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List of uniform polyhedra by spherical triangle

{{short description|none}}

{{Polyhedron types}}

There are many relations among the uniform polyhedra. This List of uniform polyhedra by spherical triangle groups them by the Wythoff symbol.

Key

class="wikitable"
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

class="wikitable"
Spherical triangle

{\pi\over p}\ {\pi\over q}\ {\pi\over r}

!p{{pipe}}q r

!q{{pipe}}p r

!r{{pipe}}p q

!q r{{pipe}}p

!p r{{pipe}}q

!p q{{pipe}}r

!p q r{{pipe}}

!{{pipe}}p q r

valign="top"

|{\pi\over 3}\ {\pi\over 3}\ {\pi\over 2}

|colspan="2" align="center"|{{Reg polyhedra db|Polyhedra smallbox2|T}}

|Octahedron

|colspan="2" align="center"|{{Semireg polyhedra db|Polyhedra smallbox2|tT}}

|Cuboctahedron

|Truncated octahedron

|Icosahedron

valign="top"

|{\pi\over 4}\ {\pi\over 3}\ {\pi\over 2}

|{{Reg polyhedra db|Polyhedra smallbox2|O}}

|{{Reg polyhedra db|Polyhedra smallbox2|C}}

|{{Semireg polyhedra db|Polyhedra smallbox2|CO}}

|{{Semireg polyhedra db|Polyhedra smallbox2|tC}}

|{{Semireg polyhedra db|Polyhedra smallbox2|tO}}

|{{Semireg polyhedra db|Polyhedra smallbox2|lrCO}}

|{{Semireg polyhedra db|Polyhedra smallbox2|grCO}}

|{{Semireg polyhedra db|Polyhedra smallbox2|nCO}}

valign="top"

