View source for Icosahedral honeycomb

{{short description|Regular tiling of hyperbolic 3-space}}
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|'''Icosahedral honeycomb'''
|-
|bgcolor=#ffffff align=center colspan=2|[[File:H3 353 CC center.png|320px]]<BR>[[Poincaré disk model]]
|-
|bgcolor=#e7dcc3|Type||[[List of regular polytopes#Tessellations of hyperbolic 3-space|Hyperbolic regular honeycomb]]<BR>[[Uniform honeycombs in hyperbolic space|Uniform hyperbolic honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||{{math|{3,5,3} }}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_1|3|node|5|node|3|node}}
|-
|bgcolor=#e7dcc3|Cells||{{math|{5,3} }} ([[regular icosahedron]]) [[File:Uniform polyhedron-53-t2.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||{{math|{3} }} ([[triangle]])
|-
|bgcolor=#e7dcc3|[[Edge figure]]||{{math|{3} }} (triangle)
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[Image:Order-3 icosahedral honeycomb verf.svg|80px]]<BR>[[regular dodecahedron|dodecahedron]]
|-
|bgcolor=#e7dcc3|[[Dual polytope|Dual]]||[[Self-dual polytope|Self-dual]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||{{math|{{overline|J}}{{sub|3}}, [3,5,3]}}
|-
|bgcolor=#e7dcc3|Properties||Regular
|}

In [[geometry]], the '''icosahedral honeycomb''' is one of four compact, [[List of regular polytopes and compounds|regular]], space-filling [[tessellation]]s (or [[honeycomb (geometry)|honeycombs]]) in [[Hyperbolic space|hyperbolic 3-space]]. With [[Schläfli symbol]] {{math|{3,5,3},}} there are three [[regular icosahedron|icosahedra]] around each [[Edge (geometry)|edge]], and 12 icosahedra around each [[Vertex (geometry)|vertex]], in a regular [[regular dodecahedron|dodecahedral]] [[vertex figure]]. It is analogous to the [[24-cell]] and the [[5-cell]].

{{Honeycomb}}

==Description==
The [[dihedral angle]] of a [[regular icosahedron]] is around 138.2°, so it is impossible to fit three icosahedra around an edge in Euclidean 3-space. However, in hyperbolic space, properly scaled icosahedra can have dihedral angles of exactly 120 degrees, so three of those can fit around an edge.

[[Image:Hyperb icosahedral hc.png|200px|thumb|Honeycomb seen in perspective outside Poincare's model disk]]

== Related regular honeycombs ==
There are four regular compact honeycombs in 3D hyperbolic space:
{{Regular compact H3 honeycombs}}

== Related regular polytopes and honeycombs ==
It is a member of a sequence of [[regular polychora]] and honeycombs {3,''p'',3} with [[deltahedron|deltrahedral]] cells:
{{Symmetric_tessellations}}

It is also a member of a sequence of [[regular polychora]] and honeycombs {''p'',5,''p''}, with [[vertex figure]]s composed of pentagons:
{{Symmetric4 tessellations}}

== Uniform honeycombs ==

There are [[Convex uniform honeycombs in hyperbolic space#.5B3.2C5.2C3.5D family|nine uniform honeycombs]] in the [3,5,3] [[Coxeter group]] family, including this regular form as well as the [[Bitruncation (geometry)|bitruncated]] form, t<sub>1,2</sub>{3,5,3}, {{CDD|node|3|node_1|5|node_1|3|node}}, also called ''truncated dodecahedral honeycomb'', each of whose cells are [[truncated dodecahedron|truncated dodecahedra]].
{{353 family}}

=== Rectified icosahedral honeycomb===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Rectified icosahedral honeycomb
<!--|-
|bgcolor=#ffffff align=center colspan=2|-->
|-
|bgcolor=#e7dcc3|Type||[[Uniform honeycombs in hyperbolic space]]
|-
|width=100 bgcolor=#e7dcc3|[[Schläfli symbol]]||r{3,5,3} or t<sub>1</sub>{3,5,3}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node|3|node_1|5|node|3|node}}
|-
|bgcolor=#e7dcc3|Cells||[[icosidodecahedron|r{3,5}]] [[File:Uniform polyhedron-53-t1.png|40px]]<BR>[[regular dodecahedron|{5,3}]] [[File:Uniform polyhedron-53-t0.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[pentagon]] {5}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Rectified icosahedral honeycomb verf.png|80px]]<BR>[[triangular prism]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||<math>\overline{J}_3</math>, [3,5,3]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive, edge-transitive
|}
The '''rectified icosahedral honeycomb''', t<sub>1</sub>{3,5,3}, {{CDD|node|3|node_1|5|node|3|node}}, has alternating [[regular dodecahedron|dodecahedron]] and [[icosidodecahedron]] cells, with a [[triangular prism]] vertex figure:
: [[File:H3 353 CC center 0100.png|320px]][[File:Rectified icosahedral honeycomb.png|240px]]<BR>[[Perspective projection]]s from center of [[Poincaré disk model]]

