View source for Runcinated 5-cell

{| class=wikitable align=right width=400
|- align=center valign=top
|[[File:4-simplex t0.svg|200px]]<BR>[[5-cell]]<BR>{{CDD|node_1|3|node|3|node|3|node}}
|[[File:4-simplex t03.svg|200px]]<BR>Runcinated 5-cell<BR>{{CDD|node_1|3|node|3|node|3|node_1}}
|- align=center valign=top
|[[File:4-simplex t013.svg|200px]]<BR>Runcitruncated 5-cell<BR>{{CDD|node_1|3|node_1|3|node|3|node_1}}
|[[File:4-simplex t0123.svg|200px]]<BR>Omnitruncated 5-cell<BR><small>(Runcicantitruncated 5-cell)</small><BR>{{CDD|node_1|3|node_1|3|node_1|3|node_1}}
|-
!colspan=2|[[Orthogonal projection]]s in A<sub>4</sub> [[Coxeter plane]]
|}
In four-dimensional [[geometry]], a '''runcinated 5-cell''' is a convex [[uniform 4-polytope]], being a [[runcination]] (a 3rd order truncation, up to [[Face (geometry)|face-planing]]) of the regular [[5-cell]].

There are 3 unique degrees of runcinations of the 5-cell, including with permutations, truncations, and cantellations.

==Runcinated 5-cell==

{| class="wikitable" align="right" style="margin-left:10px" width="250"
|-
|bgcolor=#e7dcc3 align=center colspan=3|'''Runcinated 5-cell'''
|-
|bgcolor=#ffffff align=center colspan=3|[[File:Schlegel half-solid runcinated 5-cell.png|220px]]<BR>[[Schlegel diagram]] with half of the tetrahedral cells visible.
|-
|bgcolor=#e7dcc3|Type
|colspan=2|[[Uniform 4-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]
|colspan=2|t<sub>0,3</sub>{3,3,3}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]
|colspan=2|{{CDD|node_1|3|node|3|node|3|node_1}}<BR>or {{CDD|branch|3ab|nodes_11}} or {{CDD|branch|3ab-cross|nodes_11}}
|-
|bgcolor=#e7dcc3|Cells
|30
|10 [[Tetrahedron|(''3.3.3'')]] [[File:Tetrahedron.png|20px]]<BR>20 [[Triangular prism|(''3.4.4'')]] [[File:Triangular prism.png|20px]]
|-
|bgcolor=#e7dcc3|Faces
|70
|40 [[triangle|{3}]]<BR>30 [[square (geometry)|{4}]]
|-
|bgcolor=#e7dcc3|Edges
|colspan=2|60
|-
|bgcolor=#e7dcc3|Vertices
|colspan=2|20
|-
|bgcolor=#e7dcc3|[[Vertex figure]]
|colspan=2|[[File:Runcinated 5-cell verf.png|80px]]<BR>(Elongated equilateral-triangular antiprism)
|-
|bgcolor=#e7dcc3|[[Coxeter group|Symmetry group]]
|colspan=2|[[Automorphism|Aut]](A<sub>4</sub>), {{brackets|3,3,3}}, order 240
|-
|bgcolor=#e7dcc3|Properties
|colspan=2|[[Convex polytope|convex]], [[isogonal figure|isogonal]] [[isotoxal figure|isotoxal]]
|-
|bgcolor=#e7dcc3|Uniform index
|colspan=2|''[[Cantellated 5-cell|4]]'' 5 ''[[Bitruncated 5-cell|6]]''
|}
The '''runcinated 5-cell''' or '''small prismatodecachoron''' is constructed by [[Expansion (geometry)|expanding]] the [[cell (mathematics)|cells]] of a [[5-cell]] radially and filling in the gaps with triangular [[prism (geometry)|prism]]s (which are the face prisms and edge figures) and [[tetrahedron|tetrahedra]] (cells of the dual 5-cell). It consists of 10 tetrahedra and 20 triangular prisms. The 10 tetrahedra correspond with the cells of a 5-cell and its dual.

