:
| |
| :[[File:Tetrakishexahedron.jpg|100px]] [[File:Tetrahedral reflection domains.png|100px]][[File:Sphere symmetry group td.png|100px]]
| |
| | |
| {| class="wikitable"
| |
| !rowspan=2|#
| |
| !rowspan=2|Name
| |
| !rowspan=2|Graph<BR>A<sub>3</sub>
| |
| !rowspan=2|Graph<BR>A<sub>2</sub>
| |
| !rowspan=2|Picture
| |
| !rowspan=2|Tiling
| |
| !rowspan=2|[[Vertex figure|Vertex<BR>figure]]
| |
| !rowspan=2|[[Coxeter-Dynkin diagram|Coxeter-Dynkin]]<BR>and [[Schläfli symbol#Extended for uniform polychora and 3-space honeycombs|Schläfli]]<BR>symbols
| |
| !colspan=3|Face counts by position
| |
| !colspan=3|Element counts
| |
| |-
| |
| ! Pos. 2<BR>{{CDD|node|3|node}}<BR>[3]<BR>(4)
| |
| ! Pos. 1<BR>{{CDD|node|2||node}}<BR>[2]<BR>(6)
| |
| ! Pos. 0<BR>{{CDD|2|node|3|node}}<BR>[3]<BR>(4)
| |
| ! Faces
| |
| ! Edges
| |
| ! Vertices
| |
| |- BGCOLOR="#f0e0e0" align=center
| |
| !1
| |
| |align=center|[[Tetrahedron]]
| |
| |[[File:3-simplex t0.svg|50px]]
| |
| |[[File:3-simplex t0 A2.svg|50px]]
| |
| |[[File:Uniform polyhedron-33-t0.png|50px]]
| |
| |[[File:Uniform tiling 332-t0-1-.png|50px]]
| |
| |[[File:Tetrahedron vertfig.png|50px]]
| |
| |align=center|{{CDD|node_1|3|node|3|node}}<BR>{3,3}
| |
| | [[File:Regular_polygon_3.svg|30px]]<BR>[[Triangle|{3}]]
| |
| |
| |
| |
| |
| | 4
| |
| | 6
| |
| | 4
| |
| |- BGCOLOR="#e0e0f0" align=center
| |
| ![1]
| |
| |align=center|Birectified tetrahedron<BR>(Same as [[tetrahedron]])
| |
| |[[File:3-simplex t0.svg|50px]]
| |
| |[[File:3-simplex t0 A2.svg|50px]]
| |
| |[[File:Uniform polyhedron-33-t2.png|50px]]
| |
| |[[File:Uniform tiling 332-t2.png|50px]]
| |
| |[[File:Tetrahedron vertfig.png|50px]]
| |
| |align=center|{{CDD|node|3|node|3|node_1}}<BR>t<sub>2</sub>{3,3}={3,3}
| |
| |
| |
| |
| |
| | [[File:Regular_polygon_3.svg|30px]]<BR>[[Triangle|{3}]]
| |
| | 4
| |
| | 6
| |
| | 4
| |
| |- BGCOLOR="#e0f0e0" align=center
| |
| !2
| |
| |align=center|Rectified tetrahedron<BR>(Same as [[octahedron]])
| |
| |[[File:3-simplex t1.svg|50px]]
| |
| |[[File:3-simplex t1 A2.svg|50px]]
| |
| |[[File:Uniform polyhedron-33-t1.png|50px]]
| |
| |[[File:Uniform tiling 332-t1-1-.png|50px]]
| |
| |[[File:Octahedron vertfig.png|50px]]
| |
| |align=center|{{CDD|node|3|node_1|3|node}}<BR>t<sub>1</sub>{3,3}=r{3,3}
| |
| | [[File:Regular_polygon_3.svg|30px]]<BR>[[Triangle|{3}]]
| |
| |
| |
| | [[File:Regular_polygon_3.svg|30px]]<BR>[[Triangle|{3}]]
| |
| | 8
| |
| | 12
| |
| | 6
| |
| |- BGCOLOR="#f0e0e0" align=center
| |
| !3
| |
| |align=center|[[Truncated tetrahedron]]
| |
| |[[File:3-simplex t01.svg|50px]]
| |
| |[[File:3-simplex t01 A2.svg|50px]]
| |
| |[[File:Uniform polyhedron-33-t01.png|50px]]
| |
| |[[File:Uniform tiling 332-t01-1-.png|50px]]
| |
| |[[File:Truncated tetrahedron vertfig.png|50px]]
| |
| |align=center|{{CDD|node_1|3|node_1|3|node}}<BR>t<sub>0,1</sub>{3,3}=t{3,3}
| |
| | [[File:Regular_polygon_6.svg|30px]]<BR>[[Hexagon|{6}]]
| |
| |
| |
| | [[File:Regular_polygon_3.svg|30px]]<BR>[[Triangle|{3}]]
| |
| | 8
| |
| | 18
| |
| | 12
| |
| |- BGCOLOR="#e0e0f0" align=center
| |
| ![3]
| |
| |align=center|Bitruncated tetrahedron<BR>(Same as [[truncated tetrahedron]])
| |
| |[[File:3-simplex t01.svg|50px]]
| |
| |[[File:3-simplex t01 A2.svg|50px]]
| |
| |[[File:Uniform polyhedron-33-t12.png|50px]]
| |
| |[[File:Uniform tiling 332-t12.png|50px]]
| |
| |[[File:Truncated tetrahedron vertfig.png|50px]]
| |
| |align=center|{{CDD|node|3|node_1|3|node_1}}<BR>t<sub>1,2</sub>{3,3}=t{3,3}
| |
| | [[File:Regular_polygon_3.svg|30px]]<BR>[[Triangle|{3}]]
| |
| |
| |
| | [[File:Regular_polygon_6.svg|30px]]<BR>[[Hexagon|{6}]]
| |
| | 8
| |
| | 18
| |
| | 12
| |
| |- BGCOLOR="#e0f0e0" align=center
| |
| !4
| |
| |align=center|Rhombitetratetrahedron<BR>(Same as [[cuboctahedron]])
| |
| |[[File:3-simplex t02.svg|50px]]
| |
| |[[File:3-simplex t02 A2.svg|50px]]
| |
| |[[File:Uniform polyhedron-33-t02.png|50px]]
| |
| |[[File:Uniform tiling 332-t02.png|50px]]
| |
| |[[File:Cuboctahedron vertfig.png|50px]]
| |
| |align=center|{{CDD|node_1|3|node|3|node_1}}<BR>t<sub>0,2</sub>{3,3}=rr{3,3}
| |
| | [[File:Regular_polygon_3.svg|30px]]<BR>[[Triangle|{3}]]
| |
| | [[File:Regular_polygon_4.svg|30px]]<BR>[[Square|{4}]]
| |
| | [[File:Regular_polygon_3.svg|30px]]<BR>[[Triangle|{3}]]
| |
| | 14
| |
| | 24
| |
| | 12
| |
| |- BGCOLOR="#e0f0e0" align=center
| |
| !5
| |
| |align=center|Truncated tetratetrahedron<BR>(Same as [[truncated octahedron]])
| |
| |[[File:3-simplex t012.svg|50px]]
| |
| |[[File:3-simplex t012 A2.svg|50px]]
| |
| | [[File:Uniform polyhedron-33-t012.png|50px]]
| |
| |[[File:Uniform tiling 332-t012.png|50px]]
| |
| |[[File:Truncated octahedron vertfig.png|50px]]
| |
| |align=center|{{CDD|node_1|3|node_1|3|node_1}}<BR>t<sub>0,1,2</sub>{3,3}=tr{3,3}
| |
| | [[File:Regular_polygon_6.svg|30px]]<BR>[[Hexagon|{6}]]
| |
| | [[File:Regular_polygon_4.svg|30px]]<BR>[[Square|{4}]]
| |
| | [[File:Regular_polygon_6.svg|30px]]<BR>[[Hexagon|{6}]]
| |
| | 14
| |
| | 36
| |
| | 24
| |
| |- BGCOLOR="#d0f0f0" align=center
| |
| !6
| |
| |align=center|Snub tetratetrahedron<BR>(Same as [[icosahedron]])
| |
| |[[File:icosahedron graph A3.png|50px]]
| |
| |[[File:icosahedron graph A2.png|50px]]
| |
| | [[File:Uniform polyhedron-33-s012.png|50px]]
| |
| | [[File:Spherical snub tetrahedron.png|50px]]
| |
| |[[File:Icosahedron vertfig.png|50px]]
| |
| |align=center|{{CDD|node_h|3|node_h|3|node_h}}<BR>sr{3,3}
| |
| | [[File:Regular_polygon_3.svg|30px]]<BR>[[Triangle|{3}]]
| |
| | [[File:Regular_polygon_3.svg|20px]][[File:Regular_polygon_3.svg|20px]]<BR>2 [[Triangle|{3}]]
| |
| | [[File:Regular_polygon_3.svg|30px]]<BR>[[Triangle|{3}]]
| |
| | 20
| |
| | 30
| |
| | 12
| |
| |}
| |
| | |
| === (4 3 2) O<sub>h</sub> Octahedral symmetry ===
| |
| | |
| The [[octahedral symmetry]] of the sphere generates 7 uniform polyhedra, and a 3 more by alternation. Four 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]] B<sub>2</sub> or [4,3], as well as a [[Coxeter-Dynkin diagram]]: {{CDD|node|4|node|3|node}}.
