“List of uniform polyhedra by spherical triangle”的意思、由来-开放百科全书

词条

List of uniform polyhedra by spherical triangle

释义

Key
Convex
Non-convex
a b 2 3 3 2 4 3 2 5 3 2 5 5 2 a b 3 3 3 3 4 3 3 5 3 3 4 4 3 5 5 3 a b 5 5 5 5

There are many relations among the uniform polyhedra.

Here they are grouped by the Wythoff symbol.

Key

Image

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- Kalido number, C -Coxeter number

alternative name

second alternative name

The vertex figure can be discovered by considering the Wythoff symbol:

p|q r - 2p edges, alternating q-gons and r-gons. Vertex figure (q.r)^p.
p|q 2 - p edges, q-gons (here r=2 so the r-gons are degenerate lines).
2|q r - 4 edges, alternating q-gons and r-gons
q r|p - 4 edges, 2p-gons, q-gons, 2p-gons r-gons, Vertex figure 2p.q.2p.r.
q 2|p - 3 edges, 2p-gons, q-gons, 2p-gons, Vertex figure 2p.q.2p.
p q r|- 3 edges, 2p-gons, 2q-gons, 2r-gons, vertex figure 2p.2q.2r

Convex

p\|q r	q\|p r	r\|p q	q r\|p	p r\|q	p q\|r	p q r\|	\|p q r
{{Reg polyhedra db\|Polyhedra smallbox2\|T}}		Octahedron	{{Semireg polyhedra db\|Polyhedra smallbox2\|tT}}		Cuboctahedron	Truncated octahedron	Icosahedron
Polyhedra smallbox2\|O}}	Polyhedra smallbox2\|C}}	Polyhedra smallbox2\|CO}}	Polyhedra smallbox2\|tC}}	Polyhedra smallbox2\|tO}}	Polyhedra smallbox2\|lrCO}}	Polyhedra smallbox2\|grCO}}	Polyhedra smallbox2\|nCO}}
Polyhedra smallbox2\|I}}	Polyhedra smallbox2\|D}}	Polyhedra smallbox2\|ID}}	Polyhedra smallbox2\|tD}}	Polyhedra smallbox2\|tI}}	Polyhedra smallbox2\|lrID}}	Polyhedra smallbox2\|grID}}	Polyhedra smallbox2\|nID}}

Non-convex

a b 2

3 3 2

Group

Spherical triangle	p\|q r	q\|p r	r\|p q	q r\|p	p r\|q	p q\|r	p q r\|	\|p q r
	Polyhedra smallbox2\|ThH}}

4 3 2

Group

p\|q r	q\|p r	r\|p q	q r\|p	p r\|q
octahedron	cube	Polyhedra smallbox2\|stH}}	Polyhedra smallbox2\|ugrCO}}	Polyhedra smallbox2\|lrH}}
Polyhedra smallbox2\|gtCO}}
Polyhedra smallbox2\|grH}}

5 3 2

Group

Spherical triangle	p\|q r	q\|p r	r\|p q	q r\|p	p r\|q	p q\|r
	Polyhedra smallbox2\|gI}}	Polyhedra smallbox2\|gsD}}	Polyhedra smallbox2\|gID}}	Polyhedra smallbox2\|gstD}}	Polyhedra smallbox2\|gtI}}	Polyhedra smallbox2\|ugrID}}
p q r\|	p q r\|	p q r\|	\|p q r
	Polyhedra smallbox2\|rI}}	Polyhedra smallbox2\|gtID}}	Polyhedra smallbox2\|grD}}

5 5 2

Group

Spherical triangle	p\|q r	q\|p r	r\|p q	q r\|p	p r\|q	p q\|r
	Polyhedra smallbox2\|lsD}}	Polyhedra smallbox2\|gD}}	Polyhedra smallbox2\|DD}}	Polyhedra smallbox2\|lstD}}	Polyhedra smallbox2\|tgD}}	Polyhedra smallbox2\|rDD}}
p q r\|	p q r\|	\|p q r
	Polyhedra smallbox2\|lrD}}	Polyhedra smallbox2\|tDD}}

a b 3

3 3 3

Group

Spherical triangle	p\|q r	q\|p r	r\|p q	q r\|p	p r\|q	p q\|r	p q r\|	\|p q r
	{{Uniform polyhedra db\|Polyhedra smallbox2\|OhO}}

4 3 3

Group

Spherical triangle	p\|q r	q\|p r	r\|p q	q r\|p	p r\|q	p q\|r	p q r\|	\|p q r

5 3 3

Group

Spherical triangle	p\|q r	q\|p r	q r\|p	p r\|q	p q\|r
	Polyhedra smallbox2\|gdID}}	{{Uniform polyhedra db\|Polyhedra smallbox2\|ldID}}	Polyhedra smallbox2\|gIhD}}	Polyhedra smallbox2\|lIhD}}	Polyhedra smallbox2\|gIID}}
p q r\|	p q r\|	\|p q r
	Polyhedra smallbox2\|lIID}}	Polyhedra smallbox2\|lDI}}

4 4 3

Group

Spherical triangle	p\|q r	q\|p r	r\|p q	q r\|p	p r\|q	p q\|r	p q r\|	\|p q r
	Polyhedra smallbox2\|ChO}}	Polyhedra smallbox2\|gCCO}}	Polyhedra smallbox2\|ctCO}}
	{{Uniform polyhedra db\|Polyhedra smallbox2\|lCCO}}

5 5 3

Group

p\|q r	q\|p r	r\|p q	q r\|p
Polyhedra smallbox2\|lDhI}}	Polyhedra smallbox2\|gDI}}	Polyhedra smallbox2\|lDID}}
Polyhedra smallbox2\|gDhI}}	Polyhedra smallbox2\|ldDID}}	Polyhedra smallbox2\|gdDID}}
Polyhedra smallbox2\|lDID}}	Polyhedra smallbox2\|gDID}}
Polyhedra smallbox2\|dDD}}	Polyhedra smallbox2\|IDD}}	Polyhedra smallbox2\|ldDID}}	Polyhedra smallbox2\|itDD}}

a b 5

5 5 5

Group

Spherical triangle	p\|q r	q\|p r	r\|p q	q r\|p	p r\|q	p q\|r	p q r\|	\|p q r
	{{Uniform polyhedra db\|Polyhedra smallbox2\|gDhD}}

1 : Uniform polyhedra

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