Topologically, under its highest symmetry, <nowiki>[[3,3,3]]</nowiki>, there is only one geometrical form, containing 10 tetrahedra and 20 uniform triangular prisms. The rectangles are always squares because the two pairs of edges correspond to the edges of the two sets of 5 regular tetrahedra each in dual orientation, which are made equal under extended symmetry.

[[Emanuel Lodewijk Elte|E. L. Elte]] identified it in 1912 as a semiregular polytope.

===Alternative names ===

* Runcinated [[5-cell]] ([[Norman Johnson (mathematician)|Norman Johnson]])
* Runcinated pentachoron
* Runcinated [[Simplex|4-simplex]]
* [[Expansion (geometry)|Expanded]] 5-cell/4-simplex/pentachoron
* Small prismatodecachoron (Acronym: Spid) (Jonathan Bowers)

===Structure===

Two of the ten tetrahedral cells meet at each vertex. The triangular prisms lie between them, joined to them by their triangular faces and to each other by their square faces. Each triangular prism is joined to its neighbouring triangular prisms in ''anti'' orientation (i.e., if edges A and B in the shared square face are joined to the triangular faces of one prism, then it is the other two edges that are joined to the triangular faces of the other prism); thus each pair of adjacent prisms, if rotated into the same [[hyperplane]], would form a [[gyrobifastigium]].

=== Dissection ===
The ''runcinated 5-cell'' can be dissected by a central [[cuboctahedron]] into two [[tetrahedral cupola]]. This dissection is analogous to the 3D [[cuboctahedron]] being dissected by a central hexagon into two [[triangular cupola]].
: [[File:4D Tetrahedral Cupola-perspective-cuboctahedron-first.png|160px]]

===Images ===

{{4-simplex2 Coxeter plane graphs|t03|100}}

{| class="wikitable" width=200
|[[File:Runcinated pentatope.png|200px]]<BR>View inside of a 3-sphere projection [[Schlegel diagram]] with its 10 tetrahedral cells
|[[File:Small prismatodecachoron net.png|200px]]<BR>[[Net (polyhedron)|Net]]
|}

===Coordinates===

The [[Cartesian coordinate]]s of the vertices of an origin-centered runcinated 5-cell with edge length&nbsp;2 are:

{| class=wikitable
|
:<math>\pm\left(\sqrt{\frac{5}{2}},\ \frac{1}{\sqrt{6}},\  \frac{1}{\sqrt{3}},\  \pm1\right)</math>
:<math>\pm\left(\sqrt{\frac{5}{2}},\ \frac{1}{\sqrt{6}},\  \frac{-2}{\sqrt{3}},\ 0\right)</math>
:<math>\pm\left(\sqrt{\frac{5}{2}},\ -\sqrt{\frac{3}{2}},\ 0,\                   0\right)</math>
|
:<math>\pm\left(0,\                  2\sqrt{\frac{2}{3}},\ \frac{1}{\sqrt{3}},\  \pm1\right)</math>
:<math>\pm\left(0,\                  2\sqrt{\frac{2}{3}},\ \frac{-2}{\sqrt{3}},\ 0\right)</math>
:<math>   \left(0,\                  0,\                   \pm\sqrt{3},\         \pm1\right)</math>
:<math>   \left(0,\                  0,\                   0,\                   \pm2\right)</math>
|}

An alternate simpler set of coordinates can be made in 5-space, as 20 permutations of:
: (0,1,1,1,2)

This construction exists as one of 32 [[orthant]] [[Facet (geometry)|facets]] of the [[runcinated 5-orthoplex]].