| |
| | |
| There are 48 triangles, visible in the faces of the [[disdyakis dodecahedron]] and alternately colored triangles on a sphere:
| |
| :[[File:Disdyakisdodecahedron.jpg|100px]] [[File:Octahedral reflection domains.png|100px]][[File:Sphere symmetry group oh.png|100px]]
| |
| | |
| {| class="wikitable"
| |
| !rowspan=2|#
| |
| !rowspan=2|Name
| |
| !rowspan=2|Graph<BR>B<sub>3</sub>
| |
| !rowspan=2|Graph<BR>B<sub>2</sub>
| |
| !rowspan=2|Picture
| |
| !rowspan=2|Tiling
| |
| !rowspan=2|[[Vertex figure|Vertex<BR>figure]]
| |
| !rowspan=2|[[Coxeter-Dynkin diagram|Coxeter-Dynkin]]<BR>and [[Schläfli symbol#Extended for uniform polychora and 3-space honeycombs|Schläfli]]<BR>symbols
| |
| !colspan=3|Face counts by position
| |
| !colspan=3|Element counts
| |
| |-
| |
| ! Pos. 2<BR>{{CDD|node|4|node|2}}<BR>[4]<BR>(8)
| |
| ! Pos. 1<BR>{{CDD|node|2|2|node}}<BR>[2]<BR>(12)
| |
| ! Pos. 0<BR>{{CDD|2|node|3|node}}<BR>[3]<BR>(6)
| |
| ! Faces
| |
| ! Edges
| |
| ! Vertices
| |
| |- BGCOLOR="#f0e0e0" align=center
| |
| !7
| |
| |align=center|[[Cube]]
| |
| |[[File:3-cube t0.svg|50px]]
| |
| |[[File:3-cube t0 B2.svg|50px]]
| |
| |[[File:Uniform polyhedron-43-t0.png|50px]]
| |
| |[[File:Uniform tiling 432-t0.png|50px]]
| |
| |[[File:Cube vertfig.png|50px]]
| |
| |align=center|{{CDD|node_1|4|node|3|node}}<BR>{4,3}
| |
| | [[File:Regular_polygon_4.svg|30px]]<BR>[[Square|{4}]]
| |
| |
| |
| |
| |
| | 6
| |
| | 12
| |
| | 8
| |
| |- BGCOLOR="#e0e0f0" align=center
| |
| ![2]
| |
| |align=center|[[Octahedron]]
| |
| |[[File:3-cube t2.svg|50px]]
| |
| |[[File:3-cube t2 B2.svg|50px]]
| |
| |[[File:Uniform polyhedron-43-t2.png|50px]]
| |
| |[[File:Uniform tiling 432-t2.png|50px]]
| |
| |[[File:Octahedron vertfig.png|50px]]
| |
| |align=center|{{CDD|node|4|node|3|node_1}}<BR>{3,4}
| |
| |
| |
| |
| |
| | [[File:Regular_polygon_3.svg|30px]]<BR>[[Triangle|{3}]]
| |
| | 8
| |
| | 12
| |
| | 6
| |
| |- BGCOLOR="#e0f0e0" align=center
| |
| ![4]
| |
| |align=center|rectified cube<BR>rectified octahedron<BR>([[Cuboctahedron]])
| |
| |[[File:3-cube t1.svg|50px]]
| |
| |[[File:3-cube t1 B2.svg|50px]]
| |
| |[[File:Uniform polyhedron-43-t1.png|50px]]
| |
| |[[File:Uniform tiling 432-t1.png|50px]]
| |
| |[[File:Cuboctahedron vertfig.png|50px]]
| |
| |align=center|{{CDD|node|4|node_1|3|node}}<BR>{4,3}
| |
| | [[File:Regular_polygon_4.svg|30px]]<BR>[[Square|{4}]]
| |
| |
| |
| | [[File:Regular_polygon_3.svg|30px]]<BR>[[Triangle|{3}]]
| |
| | 14
| |
| | 24
| |
| | 12
| |
| |- BGCOLOR="#f0e0e0" align=center
| |
| !8
| |
| |align=center|[[Truncated cube]]
| |
| |[[File:3-cube t01.svg|50px]]
| |
| |[[File:3-cube t01 B2.svg|50px]]
| |
| |[[File:Uniform polyhedron-43-t01.png|50px]]
| |
| |[[File:Uniform tiling 432-t01.png|50px]]
| |
| |[[File:Truncated cube vertfig.png|50px]]
| |
| |align=center|{{CDD|node_1|4|node_1|3|node}}<BR>t<sub>0,1</sub>{4,3}=t{4,3}
| |
| | [[File:Regular_polygon_8.svg|30px]]<BR>[[Octagon|{8}]]
| |
| |
| |
| | [[File:Regular_polygon_3.svg|30px]]<BR>[[Triangle|{3}]]
| |
| | 14
| |
| | 36
| |
| | 24
| |
| |- BGCOLOR="#e0e0f0" align=center
| |
| ![5]
| |
| |align=center|[[Truncated octahedron]]
| |
| |[[File:3-cube t12.svg|50px]]
| |
| |[[File:3-cube t12 B2.svg|50px]]
| |
| |[[File:Uniform polyhedron-43-t12.png|50px]]
| |
| |[[File:Uniform tiling 432-t12.png|50px]]
| |
| |[[File:Truncated octahedron vertfig.png|50px]]
| |
| |align=center|{{CDD|node|4|node_1|3|node_1}}<BR>t<sub>0,1</sub>{3,4}=t{3,4}
| |
| | [[File:Regular_polygon_4.svg|30px]]<BR>[[Square|{4}]]
| |
| |
| |
| | [[File:Regular_polygon_6.svg|30px]]<BR>[[Hexagon|{6}]]
| |
| | 14
| |
| | 36
| |
| | 24
| |
| |- BGCOLOR="#e0f0e0" align=center
| |
| !9
| |
| |align=center|Cantellated cube<BR>cantellated octahedron<BR>[[Rhombicuboctahedron]]
| |
| |[[File:3-cube t02.svg|50px]]
| |
| |[[File:3-cube t02 B2.svg|50px]]
| |
| |[[File:Uniform polyhedron-43-t02.png|50px]]
| |