|{\pi\over 5}\ {\pi\over 3}\ {\pi\over 2}

|{{Reg polyhedra db|Polyhedra smallbox2|I}}

|{{Reg polyhedra db|Polyhedra smallbox2|D}}

|{{Semireg polyhedra db|Polyhedra smallbox2|ID}}

|{{Semireg polyhedra db|Polyhedra smallbox2|tD}}

|{{Semireg polyhedra db|Polyhedra smallbox2|tI}}

|{{Semireg polyhedra db|Polyhedra smallbox2|lrID}}

|{{Semireg polyhedra db|Polyhedra smallbox2|grID}}

|{{Semireg polyhedra db|Polyhedra smallbox2|nID}}

Non-convex

= a b 2 =

== 3 3 2 ==

{a\pi\over 3}\ {b\pi\over 3}\ {c\pi\over 2} Group

class="wikitable"
Spherical triangle

{\pi\over p}\ {\pi\over q}\ {\pi\over r}

!p{{pipe}}q r

!q{{pipe}}p r

!r{{pipe}}p q

!q r{{pipe}}p

!p r{{pipe}}q

!p q{{pipe}}r

!p q r{{pipe}}

!{{pipe}}p q r

valign=top

|{\pi\over 3}\ {\pi\over 2}\ {2\pi\over 3}

|

|

|

|

|{{Uniform polyhedra db|Polyhedra smallbox2|ThH}}

|

|

==4 3 2==

{a\pi\over 4}\ {b\pi\over 3}\ {c\pi\over 2} Group

class="wikitable"
Spherical triangle

{\pi\over p}\ {\pi\over q}\ {\pi\over r}

!p{{pipe}}q r

!q{{pipe}}p r

!r{{pipe}}p q

!q r{{pipe}}p

!p r{{pipe}}q

!p q{{pipe}}r

!p q r{{pipe}}

!{{pipe}}p q r

valign=top

|{\pi\over 4}\ {2\pi\over 3}\ {\pi\over 2}

|octahedron

|cube

|

|{{Uniform polyhedra db|Polyhedra smallbox2|stH}}

|

|{{Uniform polyhedra db|Polyhedra smallbox2|ugrCO}}

|{{Uniform polyhedra db|Polyhedra smallbox2|lrH}}

|

valign=top

|{3\pi\over 4}\ {\pi\over 3}\ {\pi\over 2}

|

|

|

|

|

|

|{{Uniform polyhedra db|Polyhedra smallbox2|gtCO}}

|

valign=top

|{3\pi\over 4}\ {2\pi\over 3}\ {\pi\over 2}

|

|

|

|

|

|

|{{Uniform polyhedra db|Polyhedra smallbox2|grH}}

==5 3 2==

{a\pi\over 5}\ {b\pi\over 3}\ {c\pi\over 2} Group

class="wikitable"
Spherical triangle

{\pi\over p}\ {\pi\over q}\ {\pi\over r}

!p{{pipe}}q r

!q{{pipe}}p r

!r{{pipe}}p q

!q r{{pipe}}p

!p r{{pipe}}q

!p q{{pipe}}r

valign=top

|{2\pi\over 5}\ {\pi\over 3}\ {\pi\over 2}

|{{Reg polyhedra db|Polyhedra smallbox2|gI}}

|{{Reg polyhedra db|Polyhedra smallbox2|gsD}}

|{{Uniform polyhedra db|Polyhedra smallbox2|gID}}

|{{Uniform polyhedra db|Polyhedra smallbox2|gstD}}

|{{Uniform polyhedra db|Polyhedra smallbox2|gtI}}

|{{Uniform polyhedra db|Polyhedra smallbox2|ugrID}}

valign=top

|

!p q r{{pipe}}

!p q r{{pipe}}

!p q r{{pipe}}

!{{pipe}}p q r

!

!

!

valign=top

|{3\pi\over 5}\ {\pi\over 3}\ {\pi\over 2}

|{{Uniform polyhedra db|Polyhedra smallbox2|rI}}

|{{Uniform polyhedra db|Polyhedra smallbox2|gtID}}

|{{Uniform polyhedra db|Polyhedra smallbox2|grD}}

|

|

|

==5 5 2==

{a\pi\over 5}\ {b\pi\over 5}\ {c\pi\over 2} Group

class="wikitable"
Spherical triangle

{\pi\over p}\ {\pi\over q}\ {\pi\over r}

!p{{pipe}}q r

!q{{pipe}}p r

!r{{pipe}}p q

!q r{{pipe}}p

!p r{{pipe}}q

!p q{{pipe}}r

valign=top

|{\pi\over 5}\ {2\pi\over 5}\ {\pi\over 2}

|{{Reg polyhedra db|Polyhedra smallbox2|lsD}}

|{{Reg polyhedra db|Polyhedra smallbox2|gD}}

|{{Uniform polyhedra db|Polyhedra smallbox2|DD}}

|{{Uniform polyhedra db|Polyhedra smallbox2|lstD}}

|{{Uniform polyhedra db|Polyhedra smallbox2|tgD}}

|{{Uniform polyhedra db|Polyhedra smallbox2|rDD}}

!p q r{{pipe}}

!p q r{{pipe}}

!{{pipe}}p q r

!

!

!

valign=top

|{\pi\over 5}\ {3\pi\over 5}\ {\pi\over 2}

|{{Uniform polyhedra db|Polyhedra smallbox2|lrD}}

|{{Uniform polyhedra db|Polyhedra smallbox2|tDD}}

|

|

|

|

=a b 3=

==3 3 3==

{a\pi\over 3}\ {b\pi\over 3}\ {c\pi\over 3} Group

class="wikitable"
Spherical triangle

{\pi\over p}\ {\pi\over q}\ {\pi\over r}

!p{{pipe}}q r

!q{{pipe}}p r

!r{{pipe}}p q

!q r{{pipe}}p

!p r{{pipe}}q

!p q{{pipe}}r

!p q r{{pipe}}

!{{pipe}}p q r

valign=top

|{\pi\over 3}\ {\pi\over 3}\ {2\pi\over 3}

|

|

|

|colspan=2 align=center|{{Uniform polyhedra db|Polyhedra smallbox2|OhO}}

|

==4 3 3==

{a\pi\over 4}\ {b\pi\over 3}\ {c\pi\over 3} Group

class="wikitable"
Spherical triangle

{\pi\over p}\ {\pi\over q}\ {\pi\over r}

!p{{pipe}}q r

!q{{pipe}}p r

!r{{pipe}}p q

!q r{{pipe}}p

!p r{{pipe}}q

!p q{{pipe}}r

!p q r{{pipe}}

!{{pipe}}p q r

==5 3 3==

{a\pi\over 5}\ {b\pi\over 3}\ {c\pi\over 3} Group

class="wikitable"
Spherical triangle

{\pi\over p}\ {\pi\over q}\ {\pi\over r}

!p{{pipe}}q r

!q{{pipe}}p r

!r{{pipe}}p q

!q r{{pipe}}p

!p r{{pipe}}q

!p q{{pipe}}r

valign=top

|{3\pi\over 5}\ {\pi\over 3}\ {\pi\over 3}

|{{Uniform polyhedra db|Polyhedra smallbox2|gdID}}

|colspan=2 align=center|{{Uniform polyhedra db|Polyhedra smallbox2|ldID}}

|{{Uniform polyhedra db|Polyhedra smallbox2|gIhD}}

|{{Uniform polyhedra db|Polyhedra smallbox2|lIhD}}

|{{Uniform polyhedra db|Polyhedra smallbox2|gIID}}

|

!p q r{{pipe}}

!p q r{{pipe}}

!{{pipe}}p q r

!