A second construction in 5-space, from the center of a [[rectified 5-orthoplex]] is given by coordinate permutations of:
: (1,-1,0,0,0)

=== Root vectors ===

Its 20 vertices represent the root vectors of the [[simple Lie group]] A<sub>4</sub>. It is also the [[vertex figure]] for the [[5-cell honeycomb]] in 4-space.

===Cross-sections===

The maximal cross-section of the runcinated 5-cell with a 3-dimensional [[hyperplane]] is a [[cuboctahedron]]. This cross-section divides the runcinated 5-cell into two [[Cupola (geometry)#Hypercupolas|tetrahedral hypercupolae]] consisting of 5 tetrahedra and 10 triangular prisms each.

===Projections===

The tetrahedron-first [[orthographic projection]] of the runcinated 5-cell into 3-dimensional space has a [[cuboctahedron|cuboctahedral]] envelope. The structure of this projection is as follows:

* The cuboctahedral envelope is divided internally as follows:
:* Four flattened tetrahedra join 4 of the triangular faces of the cuboctahedron to a central tetrahedron. These are the images of 5 of the tetrahedral cells.
:* The 6 square faces of the cuboctahedron are joined to the edges of the central tetrahedron via distorted triangular prisms. These are the images of 6 of the triangular prism cells.
:* The other 4 triangular faces are joined to the central tetrahedron via 4  triangular prisms (distorted by projection). These are the images of another 4 of the triangular prism cells.
:* This accounts for half of the runcinated 5-cell (5 tetrahedra and 10 triangular prisms), which may be thought of as the 'northern hemisphere'.
* The other half, the 'southern hemisphere', corresponds to an isomorphic division of the cuboctahedron in dual orientation, in which the central tetrahedron is dual to the one in the first half. The triangular faces of the cuboctahedron join the triangular prisms in one hemisphere to the flattened tetrahedra in the other hemisphere, and vice versa. Thus, the southern hemisphere contains another 5 tetrahedra and another 10 triangular prisms, making the total of 10 tetrahedra and 20 triangular prisms.

=== Related skew polyhedron ===

The [[regular skew polyhedron]], {4,6|3}, exists in 4-space with 6 squares around each vertex, in a zig-zagging nonplanar vertex figure. These square faces can be seen on the runcinated 5-cell, using all 60 edges and 20 vertices. The 40 triangular faces of the runcinated 5-cell can be seen as removed. The dual regular skew polyhedron, {6,4|3}, is similarly related to the hexagonal faces of the [[bitruncated 5-cell]].

== Runcitruncated 5-cell==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
|-
|bgcolor=#e7dcc3 align=center colspan=3|'''Runcitruncated 5-cell'''
|-
|bgcolor=#ffffff align=center colspan=3|[[File:Schlegel half-solid runcitruncated 5-cell.png|220px]]<BR>[[Schlegel diagram]] with<BR>cuboctahedral cells shown
|-
|bgcolor=#e7dcc3|Type
|colspan=2|[[Uniform 4-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]
|colspan=2| t<sub>0,1,3</sub>{3,3,3}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]
|colspan=2|{{CDD|node_1|3|node_1|3|node|3|node_1}}
|-
|bgcolor=#e7dcc3|Cells
|30
|5 [[File:truncated tetrahedron.png|20px]][[truncated tetrahedron|(3.6.6)]]<BR>10 [[File:hexagonal prism.png|20px]][[hexagonal prism|(4.4.6)]]<BR>10 [[File:triangular prism.png|20px]][[triangular prism|(3.4.4)]]<BR>5 [[File:cuboctahedron.png|20px]][[cuboctahedron|(3.4.3.4)]]
|-
|bgcolor=#e7dcc3|Faces
|120
|40 [[Triangle|{3}]]<BR>60 [[Square (geometry)|{4}]]<BR>20 [[Hexagon|{6}]]
|-
|bgcolor=#e7dcc3|Edges
|colspan=2|150
|-
|bgcolor=#e7dcc3|Vertices
|colspan=2|60
|-
|bgcolor=#e7dcc3|[[Vertex figure]]
|colspan=2|[[File:Runcitruncated 5-cell verf.png|80px]]<BR>(Rectangular pyramid)
|-
|bgcolor=#e7dcc3|[[Coxeter group]]
|colspan=2|A<sub>4</sub>, [3,3,3], order 120
|-
|bgcolor=#e7dcc3|Properties
|colspan=2|[[Convex polytope|convex]], [[isogonal figure|isogonal]]
|-
|bgcolor=#e7dcc3|Uniform index
|colspan=2|''[[Cantitruncated 5-cell|7]]'' 8 ''[[Omnitruncated 5-cell|9]]''
|}
[[File:Prismatorhombated pentachoron net.png|thumb|right|[[Net (polyhedron)|Net]]]]
The '''runcitruncated [[5-cell]]''' or '''prismatorhombated pentachoron''' is composed of 60 vertices, 150 edges, 120 faces, and 30 cells. The cells are: 5 [[truncated tetrahedron|truncated tetrahedra]], 10 [[hexagonal prism]]s, 10 [[triangular prism]]s, and 5 [[cuboctahedron|cuboctahedra]]. Each vertex is surrounded by five cells: one truncated tetrahedron, two hexagonal prisms, one triangular prism, and one cuboctahedron; the [[vertex figure]] is a rectangular pyramid.