| |[[File:Uniform tiling 432-t02.png|50px]]
| |
| |[[File:Small rhombicuboctahedron vertfig.png|50px]]
| |
| |align=center|{{CDD|node_1|4|node|3|node_1}}<BR>t<sub>0,2</sub>{4,3}=rr{4,3}
| |
| | [[File:Regular_polygon_8.svg|30px]]<BR>[[Octagon|{8}]]
| |
| | [[File:Regular_polygon_4.svg|30px]]<BR>[[Square|{4}]]
| |
| | [[File:Regular_polygon_6.svg|30px]]<BR>[[Hexagon|{6}]]
| |
| | 26
| |
| | 48
| |
| | 24
| |
| |- BGCOLOR="#e0f0e0" align=center
| |
| !10
| |
| |align=center|Omnitruncated cube<BR>omnitruncated octahedron<BR>[[Truncated cuboctahedron]]
| |
| |[[File:3-cube t012.svg|50px]]
| |
| |[[File:3-cube t012 B2.svg|50px]]
| |
| |[[File:Uniform polyhedron-43-t012.png|50px]]
| |
| |[[File:Uniform tiling 432-t012.png|50px]]
| |
| |[[File:Great_rhombicuboctahedron_vertfig.png|50px]]
| |
| |align=center|{{CDD|node_1|4|node_1|3|node_1}}<BR>t<sub>0,1,2</sub>{4,3}=tr{4,3}
| |
| | [[File:Regular_polygon_8.svg|30px]]<BR>[[Octagon|{8}]]
| |
| | [[File:Regular_polygon_4.svg|30px]]<BR>[[Square|{4}]]
| |
| | [[File:Regular_polygon_6.svg|30px]]<BR>[[Hexagon|{6}]]
| |
| | 26
| |
| | 72
| |
| | 48
| |
| |- BGCOLOR="#d0f0f0" align=center
| |
| ![6]
| |
| |align=center|Alternated truncated octahedron<BR>(Same as [[Icosahedron]])
| |
| |[[File:3-cube h01.svg|50px]]
| |
| |[[File:3-cube h01 B2.svg|50px]]
| |
| |[[File:Uniform polyhedron-43-h01.png|50px]]
| |
| |[[File:Spherical alternated truncated octahedron.png|50px]]
| |
| |[[File:Icosahedron vertfig.png|50px]]
| |
| |align=center|{{CDD|node|4|node_h|3|node_h}}<BR>= {{CDD|nodes_hh|split2|node_h}}<BR>s{3,4}=sr{3,3}
| |
| |
| |
| | [[File:Regular_polygon_3.svg|30px]]<BR>[[Triangle|{3}]]
| |
| | [[File:Regular_polygon_3.svg|30px]]<BR>[[Triangle|{3}]]
| |
| | 20
| |
| | 30
| |
| | 12
| |
| | |
| |- BGCOLOR="#d0f0f0" align=center
| |
| ![1]
| |
| |align=center|Half cube<BR>(Same as [[tetrahedron]])
| |
| |[[File:3-simplex t0 A2.svg|50px]]
| |
| |[[File:3-simplex t0.svg|50px]]
| |
| |[[File:Uniform polyhedron-33-t2.png|50px]]
| |
| |[[File:Uniform tiling 332-t2.png|50px]]
| |
| |[[File:Tetrahedron vertfig.png|50px]]
| |
| |align=center|{{CDD|node_h1|4|node|3|node}}<BR>= {{CDD|nodes_10ru|split2|node}}<BR>h{4,3}={3,3}
| |
| | [[File:Regular_polygon_3.svg|30px]]<BR><sup>1</sup>/<sub>2</sub> [[Triangle|{3}]]
| |
| |
| |
| |
| |
| | 4
| |
| | 6
| |
| | 4
| |
| | |
| |- BGCOLOR="#d0f0f0" align=center
| |
| ![2]
| |
| |align=center|Cantic cube<BR>(Same as [[Truncated tetrahedron]])
| |
| |[[File:3-simplex t01 A2.svg|50px]]
| |
| |[[File:3-simplex t01.svg|50px]]
| |
| |[[File:Uniform polyhedron-33-t12.png|50px]]
| |
| |[[File:Uniform tiling 332-t12.png|50px]]
| |
| |[[File:Truncated tetrahedron vertfig.png|50px]]
| |
| |align=center|{{CDD|node_h1|4|node|3|node_1}}<BR>= {{CDD|nodes_10ru|split2|node_1}}<BR>h<sub>2</sub>{4,3}=t{3,3}
| |
| | [[File:Regular_polygon_6.svg|30px]]<BR> <sup>1</sup>/<sub>2</sub> [[Hexagon|{6}]]
| |
| |
| |
| | [[File:Regular_polygon_3.svg|30px]]<BR><sup>1</sup>/<sub>2</sub> [[Triangle|{3}]]
| |
| | 8
| |
| | 18
| |
| | 12
| |
| | |
| |- BGCOLOR="#d0f0f0" align=center
| |
| !11
| |
| |align=center|[[Snub cube]]
| |
| |[[File:Snub cube_A2.png|50px]]
| |
| |[[File:Snub cube_B2.png|50px]]
| |
| |[[File:Uniform polyhedron-43-s012.png|50px]]
| |
| |[[File:Spherical snub cube.png|50px]]
| |
| |[[File:Snub cube vertfig.png|50px]]
| |
| |align=center|{{CDD|node_h|4|node_h|3|node_h}}<BR>sr{4,3}
| |
| | [[File:Regular_polygon_4.svg|30px]]<BR>[[Square|{4}]]
| |
| | [[File:Regular_polygon_3.svg|20px]][[File:Regular_polygon_3.svg|20px]]<BR>2 [[Triangle|{3}]]
| |
| | [[File:Regular_polygon_3.svg|30px]]<BR>[[Triangle|{3}]]
| |
| | 38
| |
| | 60
| |
| | 24
| |
| |}
| |
| | |
| === (5 3 2) I<sub>h</sub> Icosahedral symmetry ===
| |
| | |
| 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]] G<sub>2</sub> or [5,3], as well as a [[Coxeter-Dynkin diagram]]: {{CDD|node|5|node|3|node}}.