!

!

valign=top

|{\pi\over 5}\ {2\pi\over 3}\ {\pi\over 3}

|{{Uniform polyhedra db|Polyhedra smallbox2|lIID}}

|{{Uniform polyhedra db|Polyhedra smallbox2|lDI}}

|

|

|

|

==4 4 3==

{a\pi\over 4}\ {b\pi\over 4}\ {c\pi\over 3} Group

class="wikitable"
Spherical triangle

{\pi\over p}\ {\pi\over q}\ {\pi\over r}

!p{{pipe}}q r

!q{{pipe}}p r

!r{{pipe}}p q

!q r{{pipe}}p

!p r{{pipe}}q

!p q{{pipe}}r

!p q r{{pipe}}

!{{pipe}}p q r

valign=top

|{\pi\over 4}\ {\pi\over 3}\ {3\pi\over 4}

|

|

|

|

|{{Uniform polyhedra db|Polyhedra smallbox2|ChO}}

|{{Uniform polyhedra db|Polyhedra smallbox2|gCCO}}

|{{Uniform polyhedra db|Polyhedra smallbox2|ctCO}}

valign=top

|{\pi\over 4}\ {\pi\over 4}\ {2\pi\over 3}

|

|

|

|colspan=2 align=center|{{Uniform polyhedra db|Polyhedra smallbox2|lCCO}}

|

==5 5 3==

{a\pi\over 5}\ {b\pi\over 5}\ {c\pi\over 3} Group

class="wikitable"
Spherical triangle

{\pi\over p}\ {\pi\over q}\ {\pi\over r}

!p{{pipe}}q r

!q{{pipe}}p r

!r{{pipe}}p q

!q r{{pipe}}p

!p r{{pipe}}q

!p q{{pipe}}r

!p q r{{pipe}}

!{{pipe}}p q r

valign=top

|{\pi\over 3}\ {2\pi\over 5}\ {3\pi\over 5}

|

|

|

|{{Uniform polyhedra db|Polyhedra smallbox2|lDhI}}

|{{Uniform polyhedra db|Polyhedra smallbox2|gDI}}

|{{Uniform polyhedra db|Polyhedra smallbox2|lDID}}

valign=top

|{\pi\over 3}\ {\pi\over 5}\ {4\pi\over 5}

|

|

|

|{{Uniform polyhedra db|Polyhedra smallbox2|gDhI}}

|{{Uniform polyhedra db|Polyhedra smallbox2|ldDID}}

|{{Uniform polyhedra db|Polyhedra smallbox2|gdDID}}

valign=top

|{\pi\over 5}\ {\pi\over 5}\ {2\pi\over 3}

|

|

|

|{{Uniform polyhedra db|Polyhedra smallbox2|lDID}}

|{{Uniform polyhedra db|Polyhedra smallbox2|gDID}}

valign=top

|{\pi\over 5}\ {\pi\over 3}\ {3\pi\over 5}

|

|{{Uniform polyhedra db|Polyhedra smallbox2|dDD}}

|

|{{Uniform polyhedra db|Polyhedra smallbox2|IDD}}

|{{Uniform polyhedra db|Polyhedra smallbox2|ldDID}}

|{{Uniform polyhedra db|Polyhedra smallbox2|itDD}}

=a b 5=

==5 5 5==

{a\pi\over 5}\ {b\pi\over 5}\ {c\pi\over 5} Group

class="wikitable"
Spherical triangle

{\pi\over p}\ {\pi\over q}\ {\pi\over r}

!p{{pipe}}q r

!q{{pipe}}p r

!r{{pipe}}p q

!q r{{pipe}}p

!p r{{pipe}}q

!p q{{pipe}}r

!p q r{{pipe}}

!{{pipe}}p q r

valign=top

|{2\pi\over 5}\ {3\pi\over 5}\ {3\pi\over 5}

|

|

|

|

|colspan=2 align=center|{{Uniform polyhedra db|Polyhedra smallbox2|gDhD}}

|

Category:Uniform polyhedra