Editing Icosidodecahedron (section)

==Related polytopes==
The icosidodecahedron is a [[Rectification (geometry)|rectified]] [[dodecahedron]] and also a rectified [[icosahedron]], existing as the full-edge truncation between these regular solids.

The icosidodecahedron contains 12 pentagons of the [[dodecahedron]] and 20 triangles of the [[icosahedron]]:
{{Icosahedral truncations}}

The icosidodecahedron exists in a sequence of symmetries of quasiregular polyhedra and tilings with [[vertex configuration]]s (3.''n'')<sup>2</sup>, progressing from tilings of the sphere to the Euclidean plane and into the hyperbolic plane. With [[orbifold notation]] symmetry of *''n''32 all of these tilings are [[wythoff construction]] within a [[fundamental domain]] of symmetry, with generator points at the right angle corner of the domain.<ref>[[Harold Scott MacDonald Coxeter|Coxeter]] ''[[Regular Polytopes (book)|Regular Polytopes]]'', Third edition, (1973), Dover edition, {{ISBN|0-486-61480-8}} (Chapter V: The Kaleidoscope, Section: 5.7 Wythoff's construction)</ref><ref>[http://www.google.com/search?q=Two-Dimensional+Symmetry+Mutation ''Two Dimensional symmetry Mutations'' by Daniel Huson]</ref>
{{Quasiregular3 small table}}

{{Quasiregular5 table}}

=== Related polyhedra ===
[[File:Icosidecahedron in truncated cube.png|120px|thumb|A topological icosi&shy;dodeca&shy;hedron in truncated cube, inserting 6 vertices in center of octagons, and dissecting them into 2 pentagons and 2 triangles.]]
The [[truncated cube]] can be turned into an icosidodecahedron by dividing the octagons into two pentagons and two triangles. It has [[pyritohedral symmetry]].

Eight [[uniform star polyhedron|uniform star polyhedra]] share the same [[vertex arrangement]]. Of these, two also share the same [[edge arrangement]]: the [[small icosihemidodecahedron]] (having the triangular faces in common), and the [[small dodecahemidodecahedron]] (having the pentagonal faces in common). The vertex arrangement is also shared with the [[uniform polyhedron compound|compounds]] of [[compound of five octahedra|five octahedra]] and of [[compound of five tetrahemihexahedra|five tetrahemihexahedra]].
{|class="wikitable" width="400" style="vertical-align:top;text-align:center"
|align=center|[[Image:Icosidodecahedron.png|100px]]<br>Icosidodecahedron
|align=center|[[Image:Small icosihemidodecahedron.png|100px]]<br>[[Small icosihemidodecahedron]]
|align=center|[[Image:Small dodecahemidodecahedron.png|100px]]<br>[[Small dodecahemidodecahedron]]
|-
|align=center|[[Image:Great icosidodecahedron.png|100px]]<br>[[Great icosidodecahedron]]
|align=center|[[Image:Great dodecahemidodecahedron.png|100px]]<br>[[Great dodecahemidodecahedron]]
|align=center|[[Image:Great icosihemidodecahedron.png|100px]]<br>[[Great icosihemidodecahedron]]
|-
|align=center|[[Image:Dodecadodecahedron.png|100px]]<br>[[Dodecadodecahedron]]
|align=center|[[Image:Small dodecahemicosahedron.png|100px]]<br>[[Small dodecahemicosahedron]]
|align=center|[[Image:Great dodecahemicosahedron.png|100px]]<br>[[Great dodecahemicosahedron]]
|-
|align=center|[[Image:Compound of five octahedra.png|100px]]<br>[[Compound of five octahedra]]
|align=center|[[Image:UC18-5 tetrahemihexahedron.png|100px]]<br>[[Compound of five tetrahemihexahedra]]
|}

=== Related polychora ===

In four-dimensional geometry, the icosidodecahedron appears in the [[Regular polytopes|regular]] [[600-cell]] as the equatorial slice that belongs to the vertex-first passage of the 600-cell through 3D space. In other words: the 30 vertices of the 600-cell which lie at arc distances of 90 degrees on its circumscribed [[hypersphere]] from a pair of opposite vertices, are the vertices of an icosidodecahedron. The wireframe figure of the 600-cell consists of 72 flat regular decagons. Six of these are the equatorial decagons to a pair of opposite vertices, and these six form the wireframe figure of an icosidodecahedron.

If a [[600-cell]] is [[Stereographic projection|stereographically projected]] to 3-space about any vertex and all points are normalised, the [[geodesic]]s upon which edges fall comprise the icosidodecahedron's [[Barycentric subdivision#Barycentric subdivision of a convex polytope|barycentric subdivision]].