| |
| | |
| There are 120 triangles, visible in the faces of the [[disdyakis triacontahedron]] and alternately colored triangles on a sphere:
| |
| :[[File:Disdyakistriacontahedron.jpg|100px]] [[File:Icosahedral reflection domains.png|100px]][[File:Sphere symmetry group ih.png|100px]]
| |
| | |
| {| class="wikitable"
| |
| !rowspan=2|#
| |
| !rowspan=2|Name
| |
| !rowspan=2|Graph<BR>(A<sub>2</sub>)<BR>[6]
| |
| !rowspan=2|Graph<BR>(H<sub>3</sub>)<BR>[10]
| |
| !rowspan=2|Picture
| |
| !rowspan=2|Tiling
| |
| !rowspan=2|[[Vertex figure|Vertex<BR>figure]]
| |
| !rowspan=2|[[Coxeter-Dynkin diagram|Coxeter-Dynkin]]<BR>and [[Schläfli symbol#Extended for uniform polychora and 3-space honeycombs|Schläfli]]<BR>symbols
| |
| !colspan=3|Face counts by position
| |
| !colspan=3|Element counts
| |
| |-
| |
| ! Pos. 2<BR>{{CDD|node|5|node|2}}<BR>[5]<BR>(12)
| |
| ! Pos. 1<BR>{{CDD|node|2|node}}<BR>[2]<BR>(30)
| |
| ! Pos. 0<BR>{{CDD|2|node|3|node}}<BR>[3]<BR>(20)
| |
| ! Faces
| |
| ! Edges
| |
| ! Vertices
| |
| |- BGCOLOR="#f0e0e0" align=center
| |
| !12
| |
| |align=center|[[Dodecahedron]]
| |
| |[[Image:Dodecahedron t0 A2.png|50px]]
| |
| |[[Image:Dodecahedron t0 H3.png|50px]]
| |
| |[[File:Uniform polyhedron-53-t0.png|50px]]
| |
| |[[File:Uniform tiling 532-t0.png|50px]]
| |
| |[[File:Dodecahedron vertfig.png|50px]]
| |
| |align=center|{{CDD|node_1|5|node|3|node}}<BR>{5,3}
| |
| | [[File:Regular_polygon_5.svg|30px]]<BR>[[Pentagon|{5}]]
| |
| |
| |
| |
| |
| | 12
| |
| | 30
| |
| | 20
| |
| |- BGCOLOR="#e0e0f0" align=center
| |
| ![6]
| |
| |align=center|[[Icosahedron]]
| |
| |[[Image:Icosahedron t0 A2.png|50px]]
| |
| |[[Image:Icosahedron t0 H3.png|50px]]
| |
| |[[File:Uniform polyhedron-53-t2.png|50px]]
| |
| |[[File:Uniform tiling 532-t2.png|50px]]
| |
| |[[File:Icosahedron vertfig.png|50px]]
| |
| |align=center|{{CDD|node|5|node|3|node_1}}<BR>{3,5}
| |
| |
| |
| |
| |
| | [[File:Regular_polygon_3.svg|30px]]<BR>[[Triangle|{3}]]
| |
| | 20
| |
| | 30
| |
| | 12
| |
| |- BGCOLOR="#e0f0e0" align=center
| |
| !13
| |
| |align=center|Rectified dodecahedron<BR>Rectified icosahedron<BR>[[Icosidodecahedron]]
| |
| |[[Image:Dodecahedron t1 A2.png|50px]]
| |
| |[[Image:Dodecahedron t1 H3.png|50px]]
| |
| |[[File:Uniform polyhedron-53-t1.png|50px]]
| |
| |[[File:Uniform tiling 532-t1.png|50px]]
| |
| |[[File:Icosidodecahedron vertfig.png|50px]]
| |
| |align=center|{{CDD|node|5|node_1|3|node}}<BR>t<sub>1</sub>{5,3}=r{5,3}
| |
| | [[File:Regular_polygon_5.svg|30px]]<BR>[[Pentagon|{5}]]
| |
| |
| |
| | [[File:Regular_polygon_3.svg|30px]]<BR>[[Triangle|{3}]]
| |
| | 32
| |
| | 60
| |
| | 30
| |
| |- BGCOLOR="#f0e0e0" align=center
| |
| !14
| |
| |align=center|[[Truncated dodecahedron]]
| |
| |[[Image:Dodecahedron t01 A2.png|50px]]
| |
| |[[Image:Dodecahedron t01 H3.png|50px]]
| |
| |[[File:Uniform polyhedron-53-t01.png|50px]]
| |
| |[[File:Uniform tiling 532-t01.png|50px]]
| |
| |[[File:Truncated dodecahedron vertfig.png|50px]]
| |
| |align=center|{{CDD|node_1|5|node_1|3|node}}<BR>t<sub>0,1</sub>{5,3}=t{5,3}
| |
| | [[File:Regular_polygon_5.svg|30px]]<BR>[[Decagon|{10}]]
| |
| |
| |
| | [[File:Regular_polygon_3.svg|30px]]<BR>[[Triangle|{3}]]
| |
| | 32
| |
| | 90
| |
| | 60
| |
| |- BGCOLOR="#e0e0f0" align=center
| |
| !15
| |
| |align=center|[[Truncated icosahedron]]
| |
| |[[Image:Icosahedron t01 A2.png|50px]]
| |
| |[[Image:Icosahedron t01 H3.png|50px]]
| |
| |[[File:Uniform polyhedron-53-t12.png|50px]]
| |
| |[[File:Uniform tiling 532-t12.png|50px]]
| |
| |[[File:Truncated icosahedron vertfig.png|50px]]
| |
| |align=center|{{CDD|node|5|node_1|3|node_1}}<BR>t<sub>0,1</sub>{3,5}=t{3,5}
| |
| | [[File:Regular_polygon_5.svg|30px]]<BR>[[Pentagon|{5}]]
| |
| |
| |
| | [[File:Regular_polygon_6.svg|30px]]<BR>[[Hexagon|{6}]]
| |
| | 32
| |
| | 90
| |
| | 60
| |
| |- BGCOLOR="#e0f0e0" align=center
| |
| !16
| |
| |align=center|Cantellated dodecahedron<BR>Cantellated icosahedron<BR>[[Rhombicosidodecahedron]]
| |
| |[[Image:Dodecahedron t02 A2.png|50px]]
| |
| |[[Image:Dodecahedron t02 H3.png|50px]]
| |
| |[[File:Uniform polyhedron-53-t02.png|50px]]
| |
| |[[File:Uniform tiling 532-t02.png|50px]]
| |
| |[[File:Small rhombicosidodecahedron vertfig.png|50px]]
| |
| |align=center|{{CDD|node_1|5|node|3|node_1}}<BR>t<sub>0,2</sub>{5,3}=rr{5,3}
| |
| | [[File:Regular_polygon_5.svg|30px]]<BR>[[Pentagon|{5}]]
| |
| | [[File:Regular_polygon_4.svg|30px]]<BR>[[Square|{4}]]
| |
| | [[File:Regular_polygon_3.svg|30px]]<BR>[[Triangle|{3}]]
| |
| | 62
| |
| | 120
| |
| | 60
| |
| |- BGCOLOR="#e0f0e0" align=center
| |
| !17
| |
| |align=center|Omnitruncated dodecahedron<BR>Omnitruncated icosahedron<BR>[[Truncated icosidodecahedron]]
| |
| |[[Image:Dodecahedron t012 A2.png|50px]]
| |
| |[[Image:Dodecahedron t012 H3.png|50px]]
| |
| |[[File:Uniform polyhedron-53-t012.png|50px]]
| |
| |[[File:Uniform tiling 532-t012.png|50px]]
| |
| |[[File:Great rhombicosidodecahedron vertfig.png|50px]]
| |
| |align=center|{{CDD|node_1|5|node_1|3|node_1}}<BR>t<sub>0,1,2</sub>{5,3}=tr{5,3}
| |
| | [[File:Regular_polygon_10.svg|30px]]<BR>[[Decagon|{10}]]
| |
| | [[File:Regular_polygon_4.svg|30px]]<BR>[[Square|{4}]]
| |
| | [[File:Regular_polygon_6.svg|30px]]<BR>[[Hexagon|{6}]]
| |
| | 62
| |
| | 180
| |
| | 120
| |
| |- BGCOLOR="#d0f0f0" align=center
| |
| !18
| |
| |align=center|[[Snub dodecahedron]]<BR>Snub icosahedron
| |
| |[[Image:Snub dodecahedron A2.png|50px]]
| |
| |[[Image:Snub dodecahedron H2.png|50px]]
| |
| |[[File:Uniform polyhedron-53-s012.png|50px]]
| |
| |[[File:Spherical snub dodecahedron.png|50px]]
| |
| |[[File:Snub dodecahedron vertfig.png|50px]]
| |
| |align=center|{{CDD|node_h|5|node_h|3|node_h}}<BR>sr{5,3}
| |
| | [[File:Regular_polygon_5.svg|30px]]<BR>[[Pentagon|{5}]]
| |
| | [[File:Regular_polygon_3.svg|20px]][[File:Regular_polygon_3.svg|20px]]<BR>2 [[Triangle|{3}]]
| |
| | [[File:Regular_polygon_3.svg|30px]]<BR>[[Triangle|{3}]]
| |
| | 92
| |
| | 150
| |
| | 60
| |
| |}
| |
| | |
| === (p 2 2) Prismatic [p,2], I<sub>2</sub>(p) family (D<sub>''p''h</sub> Dihedral symmetry) ===
| |
| | |
| {{main|Prismatic uniform polyhedron}}
| |
| | |
| The [[dihedral symmetry]] of the sphere generates two infinite sets of uniform polyhedra, prisms and antiprisms, and two more infinite set of degenerate polygons, the hosohedrons and dihedrons which exists 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]] I<sub>2</sub>(p) or [n,2], as well as a prismatic [[Coxeter-Dynkin diagram]]: {{CDD|node|p|node|2|node}}.