==== Related honeycomb====
There are four rectified compact regular honeycombs:
{{Rectified compact H3 honeycombs}}
{{Clear}}

=== Truncated icosahedral honeycomb===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Truncated icosahedral honeycomb
<!--|-
|bgcolor=#ffffff align=center colspan=2|-->
|-
|bgcolor=#e7dcc3|Type||[[Uniform honeycombs in hyperbolic space]]
|-
|width=100 bgcolor=#e7dcc3|[[Schläfli symbol]]||t{3,5,3} or t<sub>0,1</sub>{3,5,3}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_1|3|node_1|5|node|3|node}}
|-
|bgcolor=#e7dcc3|Cells||[[Truncated icosahedron|t{3,5}]] [[File:Uniform polyhedron-53-t12.png|40px]]<BR>[[regular dodecahedron|{5,3}]] [[File:Uniform polyhedron-53-t0.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[pentagon]] {5}<BR>[[hexagon]] {6}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Truncated icosahedral honeycomb verf.png|80px]]<BR>[[triangular pyramid]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||<math>\overline{J}_3</math>, [3,5,3]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive
|}
The '''truncated icosahedral honeycomb''', t<sub>0,1</sub>{3,5,3}, {{CDD|node_1|3|node_1|5|node|3|node}}, has alternating [[regular dodecahedron|dodecahedron]] and [[truncated icosahedron]] cells, with a [[triangular pyramid]] vertex figure.

[[File:H3 353-0011 center ultrawide.png|480px]]

==== Related honeycombs ====
{{Truncated compact H3 honeycombs}}

{{Clear}}

=== Bitruncated icosahedral honeycomb===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Bitruncated icosahedral honeycomb
<!--|-
|bgcolor=#ffffff align=center colspan=2|-->
|-
|bgcolor=#e7dcc3|Type||[[Uniform honeycombs in hyperbolic space]]
|-
|width=100 bgcolor=#e7dcc3|[[Schläfli symbol]]||2t{3,5,3} or t<sub>1,2</sub>{3,5,3}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node|3|node_1|5|node_1|3|node}}
|-
|bgcolor=#e7dcc3|Cells||[[Truncated dodecahedron|t{5,3}]] [[File:Uniform polyhedron-53-t01.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[decagon]] {10}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Bitruncated icosahedral honeycomb verf.png|80px]]<BR>[[tetragonal disphenoid]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||<math>2\times\overline{J}_3</math>, <nowiki>[[</nowiki>3,5,3]]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive, edge-transitive, cell-transitive
|}
The '''bitruncated icosahedral honeycomb''', t<sub>1,2</sub>{3,5,3}, {{CDD|node|3|node_1|5|node_1|3|node}}, has [[truncated dodecahedron]] cells with a [[tetragonal disphenoid]] vertex figure.

[[File:H3 353-0110 center ultrawide.png|480px]]

