View source for List of uniform polyhedra by spherical triangle

{{short description|none}}
{{Polyhedron types}}
There are many relations among the [[uniform polyhedron|uniform polyhedra]]. This ''List of uniform polyhedra by spherical triangle'' groups them by the [[Wythoff symbol]].

==Key==

{|class="wikitable"
|-
|
Image<br>
Name<br>
Bowers pet name<br>
V Number of vertices,E Number of edges,F Number of faces=Face configuration<br>
''?''=Euler characteristic, group=Symmetry group<br>
Wythoff symbol - Vertex figure<br>
W - Wenninger number, U - Uniform number, K- Kalido number, C -Coxeter number<br>
alternative name<br>
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)<sup>p</sup>.
* 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<br>
<math>{\pi\over p}\ {\pi\over q}\ {\pi\over r}</math>
!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"
|<math>{\pi\over 3}\ {\pi\over 3}\ {\pi\over 2}</math>
|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"
|<math>{\pi\over 4}\ {\pi\over 3}\ {\pi\over 2}</math>
|{{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"
|<math>{\pi\over 5}\ {\pi\over 3}\ {\pi\over 2}</math>
|{{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 ====
<math>{a\pi\over 3}\ {b\pi\over 3}\ {c\pi\over 2}</math> Group
{|class="wikitable"
|-
!Spherical triangle<br>
<math>{\pi\over p}\ {\pi\over q}\ {\pi\over r}</math>
!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
|<math>{\pi\over 3}\ {\pi\over 2}\ {2\pi\over 3}</math>
|
|
|
|
|{{Uniform polyhedra db|Polyhedra smallbox2|ThH}}
|
|
|}
====4 3 2====
<math>{a\pi\over 4}\ {b\pi\over 3}\ {c\pi\over 2}</math> Group
{|class="wikitable"
|-
!Spherical triangle<br>
<math>{\pi\over p}\ {\pi\over q}\ {\pi\over r}</math>
!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
|<math>{\pi\over 4}\ {2\pi\over 3}\ {\pi\over 2}</math>
|octahedron
|cube
|
|{{Uniform polyhedra db|Polyhedra smallbox2|stH}}
|
|{{Uniform polyhedra db|Polyhedra smallbox2|ugrCO}}
|{{Uniform polyhedra db|Polyhedra smallbox2|lrH}}
|
|- valign=top
|<math>{3\pi\over 4}\ {\pi\over 3}\ {\pi\over 2}</math>
|
|
|
|
|
|
|{{Uniform polyhedra db|Polyhedra smallbox2|gtCO}}
|
|- valign=top
|<math>{3\pi\over 4}\ {2\pi\over 3}\ {\pi\over 2}</math>
|
|
|
|
|
|
|{{Uniform polyhedra db|Polyhedra smallbox2|grH}}
|}

====5 3 2====
<math>{a\pi\over 5}\ {b\pi\over 3}\ {c\pi\over 2}</math> Group
{|class="wikitable"
|-
!Spherical triangle<br>
<math>{\pi\over p}\ {\pi\over q}\ {\pi\over r}</math>
!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
|<math>{2\pi\over 5}\ {\pi\over 3}\ {\pi\over 2}</math>
|{{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
|<math>{3\pi\over 5}\ {\pi\over 3}\ {\pi\over 2}</math>
|{{Uniform polyhedra db|Polyhedra smallbox2|rI}}
|{{Uniform polyhedra db|Polyhedra smallbox2|gtID}}
|{{Uniform polyhedra db|Polyhedra smallbox2|grD}}
|
|
|
|}

====5 5 2====
<math>{a\pi\over 5}\ {b\pi\over 5}\ {c\pi\over 2}</math> Group
{|class="wikitable"
|-
!Spherical triangle<br>
<math>{\pi\over p}\ {\pi\over q}\ {\pi\over r}</math>
!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
|<math>{\pi\over 5}\ {2\pi\over 5}\ {\pi\over 2}</math>
|{{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
|<math>{\pi\over 5}\ {3\pi\over 5}\ {\pi\over 2}</math>
|{{Uniform polyhedra db|Polyhedra smallbox2|lrD}}
|{{Uniform polyhedra db|Polyhedra smallbox2|tDD}}
|
|
|
|
|}