| |
| | |
| Below are the first five dihedral symmetries: D<sub>2</sub> ... D<sub>6</sub>. The dihedral symmetry D<sub>p</sub> 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:
| |
| :[[File:Octahedron.svg|100px]] [[File:Sphere symmetry group d2h.png|100px]]
| |
| | |
| {| class="wikitable"
| |
| !rowspan=2|#
| |
| !rowspan=2|Name
| |
| !rowspan=2|Picture
| |
| !rowspan=2|Tiling
| |
| !rowspan=2|[[Vertex figure|Vertex<BR>figure]]
| |
| !rowspan=2|[[Coxeter-Dynkin diagram|Coxeter-Dynkin]]<BR>and [[Schläfli symbol#Extended for uniform polychora and 3-space honeycombs|Schläfli]]<BR>symbols
| |
| !colspan=3|Face counts by position
| |
| !colspan=3|Element counts
| |
| |-
| |
| ! Pos. 2<BR>{{CDD|node|2|node|2|}}<BR>[2]<BR>(2)
| |
| ! Pos. 1<BR>{{CDD|node|2|node}}<BR>[2]<BR>(2)
| |
| ! Pos. 0<BR>{{CDD|2|node|2|node}}<BR>[2]<BR>(2)
| |
| ! Faces
| |
| ! Edges
| |
| ! Vertices
| |
| |- BGCOLOR="#f0e0e0"
| |
| !D<sub>2</sub><BR>H<sub>2</sub>
| |
| |align=center|[[digonal dihedron]]<BR>[[digonal hosohedron]]
| |
| |
| |
| |[[File:digonal dihedron.png|50px]]
| |
| |
| |
| |align=center|{{CDD|node_1|2|node|2|node}}<BR>{2,2}
| |
| | [[File:Regular digon in spherical geometry-2.svg|30px]]<BR>[[Digon|{2}]]
| |
| |
| |
| |
| |
| | 2
| |
| | 2
| |
| | 2
| |
| |- BGCOLOR="#e0f0e0"
| |
| !D<sub>4</sub>
| |
| |align=center|truncated digonal dihedron<BR>(Same as [[square dihedron]])
| |
| |
| |
| |
| |
| |
| |
| |align=center|{{CDD|node_1|2|node_1|2|node}}<BR>t{2,2}={4,2}
| |
| | [[File:Regular_polygon_4.svg|30px]]<BR>[[Square|{4}]]
| |
| |
| |
| |
| |
| | 2
| |
| | 4
| |
| | 4
| |
| |- BGCOLOR="#f0e0e0"
| |
| !P<sub>4</sub><BR>[7]
| |
| |align=center|omnitruncated digonal dihedron<BR>(Same as [[cube]])
| |
| |[[File:Uniform polyhedron 222-t012.png|50px]]
| |
| |
| |
| |[[File:Cube vertfig.png|50px]]
| |
| |align=center|{{CDD|node_1|2|node_1|2|node_1}}<BR>t<sub>0,1,2</sub>{2,2}=tr{2,2}
| |
| | [[File:Regular_polygon_4.svg|30px]]<BR>[[Square|{4}]]
| |
| | [[File:Regular_polygon_4.svg|30px]]<BR>[[Square|{4}]]
| |
| | [[File:Regular_polygon_4.svg|30px]]<BR>[[Square|{4}]]
| |
| | 6
| |
| | 12
| |
| | 8
| |
| |- BGCOLOR="#d0f0f0"
| |
| !A<sub>2</sub><BR>[1]
| |
| |align=center|snub digonal dihedron<BR>(Same as [[tetrahedron]])
| |
| |[[File:Uniform polyhedron-33-t2.png|50px]]
| |
| |
| |
| |[[File:Tetrahedron vertfig.png|50px]]
| |
| |align=center|{{CDD|node_h|2x|node_h|2x|node_h}}<BR>sr{2,2}
| |
| |
| |
| | [[File:Regular_polygon_3.svg|20px]][[File:Regular_polygon_3.svg|20px]]<BR>2 [[Triangle|{3}]]
| |
| |
| |
| | 4
| |
| | 6
| |
| | 4
| |
| |}
| |
| | |
| ==== (3 2 2) D<sub>3h</sub>dihedral symmetry ====
| |
| | |
| There are 12 fundamental triangles, visible in the faces of the [[hexagonal bipyramid]] and alternately colored triangles on a sphere:
| |
| :[[File:Hexagonale bipiramide.png|100px]] [[File:Sphere symmetry group d3h.png|100px]]
| |
| | |
| {| class="wikitable"
| |
| !rowspan=2|#
| |
| !rowspan=2|Name
| |
| !rowspan=2|Picture
| |
| !rowspan=2|Tiling
| |
| !rowspan=2|[[Vertex figure|Vertex<BR>figure]]
| |
| !rowspan=2|[[Coxeter-Dynkin diagram|Coxeter-Dynkin]]<BR>and [[Schläfli symbol#Extended for uniform polychora and 3-space honeycombs|Schläfli]]<BR>symbols
| |
| !colspan=3|Face counts by position
| |
| !colspan=3|Element counts
| |
| |-
| |
| ! Pos. 2<BR>{{CDD|node|3|node|2}}<BR>[3]<BR>(2)
| |
| ! Pos. 1<BR>{{CDD|node|2|node}}<BR>[2]<BR>(3)
| |
| ! Pos. 0<BR>{{CDD|2|node|2|node}}<BR>[2]<BR>(3)
| |
| ! Faces
| |
| ! Edges
| |
| ! Vertices
| |
| |- BGCOLOR="#f0e0e0"
| |
| !D<sub>3</sub>
| |
| |align=center|[[Trigonal dihedron]]
| |
| |
| |
| |[[File:Trigonal dihedron.png|50px]]
| |
| |
| |
| |align=center|{{CDD|node_1|3|node|2|node}}<BR>{3,2}
| |
| | [[File:Regular_polygon_3.svg|30px]]<BR>[[Triangle|{3}]]
| |
| |
| |
| |
| |
| | 2
| |
| | 3
| |
| | 3
| |
| |- BGCOLOR="#e0e0f0"
| |
| !H<sub>3</sub>
| |
| |align=center|[[Trigonal hosohedron]]
| |
| |
| |
| |[[File:Trigonal hosohedron.png|50px]]
| |
| |
| |
| |align=center|{{CDD|node|3|node|2|node_1}}<BR>{2,3}
| |
| |
| |
| |
| |
| | [[File:Regular digon in spherical geometry-2.svg|30px]]<BR>[[Digon|{2}]]
| |
| | 3
| |
| | 3
| |
| | 2
| |
| |- BGCOLOR="#e0f0e0"
| |
| !D<sub>6</sub>
| |
| |align=center|Truncated trigonal dihedron<BR>(Same as [[hexagonal dihedron]])