==== Related honeycombs====
{{Bitruncated compact H3 honeycombs}}
{{Clear}}

=== Cantellated icosahedral honeycomb===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Cantellated icosahedral honeycomb
<!--|-
|bgcolor=#ffffff align=center colspan=2|-->
|-
|bgcolor=#e7dcc3|Type||[[Uniform honeycombs in hyperbolic space]]
|-
|width=100 bgcolor=#e7dcc3|[[Schläfli symbol]]||rr{3,5,3} or t<sub>0,2</sub>{3,5,3}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_1|3|node|5|node_1|3|node}}
|-
|bgcolor=#e7dcc3|Cells||[[rhombicosidodecahedron|rr{3,5}]] [[File:Uniform polyhedron-53-t02.png|40px]]<BR>[[icosidodecahedron|r{5,3}]] [[File:Uniform polyhedron-53-t1.png|40px]]<BR>[[triangular prism|{}x{3}]] [[File:triangular prism.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[square]] {4}<BR>[[pentagon]] {5}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Cantellated icosahedral honeycomb verf.png|80px]]<BR>[[wedge (geometry)|wedge]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||<math>\overline{J}_3</math>, [3,5,3]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive
|}
The '''cantellated icosahedral honeycomb''', t<sub>0,2</sub>{3,5,3}, {{CDD|node_1|3|node|5|node_1|3|node}}, has [[rhombicosidodecahedron]], [[icosidodecahedron]], and [[triangular prism]] cells, with a [[wedge (geometry)|wedge]] vertex figure.

[[File:H3 353-1010 center ultrawide.png|480px]]

==== Related honeycombs====
{{Cantellated compact H3 honeycombs}}
{{Clear}}

=== Cantitruncated icosahedral honeycomb===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Cantitruncated icosahedral honeycomb
<!--|-
|bgcolor=#ffffff align=center colspan=2|-->
|-
|bgcolor=#e7dcc3|Type||[[Uniform honeycombs in hyperbolic space]]
|-
|width=100 bgcolor=#e7dcc3|[[Schläfli symbol]]||tr{3,5,3} or t<sub>0,1,2</sub>{3,5,3}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_1|3|node_1|5|node_1|3|node}}
|-
|bgcolor=#e7dcc3|Cells||[[truncated icosidodecahedron|tr{3,5}]] [[File:Uniform polyhedron-53-t012.png|40px]]<BR>[[truncated dodecahedron|t{5,3}]] [[File:Uniform polyhedron-53-t01.png|40px]]<BR>[[Triangular prism|{}x{3}]] [[File:Triangular prism.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[square]] {4}<BR>[[hexagon]] {6}<BR>[[decagon]] {10}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Cantitruncated icosahedral honeycomb verf.png|80px]]<BR>[[mirrored sphenoid]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||<math>\overline{J}_3</math>, [3,5,3]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive
|}
The '''cantitruncated icosahedral honeycomb''', t<sub>0,1,2</sub>{3,5,3}, {{CDD|node_1|3|node_1|5|node_1|3|node}}, has [[truncated icosidodecahedron]], [[truncated dodecahedron]], and [[triangular prism]] cells, with a [[mirrored sphenoid]] [[vertex figure]].

[[File:H3 353-1110 center ultrawide.png|480px]]

==== Related honeycombs====
{{Cantitruncated compact H3 honeycombs}}
{{Clear}}

=== Runcinated icosahedral honeycomb===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Runcinated icosahedral honeycomb
<!--|-
|bgcolor=#ffffff align=center colspan=2|-->
|-
|bgcolor=#e7dcc3|Type||[[Uniform honeycombs in hyperbolic space]]
|-
|width=100 bgcolor=#e7dcc3|[[Schläfli symbol]]||t<sub>0,3</sub>{3,5,3} 
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_1|3|node|5|node|3|node_1}}
|-
|bgcolor=#e7dcc3|Cells||[[regular icosahedron|{3,5}]] [[File:Uniform polyhedron-53-t2.png|40px]]<BR>[[triangular prism|{}×{3}]] [[File:Triangular prism.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[square]] {4}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Runcinated icosahedral honeycomb verf.png|80px]]<BR>[[pentagonal antiprism]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||<math>2\times\overline{J}_3</math>, <nowiki>[[</nowiki>3,5,3]]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive, edge-transitive
|}
The '''runcinated icosahedral honeycomb''', t<sub>0,3</sub>{3,5,3}, {{CDD|node_1|3|node|5|node|3|node_1}}, has [[regular icosahedron|icosahedron]] and [[triangular prism]] cells, with a [[pentagonal antiprism]] vertex figure.