===a b 3===
====3 3 3====
<math>{a\pi\over 3}\ {b\pi\over 3}\ {c\pi\over 3}</math> Group
{|class="wikitable"
|-
!Spherical triangle<br>
<math>{\pi\over p}\ {\pi\over q}\ {\pi\over r}</math>
!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
|<math>{\pi\over 3}\ {\pi\over 3}\ {2\pi\over 3}</math>
|
|
|
|colspan=2 align=center|{{Uniform polyhedra db|Polyhedra smallbox2|OhO}}
|
|}
====4 3 3====
<math>{a\pi\over 4}\ {b\pi\over 3}\ {c\pi\over 3}</math> Group
{|class="wikitable"
|-
!Spherical triangle<br>
<math>{\pi\over p}\ {\pi\over q}\ {\pi\over r}</math>
!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====
<math>{a\pi\over 5}\ {b\pi\over 3}\ {c\pi\over 3}</math> Group
{|class="wikitable"
|-
!Spherical triangle<br>
<math>{\pi\over p}\ {\pi\over q}\ {\pi\over r}</math>
!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
|<math>{3\pi\over 5}\ {\pi\over 3}\ {\pi\over 3}</math>
|{{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
|<math>{\pi\over 5}\ {2\pi\over 3}\ {\pi\over 3}</math>
|{{Uniform polyhedra db|Polyhedra smallbox2|lIID}}
|{{Uniform polyhedra db|Polyhedra smallbox2|lDI}}
|
|
|
|
|}

====4 4 3====
<math>{a\pi\over 4}\ {b\pi\over 4}\ {c\pi\over 3}</math> Group
{|class="wikitable"
|-
!Spherical triangle<br>
<math>{\pi\over p}\ {\pi\over q}\ {\pi\over r}</math>
!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
|<math>{\pi\over 4}\ {\pi\over 3}\ {3\pi\over 4}</math>
|
|
|
|
|{{Uniform polyhedra db|Polyhedra smallbox2|ChO}}
|{{Uniform polyhedra db|Polyhedra smallbox2|gCCO}}
|{{Uniform polyhedra db|Polyhedra smallbox2|ctCO}}
|- valign=top
|<math>{\pi\over 4}\ {\pi\over 4}\ {2\pi\over 3}</math>
|
|
|
|colspan=2 align=center|{{Uniform polyhedra db|Polyhedra smallbox2|lCCO}}
|
|}

====5 5 3====
<math>{a\pi\over 5}\ {b\pi\over 5}\ {c\pi\over 3}</math> Group
{|class="wikitable"
|-
!Spherical triangle<br>
<math>{\pi\over p}\ {\pi\over q}\ {\pi\over r}</math>
!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
|<math>{\pi\over 3}\ {2\pi\over 5}\ {3\pi\over 5}</math>
|
|
|
|{{Uniform polyhedra db|Polyhedra smallbox2|lDhI}}
|{{Uniform polyhedra db|Polyhedra smallbox2|gDI}}
|{{Uniform polyhedra db|Polyhedra smallbox2|lDID}}
|- valign=top
|<math>{\pi\over 3}\ {\pi\over 5}\ {4\pi\over 5}</math>
|
|
|
|{{Uniform polyhedra db|Polyhedra smallbox2|gDhI}}
|{{Uniform polyhedra db|Polyhedra smallbox2|ldDID}}
|{{Uniform polyhedra db|Polyhedra smallbox2|gdDID}}
|- valign=top
|<math>{\pi\over 5}\ {\pi\over 5}\ {2\pi\over 3}</math>
|
|
|
|{{Uniform polyhedra db|Polyhedra smallbox2|lDID}}
|{{Uniform polyhedra db|Polyhedra smallbox2|gDID}}
|- valign=top
|<math>{\pi\over 5}\ {\pi\over 3}\ {3\pi\over 5}</math>
|
|{{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====
<math>{a\pi\over 5}\ {b\pi\over 5}\ {c\pi\over 5}</math> Group
{|class="wikitable"
|-
!Spherical triangle<br>
<math>{\pi\over p}\ {\pi\over q}\ {\pi\over r}</math>
!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
|<math>{2\pi\over 5}\ {3\pi\over 5}\ {3\pi\over 5}</math>
|
|
|
|
|colspan=2 align=center|{{Uniform polyhedra db|Polyhedra smallbox2|gDhD}}
|
|}

[[Category:Uniform polyhedra]]