| |
| |
| |
| |[[File:Hexagonal dihedron.png|50px]]
| |
| |
| |
| |align=center|{{CDD|node_1|3|node_1|2|node}}<BR>t{3,2}
| |
| | [[File:Regular_polygon_6.svg|30px]]<BR>[[Hexagon|{6}]]
| |
| |
| |
| |
| |
| | 2
| |
| | 6
| |
| | 6
| |
| |- BGCOLOR="#e0e0f0"
| |
| !P<sub>3</sub>
| |
| |align=center|Truncated trigonal hosohedron<BR>([[Triangular prism]])
| |
| |[[File:Triangular prism.png|50px]]
| |
| |
| |
| |[[File:Triangular prism vertfig.png|50px]]
| |
| |align=center|{{CDD|node|3|node_1|2|node_1}}<BR>t{2,3}
| |
| | [[File:Regular_polygon_3.svg|30px]]<BR>[[Triangle|{3}]]
| |
| | [[File:Regular_polygon_4.svg|30px]]<BR>[[Square|{4}]]
| |
| |
| |
| | 5
| |
| | 9
| |
| | 6
| |
| |- BGCOLOR="#e0f0e0"
| |
| !P<sub>6</sub>
| |
| |align=center|Omnitruncated trigonal dihedron<BR>([[Hexagonal prism]])
| |
| |[[File:Hexagonal prism.png|50px]]
| |
| |
| |
| |[[File:Hexagonal prism vertfig.png|50px]]
| |
| |align=center|{{CDD|node_1|3|node_1|2|node_1}}<BR>t<sub>0,1,2</sub>{2,3}=tr{2,3}
| |
| | [[File:Regular_polygon_6.svg|30px]]<BR>[[Hexagon|{6}]]
| |
| | [[File:Regular_polygon_4.svg|30px]]<BR>[[Square|{4}]]
| |
| | [[File:Regular_polygon_4.svg|30px]]<BR>[[Square|{4}]]
| |
| | 8
| |
| | 18
| |
| | 12
| |
| |- BGCOLOR="#d0f0f0"
| |
| !A<sub>3</sub><BR>[2]
| |
| |align=center|Snub trigonal dihedron<BR>(Same as [[Triangular antiprism]])<BR>(Same as [[octahedron]])
| |
| |[[File:Trigonal antiprism.png|50px]]
| |
| |
| |
| |[[File:Octahedron vertfig.png|50px]]
| |
| |align=center|{{CDD|node_h|3|node_h|2x|node_h}}<BR>sr{2,3}
| |
| | [[File:Regular_polygon_3.svg|30px]]<BR>[[Triangle|{3}]]
| |
| | [[File:Regular_polygon_3.svg|20px]][[File:Regular_polygon_3.svg|20px]]<BR>2 [[Triangle|{3}]]
| |
| |
| |
| | 8
| |
| | 12
| |
| | 6
| |
| |}
| |
| | |
| ==== (4 2 2) D<sub>4h</sub>dihedral symmetry ====
| |
| | |
| There are 16 fundamental triangles, visible in the faces of the [[octagonal bipyramid]] and alternately colored triangles on a sphere:
| |
| :[[File:Octagonal bipyramid.png|80px]]
| |
| | |
| {| class="wikitable"
| |
| !rowspan=2|#
| |
| !rowspan=2|Name
| |
| !rowspan=2|Picture
| |
| !rowspan=2|Tiling
| |
| !rowspan=2|[[Vertex figure|Vertex<BR>figure]]
| |
| !rowspan=2|[[Coxeter-Dynkin diagram|Coxeter-Dynkin]]<BR>and [[Schläfli symbol#Extended for uniform polychora and 3-space honeycombs|Schläfli]]<BR>symbols
| |
| !colspan=3|Face counts by position
| |
| !colspan=3|Element counts
| |
| |-
| |
| ! Pos. 2<BR>{{CDD|node|4|node|2}}<BR>[4]<BR>(2)
| |
| ! Pos. 1<BR>{{CDD|node|2|node}}<BR>[2]<BR>(4)
| |
| ! Pos. 0<BR>{{CDD|2|node|2|node}}<BR>[2]<BR>(4)
| |
| ! Faces
| |
| ! Edges
| |
| ! Vertices
| |
| |- BGCOLOR="#f0e0e0"
| |
| !D<sub>4</sub>
| |
| |align=center|[[square dihedron]]
| |
| |
| |
| |
| |
| |
| |
| |align=center|{{CDD|node_1|4|node|2|node}}<BR>{4,2}
| |
| | [[File:Regular_polygon_4.svg|30px]]<BR>[[Square|{4}]]
| |
| |
| |
| |
| |
| | 2
| |
| | 4
| |
| | 4
| |
| |- BGCOLOR="#e0e0f0"
| |
| !H<sub>4</sub>
| |
| |align=center|[[square hosohedron]]
| |
| |
| |
| |
| |
| |
| |
| |align=center|{{CDD|node|4|node|2|node_1}}<BR>{2,4}
| |
| |
| |
| |
| |
| | [[File:Regular digon in spherical geometry-2.svg|30px]]<BR>[[Digon|{2}]]
| |
| | 4
| |
| | 4
| |
| | 2
| |
| |- BGCOLOR="#e0f0e0"
| |
| !D<sub>8</sub>
| |
| |align=center|Truncated square dihedron<BR>(Same as [[octagonal dihedron]])
| |
| |
| |
| |
| |
| |
| |
| |align=center|{{CDD|node_1|4|node_1|2|node}}<BR>t{4,2}
| |
| | [[File:Regular_polygon_8.svg|30px]]<BR>[[Octagon|{8}]]
| |
| |
| |
| |
| |
| | 2
| |
| | 8
| |
| | 8
| |
| |- BGCOLOR="#e0e0f0"
| |
| !P<sub>4</sub><BR>[7]
| |
| |align=center|Truncated square hosohedron<BR>([[Cube]])
| |
| |[[File:Tetragonal prism.png|50px]]
| |
| |
| |
| |[[File:Cube vertfig.png|50px]]
| |
| |align=center|{{CDD|node|4|node_1|2|node_1}}<BR>t{2,4}
| |
| | [[File:Regular_polygon_4.svg|30px]]<BR>[[Square|{4}]]
| |
| | [[File:Regular_polygon_4.svg|30px]]<BR>[[Square|{4}]]
| |
| |
| |
| | 6
| |
| | 12
| |
| | 8
| |
| |- BGCOLOR="#e0f0e0"
| |
| !D<sub>8</sub>
| |
| |align=center|Omnitruncated square dihedron<BR>([[Octagonal prism]])
| |
| |[[File:Octagonal prism.png|50px]]
| |
| |
| |
| |[[File:Octagonal prism vertfig.png|50px]]
| |
| |align=center|{{CDD|node_1|4|node_1|2|node_1}}<BR>t<sub>0,1,2</sub>{2,4}=tr{2,4}