[[File:H3 353-1001 center ultrawide.png|480px]]
: Viewed from center of triangular prism

==== Related honeycombs====
{{Runcinated compact H3 honeycombs}}
{{Clear}}

=== Runcitruncated icosahedral honeycomb===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Runcitruncated icosahedral honeycomb
<!--|-
|bgcolor=#ffffff align=center colspan=2|-->
|-
|bgcolor=#e7dcc3|Type||[[Uniform honeycombs in hyperbolic space]]
|-
|width=100 bgcolor=#e7dcc3|[[Schläfli symbol]]||t<sub>0,1,3</sub>{3,5,3}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_1|3|node_1|5|node|3|node_1}}
|-
|bgcolor=#e7dcc3|Cells||[[truncated icosahedron|t{3,5}]] [[File:Uniform polyhedron-53-t12.png|40px]]<BR>[[rhombicosidodecahedron|rr{3,5}]] [[File:Uniform polyhedron-53-t02.png|40px]]<BR>[[triangular prism|{}×{3}]] [[File:Triangular prism.png|40px]]<BR>[[hexagonal prism|{}×{6}]] [[File:Hexagonal prism.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[square]] {4}<BR>[[pentagon]] {5}<BR>[[hexagon]] {6}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Runcitruncated icosahedral honeycomb verf.png|80px]]<BR>[[isosceles trapezoid|isosceles-trapezoidal]] [[pyramid (geometry)|pyramid]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||<math>\overline{J}_3</math>, [3,5,3]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive
|}
The '''runcitruncated icosahedral honeycomb''', t<sub>0,1,3</sub>{3,5,3}, {{CDD|node_1|3|node_1|5|node|3|node_1}}, has [[truncated icosahedron]], [[rhombicosidodecahedron]], [[hexagonal prism]], and [[triangular prism]] cells, with an [[isosceles trapezoid|isosceles-trapezoidal]] [[pyramid (geometry)|pyramid]] [[vertex figure]].

The '''runcicantellated icosahedral honeycomb''' is equivalent to the runcitruncated icosahedral honeycomb.

[[File:H3 353-1101 center ultrawide.png|480px]]
: Viewed from center of triangular prism

==== Related honeycombs====
{{Runcitruncated compact H3 honeycombs}}
{{Clear}}

=== Omnitruncated icosahedral honeycomb===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Omnitruncated icosahedral honeycomb
<!--|-
|bgcolor=#ffffff align=center colspan=2|-->
|-
|bgcolor=#e7dcc3|Type||[[Uniform honeycombs in hyperbolic space]]
|-
|width=100 bgcolor=#e7dcc3|[[Schläfli symbol]]||t<sub>0,1,2,3</sub>{3,5,3}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_1|3|node_1|5|node_1|3|node_1}}
|-
|bgcolor=#e7dcc3|Cells||[[truncated icosidodecahedron|tr{3,5}]] [[File:Uniform polyhedron-53-t012.png|40px]]<BR>[[hexagonal prism|{}×{6}]] [[File:Hexagonal prism.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[square]] {4}<BR>[[hexagon]] {6}<BR>[[dodecagon]] {10}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Omnitruncated icosahedral honeycomb verf.png|80px]]<BR>[[phyllic disphenoid]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||<math>2\times\overline{J}_3</math>, <nowiki>[[</nowiki>3,5,3]]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive
|}
The '''omnitruncated icosahedral honeycomb''', t<sub>0,1,2,3</sub>{3,5,3}, {{CDD|node_1|3|node_1|5|node_1|3|node_1}}, has [[truncated icosidodecahedron]] and [[hexagonal prism]] cells, with a [[phyllic disphenoid]] vertex figure.

[[File:H3 353-1111 center ultrawide.png|480px]]
: Centered on hexagonal prism

==== Related honeycombs====
{{Omnitruncated compact H3 honeycombs}}

{{Clear}}

=== Omnisnub icosahedral honeycomb===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Omnisnub icosahedral honeycomb
<!--|-
|bgcolor=#ffffff align=center colspan=2|-->
|-
|bgcolor=#e7dcc3|Type||[[Uniform honeycombs in hyperbolic space]]
|-
|width=100 bgcolor=#e7dcc3|[[Schläfli symbol]]||h(t<sub>0,1,2,3</sub>{3,5,3}) 
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_h|3|node_h|5|node_h|3|node_h}}
|-
|bgcolor=#e7dcc3|Cells||[[Snub dodecahedron|sr{3,5}]] [[File:Uniform polyhedron-53-s012.png|40px]]<BR>[[octahedron|s{2,3}]] [[File:Trigonal antiprism.png|40px]]<BR>irr. [[Tetrahedron|{3,3}]] [[File:tetrahedron.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[pentagon]] {5}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Snub icosahedral honeycomb verf.png|80px]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||<nowiki>[[</nowiki>3,5,3]]<sup>+</sup>
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive
|}
The '''omnisnub icosahedral honeycomb''', h(t<sub>0,1,2,3</sub>{3,5,3}), {{CDD|node_h|3|node_h|5|node_h|3|node_h}}, has [[snub dodecahedron]], [[octahedron]], and [[tetrahedron]] cells, with an irregular vertex figure. It is [[vertex-transitive]], but cannot be made with uniform cells.
{{Clear}}

