- Key
- Regular Convex Non-convex
- Quasi-regular
- Wythoff p q|r Truncated regular forms Hemipolyhedra Rhombic quasi-regular
- Even-sided forms Wythoff p q r| Wythoff p q (r s)|
- Snub polyhedra Wythoff |p q r Wythoff |p q r s
{{Polyhedron types}}There are many relations among the uniform polyhedra. Here they are grouped by the Wythoff symbol. KeyImage
Name
Bowers pet name
V Number of vertices,E Number of edges,F Number of faces=Face configuration
?=Euler characteristic, group=Symmetry group
Wythoff symbol - Vertex figure
W - Wenninger number, U - Uniform number, K- Kaleido number, C -Coxeter number
alternative name
second alternative name
|
RegularAll the faces are identical, each edge is identical and each vertex is identical. They all have a Wythoff symbol of the form p|q 2. ConvexThe Platonic solids. {{Reg polyhedra db|Polyhedra smallbox2|T}}{{Reg polyhedra db|Polyhedra smallbox2|O}}{{Reg polyhedra db|Polyhedra smallbox2|C}}{{Reg polyhedra db|Polyhedra smallbox2|I}}{{Reg polyhedra db|Polyhedra smallbox2|D}}
Non-convexThe Kepler-Poinsot solids. {{Reg polyhedra db|Polyhedra smallbox2|gI}}{{Reg polyhedra db|Polyhedra smallbox2|gD}}{{Reg polyhedra db|Polyhedra smallbox2|lsD}}{{Reg polyhedra db|Polyhedra smallbox2|gsD}}
Quasi-regularEach edge is identical and each vertex is identical. There are two types of faces which appear in an alternating fashion around each vertex. The first row are semi-regular with 4 faces around each vertex. They have Wythoff symbol 2|p q. The second row are ditrigonal with 6 faces around each vertex. They have Wythoff symbol 3|p q or 3/2|p q. {{Semireg polyhedra db|Polyhedra smallbox2|CO}}{{Semireg polyhedra db|Polyhedra smallbox2|ID}}{{Uniform polyhedra db|Polyhedra smallbox2|gID}}{{Uniform polyhedra db|Polyhedra smallbox2|DD}} {{Uniform polyhedra db|Polyhedra smallbox2|ldID}}{{Uniform polyhedra db|Polyhedra smallbox2|dDD}}{{Uniform polyhedra db|Polyhedra smallbox2|gdID}}
Wythoff p q|rTruncated regular formsEach vertex has three faces surrounding it, two of which are identical. These all have Wythoff symbols 2 p|q, some are constructed by truncating the regular solids. {{Semireg polyhedra db|Polyhedra smallbox2|tT}}{{Semireg polyhedra db|Polyhedra smallbox2|tO}}{{Semireg polyhedra db|Polyhedra smallbox2|tC}}{{Semireg polyhedra db|Polyhedra smallbox2|tI}}{{Semireg polyhedra db|Polyhedra smallbox2|tD}} {{Uniform polyhedra db|Polyhedra smallbox2|tgD}}{{Uniform polyhedra db|Polyhedra smallbox2|gtI}}{{Uniform polyhedra db|Polyhedra smallbox2|stH}}{{Uniform polyhedra db|Polyhedra smallbox2|lstD}}{{Uniform polyhedra db|Polyhedra smallbox2|gstD}}