| |
| | [[File:Regular_polygon_8.svg|30px]]<BR>[[Octagon|{8}]]
| |
| | [[File:Regular_polygon_4.svg|30px]]<BR>[[Square|{4}]]
| |
| | [[File:Regular_polygon_4.svg|30px]]<BR>[[Square|{4}]]
| |
| | 10
| |
| | 24
| |
| | 16
| |
| |- BGCOLOR="#d0f0f0"
| |
| !A<sub>4</sub>
| |
| |align=center|Snub square dihedron<BR>([[Square antiprism]])
| |
| |[[File:Square antiprism.png|50px]]
| |
| |
| |
| |[[File:Square antiprism vertfig.png|50px]]
| |
| |align=center|{{CDD|node_h|4|node_h|2x|node_h}}<BR>sr{2,4}
| |
| | [[File:Regular_polygon_4.svg|30px]]<BR>[[Square|{4}]]
| |
| | [[File:Regular_polygon_3.svg|20px]][[File:Regular_polygon_3.svg|20px]]<BR>2 [[Triangle|{3}]]
| |
| |
| |
| | 10
| |
| | 16
| |
| | 8
| |
| |}
| |
| | |
| ==== (5 2 2) D<sub>5h</sub> dihedral symmetry ====
| |
| | |
| There are 20 fundamental triangles, visible in the faces of the [[decagonal bipyramid]] and alternately colored triangles on a sphere:
| |
| :[[File:Decagonal bipyramid.png|60px]]
| |
| | |
| {| class="wikitable"
| |
| !rowspan=2|#
| |
| !rowspan=2|Name
| |
| !rowspan=2|Picture
| |
| !rowspan=2|Tiling
| |
| !rowspan=2|[[Vertex figure|Vertex<BR>figure]]
| |
| !rowspan=2|[[Coxeter-Dynkin diagram|Coxeter-Dynkin]]<BR>and [[Schläfli symbol#Extended for uniform polychora and 3-space honeycombs|Schläfli]]<BR>symbols
| |
| !colspan=3|Face counts by position
| |
| !colspan=3|Element counts
| |
| |-
| |
| ! Pos. 2<BR>{{CDD|node|5|node|2}}<BR>[5]<BR>(2)
| |
| ! Pos. 1<BR>{{CDD|node|2|node}}<BR>[2]<BR>(5)
| |
| ! Pos. 0<BR>{{CDD|2|node|2|node}}<BR>[2]<BR>(5)
| |
| ! Faces
| |
| ! Edges
| |
| ! Vertices
| |
| |- BGCOLOR="#f0e0e0"
| |
| !D<sub>5</sub>
| |
| |align=center|[[Pentagonal dihedron]]
| |
| |
| |
| |
| |
| |
| |
| |align=center|{{CDD|node_1|5|node|2|node}}<BR>{5,2}
| |
| | [[File:Regular_polygon_5.svg|30px]]<BR>[[Pentagon|{5}]]
| |
| |
| |
| |
| |
| | 2
| |
| | 5
| |
| | 5
| |
| |- BGCOLOR="#e0e0f0"
| |
| !H<sub>5</sub>
| |
| |align=center|[[Pentagonal hosohedron]]
| |
| |
| |
| |
| |
| |
| |
| |align=center|{{CDD|node|5|node|2|node_1}}<BR>{2,5}
| |
| |
| |
| |
| |
| | [[File:Regular digon in spherical geometry-2.svg|30px]]<BR>[[Digon|{2}]]
| |
| | 5
| |
| | 5
| |
| | 2
| |
| |- BGCOLOR="#e0f0e0"
| |
| !D<sub>10</sub>
| |
| |align=center|Truncated pentagonal dihedron<BR>(Same as [[decagonal dihedron]])
| |
| |
| |
| |
| |
| |
| |
| |align=center|{{CDD|node_1|5|node_1|2|node}}<BR>t{5,2}
| |
| | [[File:Regular_polygon_10.svg|30px]]<BR>[[Decagon|{10}]]
| |
| |
| |
| |
| |
| | 2
| |
| | 10
| |
| | 10
| |
| |- BGCOLOR="#e0e0f0"
| |
| !P<sub>5</sub>
| |
| |align=center|Truncated pentagonal hosohedron<BR>(Same as [[pentagonal prism]])
| |
| |[[File:Pentagonal prism.png|50px]]
| |
| |
| |
| |[[File:Pentagonal prism vertfig.png|50px]]
| |
| |align=center|{{CDD|node|5|node_1|2|node_1}}<BR>t{2,5}
| |
| | [[File:Regular_polygon_5.svg|30px]]<BR>[[Pentagon|{5}]]
| |
| | [[File:Regular_polygon_4.svg|30px]]<BR>[[Square|{4}]]
| |
| |
| |
| | 7
| |
| | 15
| |
| | 10
| |
| |- BGCOLOR="#e0f0e0"
| |
| !P<sub>10</sub>
| |
| |align=center|Omnitruncated pentagonal dihedron<BR>([[Decagonal prism]])
| |
| |[[File:Decagonal prism.png|50px]]
| |
| |
| |
| |[[File:Decagonal prism vf.png|50px]]
| |
| |align=center|{{CDD|node_1|5|node_1|2|node_1}}<BR>t<sub>0,1,2</sub>{2,5}=tr{2,5}
| |
| | [[File:Regular_polygon_10.svg|30px]]<BR>[[Decagon|{10}]]
| |
| | [[File:Regular_polygon_4.svg|30px]]<BR>[[Square|{4}]]
| |
| | [[File:Regular_polygon_4.svg|30px]]<BR>[[Square|{4}]]
| |
| | 12
| |
| | 30
| |
| | 20
| |
| |- BGCOLOR="#d0f0f0"
| |
| !A<sub>5</sub>
| |
| |align=center|Snub pentagonal dihedron<BR>([[Pentagonal antiprism]])
| |
| |[[File:Pentagonal antiprism.png|50px]]
| |
| |
| |
| |[[File:Pentagonal antiprism vertfig.png|50px]]
| |
| |align=center|{{CDD|node_h|5|node_h|2x|node_h}}<BR>sr{2,5}
| |
| | [[File:Regular_polygon_5.svg|30px]]<BR>[[Pentagon|{5}]]
| |
| | [[File:Regular_polygon_3.svg|20px]][[File:Regular_polygon_3.svg|20px]]<BR>2 [[Triangle|{3}]]
| |
| |
| |
| | 12
| |
| | 20
| |
| | 10
| |
| |}
| |
| | |
| ==== (6 2 2) D<sub>6h</sub>dihedral symmetry ====
| |
| | |
| There are 24 fundamental triangles, visible in the faces of the [[dodecagonal bipyramid]] and alternately colored triangles on a sphere.