=== Partially diminished icosahedral honeycomb===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Partially diminished icosahedral honeycomb<BR>Parabidiminished icosahedral honeycomb
<!--|-
|bgcolor=#ffffff align=center colspan=2|-->
|-
|bgcolor=#e7dcc3|Type||[[Uniform honeycombs in hyperbolic space|Uniform honeycombs]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||pd{3,5,3}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]|| -
|-
|bgcolor=#e7dcc3|Cells||[[regular dodecahedron|{5,3}]] [[File:Uniform polyhedron-53-t0.png|40px]]<BR>[[pentagonal antiprism|s{2,5}]] [[File:Pentagonal antiprism.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[pentagon]] {5}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Partial truncation order-3 icosahedral honeycomb verf.png|80px]]<BR>[[tetrahedrally diminished dodecahedron|tetrahedrally diminished<BR>dodecahedron]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||<sup>1</sup>/<sub>5</sub>[3,5,3]<sup>+</sup>
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive
|}
The '''partially diminished icosahedral honeycomb''' or '''[[parabidiminished icosahedron|parabidiminished icosahedral]] honeycomb''', pd{3,5,3}, is a non-Wythoffian uniform honeycomb with [[regular dodecahedron|dodecahedron]] and [[pentagonal antiprism]] cells, with a [[tetrahedrally diminished dodecahedron]] vertex figure. The icosahedral cells of the {3,5,3} are [[diminishment (geometry)|diminished]] at opposite vertices (parabidiminished), leaving a [[pentagonal antiprism]] ([[parabidiminished icosahedron]]) core, and creating new dodecahedron cells above and below.<ref>Wendy Y. Krieger, Walls and bridges: The view from six dimensions, ''Symmetry: Culture and Science'' Volume 16, Number 2, pages 171–192 (2005) [http://symmetry.hu/content/aus_journal_content_abs_2005_16_2.html] {{Webarchive|url=https://web.archive.org/web/20131007060826/http://symmetry.hu/content/aus_journal_content_abs_2005_16_2.html |date=2013-10-07 }}</ref><ref>{{Cite web|url=http://www.bendwavy.org/klitzing/incmats/pt353.htm|title = Pd{3,5,3}|author=Dr. Richard Klitzing|publisher=bendwavy.org}}</ref>

[[File:H3 353-pd center ultrawide.png|320px]]

[[File:H3 353-pd center ultrawide2.png|320px]]
{{Clear}}

== See also ==
* [[Convex uniform honeycombs in hyperbolic space]]
* [[List of regular polytopes#Tessellations of hyperbolic 3-space|Regular tessellations of hyperbolic 3-space]]
* [[Seifert–Weber space]]
* [[11-cell]] - An [[abstract polytope|abstract regular polychoron]] which shares the {3,5,3} [[Schläfli symbol]].

== References ==
{{reflist}}

*[[H.S.M. Coxeter|Coxeter]], ''[[Regular Polytopes (book)|Regular Polytopes]]'', 3rd. ed., Dover Publications, 1973. {{ISBN|0-486-61480-8}}. (Tables I and II: Regular polytopes and honeycombs, pp.&nbsp;294–296)
*[[H.S.M. Coxeter|Coxeter]], ''The Beauty of Geometry: Twelve Essays'', Dover Publications, 1999 {{ISBN|0-486-40919-8}} (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II, III, IV, V, p212-213)
* [[Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript
** [[Norman Johnson (mathematician)|N.W. Johnson]]: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966 
** N.W. Johnson: ''Geometries and Transformations'', (2018) Chapter 13: Hyperbolic Coxeter groups
* {{KlitzingPolytopes|../incmats/x3o5o3o.htm|Hyperbolic H3 honeycombs|hyperbolic order 3 icosahedral tesselation}}

[[Category:Regular 3-honeycombs]]
[[Category:Self-dual tilings]]