HemipolyhedraThe hemipolyhedra all have faces which pass through the origin. Their Wythoff symbols are of the form p p/m|q or p/m p/n|q. With the exception of the tetrahemihexahedron they occur in pairs, and are closely related to the semi-regular polyhedra, like the cuboctohedron. {{Uniform polyhedra db|Polyhedra smallbox2|ThH}}{{Uniform polyhedra db|Polyhedra smallbox2|OhO}}{{Uniform polyhedra db|Polyhedra smallbox2|ChO}}{{Uniform polyhedra db|Polyhedra smallbox2|lIhD}}{{Uniform polyhedra db|Polyhedra smallbox2|lDhD}} {{Uniform polyhedra db|Polyhedra smallbox2|gIhD}}{{Uniform polyhedra db|Polyhedra smallbox2|gDhD}}{{Uniform polyhedra db|Polyhedra smallbox2|gDhI}}{{Uniform polyhedra db|Polyhedra smallbox2|lDhI}}
Rhombic quasi-regularFour faces around the vertex in the pattern p.q.r.q. The name rhombic stems from inserting a square in the cuboctahedron and icosidodecahedron. The Wythoff symbol is of the form p q|r. | Polyhedra smallbox2|lrCO}} | Polyhedra smallbox2|lCCO}} | Polyhedra smallbox2|gCCO}} | Polyhedra smallbox2|ugrCO}} | | Polyhedra smallbox2|lrID}} | Polyhedra smallbox2|lDID}} | Polyhedra smallbox2|gDID}} | Polyhedra smallbox2|ugrID}} | | Polyhedra smallbox2|lIID}} | Polyhedra smallbox2|ldDID}} | Polyhedra smallbox2|rDD}} | Polyhedra smallbox2|IDD}} | | Polyhedra smallbox2|gdDID}} | Polyhedra smallbox2|gIID}} |
Even-sided formsWythoff p q r|These have three different faces around each vertex, and the vertices do not lie on any plane of symmetry. They have Wythoff symbol p q r|, and vertex figures 2p.2q.2r. {{Semireg polyhedra db|Polyhedra smallbox2|grCO}}{{Uniform polyhedra db|Polyhedra smallbox2|gtCO}}{{Uniform polyhedra db|Polyhedra smallbox2|ctCO}} {{Semireg polyhedra db|Polyhedra smallbox2|grID}}{{Uniform polyhedra db|Polyhedra smallbox2|gtID}}{{Uniform polyhedra db|Polyhedra smallbox2|itDD}}{{Uniform polyhedra db|Polyhedra smallbox2|tDD}}
Wythoff p q (r s)|Vertex figure p.q.-p.-q. Wythoff p q (r s)|, mixing pqr| and pqs|. {{Uniform polyhedra db|Polyhedra smallbox2|lrH}}{{Uniform polyhedra db|Polyhedra smallbox2|grH}}{{Uniform polyhedra db|Polyhedra smallbox2|rI}}{{Uniform polyhedra db|Polyhedra smallbox2|grD}} {{Uniform polyhedra db|Polyhedra smallbox2|gDI}}{{Uniform polyhedra db|Polyhedra smallbox2|lrD}}{{Uniform polyhedra db|Polyhedra smallbox2|lDI}}
Snub polyhedraThese have Wythoff symbol |p q r, and one non-Wythoffian construction is given |p q r s. Wythoff |p q r| Symmetry group |
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| O{{Semireg polyhedra db|Polyhedra smallbox2|nCO}} | | Ih{{Uniform polyhedra db|Polyhedra smallbox2|Seside}}{{Uniform polyhedra db|Polyhedra smallbox2|Sirsid}} | | I{{Semireg polyhedra db|Polyhedra smallbox2|nID}}{{Uniform polyhedra db|Polyhedra smallbox2|Siddid}}{{Uniform polyhedra db|Polyhedra smallbox2|Isdid}} | | I{{Uniform polyhedra db|Polyhedra smallbox2|Gosid}}{{Uniform polyhedra db|Polyhedra smallbox2|Gisid}}{{Uniform polyhedra db|Polyhedra smallbox2|Girsid}} | | I{{Uniform polyhedra db|Polyhedra smallbox2|Sided}}{{Uniform polyhedra db|Polyhedra smallbox2|Gisdid}} |
Wythoff |p q r s| Symmetry group |
|---|
| Ih{{Uniform polyhedra db|Polyhedra smallbox2|Gidrid}} |
1 : Uniform polyhedra |