| |
| | |
| {| class="wikitable"
| |
| !rowspan=2|#
| |
| !rowspan=2|Name
| |
| !rowspan=2|Picture
| |
| !rowspan=2|Tiling
| |
| !rowspan=2|[[Vertex figure|Vertex<BR>figure]]
| |
| !rowspan=2|[[Coxeter-Dynkin diagram|Coxeter-Dynkin]]<BR>and [[Schläfli symbol#Extended for uniform polychora and 3-space honeycombs|Schläfli]]<BR>symbols
| |
| !colspan=3|Face counts by position
| |
| !colspan=3|Element counts
| |
| |-
| |
| ! Pos. 2<BR>{{CDD|node|6|node|2}}<BR>[6]<BR>(2)
| |
| ! Pos. 1<BR>{{CDD|node|2|node}}<BR>[2]<BR>(6)
| |
| ! Pos. 0<BR>{{CDD|2|node|2|node}}<BR>[2]<BR>(6)
| |
| ! Faces
| |
| ! Edges
| |
| ! Vertices
| |
| |- BGCOLOR="#f0e0e0"
| |
| !D<sub>6</sub>
| |
| |align=center|[[Hexagonal dihedron]]
| |
| |
| |
| |[[File:Hexagonal dihedron.png|50px]]
| |
| |
| |
| |align=center|{{CDD|node_1|6|node|2|node}}<BR>{6,2}
| |
| | [[File:Regular_polygon_6.svg|30px]]<BR>[[Hexagon|{6}]]
| |
| |
| |
| |
| |
| | 2
| |
| | 6
| |
| | 6
| |
| |- BGCOLOR="#e0e0f0"
| |
| !H<sub>6</sub>
| |
| |align=center|[[Hexagonal hosohedron]]
| |
| |
| |
| |[[File:Hexagonal hosohedron.png|50px]]
| |
| |
| |
| |align=center|{{CDD|node|6|node|2|node_1}}<BR>{2,6}
| |
| |
| |
| |
| |
| | [[File:Regular digon in spherical geometry-2.svg|30px]]<BR>[[Digon|{2}]]
| |
| | 6
| |
| | 6
| |
| | 2
| |
| |- BGCOLOR="#e0f0e0"
| |
| !D<sub>12</sub>
| |
| |align=center|Truncated hexagonal dihedron<BR>(Same as [[dodecagonal dihedron]])
| |
| |
| |
| |[[File:Dodecagonal dihedron.png|60px]]
| |
| |
| |
| |align=center|{{CDD|node_1|6|node_1|2|node}}<BR>t{6,2}
| |
| | [[File:Regular_polygon_10.svg|30px]]<BR>[[Dodecagon|{12}]]
| |
| |
| |
| |
| |
| | 2
| |
| | 12
| |
| | 12
| |
| |- BGCOLOR="#e0e0f0"
| |
| !H<sub>6</sub>
| |
| |align=center|Truncated hexagonal hosohedron<BR>(Same as [[hexagonal prism]])
| |
| |[[File:Hexagonal prism.png|60px]]
| |
| |[[File:Spherical hexagonal prism.png|60px]]
| |
| |[[File:Hexagonal prism vertfig.png|60px]]
| |
| |align=center|{{CDD|node|6|node_1|2|node_1}}<BR>t{2,6}
| |
| | [[File:Regular_polygon_6.svg|30px]]<BR>[[Hexagon|{6}]]
| |
| | [[File:Regular_polygon_4.svg|30px]]<BR>[[Square|{4}]]
| |
| |
| |
| | 8
| |
| | 18
| |
| | 12
| |
| |- BGCOLOR="#e0f0e0"
| |
| !P<sub>12</sub>
| |
| |align=center|Omnitruncated hexagonal dihedron<BR>([[Dodecagonal prism]])
| |
| |[[File:Dodecagonal prism.png|60px]]
| |
| |[[File:Spherical truncated hexagonal prism.png|60px]]
| |
| |[[File:Dodecagonal prism vf.png|60px]]
| |
| |align=center|{{CDD|node_1|6|node_1|2|node_1}}<BR>t<sub>0,1,2</sub>{2,6}=tr{2,6}
| |
| | [[File:Regular_polygon_10.svg|30px]]<BR>[[Dodecagon|{12}]]
| |
| | [[File:Regular_polygon_4.svg|30px]]<BR>[[Square|{4}]]
| |
| | [[File:Regular_polygon_4.svg|30px]]<BR>[[Square|{4}]]
| |
| | 14
| |
| | 36
| |
| | 24
| |
| |- BGCOLOR="#d0f0f0"
| |
| !A<sub>6</sub>
| |
| |align=center|Snub hexagonal dihedron<BR>([[Hexagonal antiprism]])
| |
| |[[File:Hexagonal antiprism.png|60px]]
| |
| |[[File:Spherical hexagonal antiprism.png|60px]]
| |
| |[[File:Hexagonal antiprism vertfig.png|60px]]
| |
| |align=center|{{CDD|node_h|6|node_h|2x|node_h}}<BR>sr{2,6}
| |
| | [[File:Regular_polygon_6.svg|30px]]<BR>[[Hexagon|{6}]]
| |
| | [[File:Regular_polygon_3.svg|20px]][[File:Regular_polygon_3.svg|20px]]<BR>2 [[Triangle|{3}]]
| |
| |
| |
| | 14
| |
| | 24
| |
| | 12
| |
| |}
| |
| | |
| == See also ==
| |
| *[[Polyhedron]]
| |
| **[[Regular polyhedron]]
| |
| **[[Quasiregular polyhedron]]
| |
| **[[Semiregular polyhedron]]
| |
| *[[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]]
| |
| *[[List of uniform polyhedra by Schwarz triangle]]
| |
| *[[Uniform tiling]]
| |
| *[[Uniform tilings in hyperbolic plane]]
| |
| | |
| == Notes ==
| |
| {{reflist}}
| |
| | |
| == References ==
| |
| *Brückner, M. ''Vielecke und vielflache. Theorie und geschichte.''. Leipzig, Germany: Teubner, 1900. [http://www.hti.umich.edu/cgi/b/bib/bibperm?q1=ABN8316.0001.001]
| |
| *{{Cite journal | last1=Coxeter | first1=Harold Scott MacDonald | author1-link=Harold Scott MacDonald Coxeter | last2=Longuet-Higgins | first2=M. S. | last3=Miller | first3=J. C. P. | title=Uniform polyhedra | jstor=91532 | mr=0062446 | year=1954 | journal=Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences | issn=0080-4614 | volume=246 |issue=916 | pages=401–450 | publisher=The Royal Society | doi=10.1098/rsta.1954.0003 | ref=harv}}
| |
| *{{Cite journal | last1=Sopov | first1=S. P. | title=A proof of the completeness on the list of elementary homogeneous polyhedra | mr=0326550 | year=1970 | journal=Ukrainskiui Geometricheskiui Sbornik | issue=8 | pages=139–156 | ref=harv | postscript=<!--None-->}}
| |
| * {{cite book | first=Magnus | last=Wenninger | authorlink=Magnus Wenninger | title=Polyhedron Models | publisher=Cambridge University Press | year=1974 | isbn=0-521-09859-9 }}
| |
| *{{Cite journal | last1=Skilling | first1=J. | title=The complete set of uniform polyhedra | jstor=74475 | mr=0365333 | year=1975 | journal=Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences | issn=0080-4614 | volume=278 | pages=111–135 | doi=10.1098/rsta.1975.0022 | ref=harv | postscript=<!--None--> | issue=1278}}
| |
| * Har'El, Z. [http://www.math.technion.ac.il/~rl/docs/uniform.pdf ''Uniform Solution for Uniform Polyhedra.''], Geometriae Dedicata 47, 57-110, 1993. [http://www.math.technion.ac.il/~rl Zvi Har’El], [http://www.math.technion.ac.il/~rl/kaleido Kaleido software], [http://www.math.technion.ac.il/~rl/kaleido/poly.html Images], [http://www.math.technion.ac.il/~rl/kaleido/dual.html dual images]
| |
| * [http://www.mathconsult.ch/showroom/unipoly Mäder, R. E.] ''Uniform Polyhedra.'' Mathematica J. 3, 48-57, 1993. [http://library.wolfram.com/infocenter/Articles/2254]
| |
| *Messer, Peter W. [http://www.springerlink.com/content/me48wm7823jhdcpe/?p=baeede46029e489f9df9a31526cd8f6&pi=2 ''Closed-Form Expressions for Uniform Polyhedra and Their Duals.''], Discrete & Computational Geometry 27:353-375 (2002).
| |
| | |
| ==External links==
| |
| * {{MathWorld | urlname=UniformPolyhedron | title=Uniform Polyhedron}}
| |
| *[http://www.math.technion.ac.il/~rl/docs/uniform.pdf Uniform Solution for Uniform Polyhedra]
| |
| *[http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
| |
| *[http://www.georgehart.com/virtual-polyhedra/uniform-info.html Virtual Polyhedra] Uniform Polyhedra
| |
| | |
| {{Polytopes}}
| |
| | |
| [[Category:Uniform polyhedra|*]]
| |