=== Alternative names ===
* Runcitruncated pentachoron
* Runcitruncated [[Simplex|4-simplex]]
* Diprismatodispentachoron
* Prismatorhombated pentachoron (Acronym: prip) (Jonathan Bowers)

=== Images ===
{{4-simplex Coxeter plane graphs|t013|100}}

{| class=wikitable width=500
|[[File:Runcitruncated 5cell.png|250px]]<BR>[[Schlegel diagram]] with its 40 blue triangular faces and its 60 green quad faces.
|[[File:Runcitruncated 5cell part.png|250px]]<BR>Central part of Schlegel diagram.
|}

=== Coordinates ===

The [[Cartesian coordinate]]s of an origin-centered runcitruncated 5-cell having edge length&nbsp;2 are:

{| class="wikitable collapsible collapsed"
! colspan=2|Coordinates
|-
|
:<math>\left(\frac{7}{\sqrt{10}},\ \sqrt{\frac{3}{2}},\    \pm\sqrt{3},\         \pm1\right)</math>
:<math>\left(\frac{7}{\sqrt{10}},\ \sqrt{\frac{3}{2}},\    0,\                   \pm2\right)</math>
:<math>\left(\frac{7}{\sqrt{10}},\ \frac{-1}{\sqrt{6}},\   \frac{2}{\sqrt{3}},\  \pm2\right)</math>
:<math>\left(\frac{7}{\sqrt{10}},\ \frac{-1}{\sqrt{6}},\   \frac{-4}{\sqrt{3}},\ 0\right)</math>
:<math>\left(\frac{7}{\sqrt{10}},\ \frac{-5}{\sqrt{6}},\   \frac{1}{\sqrt{3}},\  \pm1\right)</math>
:<math>\left(\frac{7}{\sqrt{10}},\ \frac{-5}{\sqrt{6}},\   \frac{-2}{\sqrt{3}},\ 0\right)</math>
:<math>\left(\sqrt{\frac{2}{5}},\  \pm\sqrt{6},\           \pm\sqrt{3},\         \pm1\right)</math>
:<math>\left(\sqrt{\frac{2}{5}},\  \pm\sqrt{6},\           0,\                   \pm2\right)</math>
:<math>\left(\sqrt{\frac{2}{5}},\  \sqrt{\frac{2}{3}},\    \frac{5}{\sqrt{3}},\  \pm1\right)</math>
|
:<math>\left(\sqrt{\frac{2}{5}},\  \sqrt{\frac{2}{3}},\    \frac{-1}{\sqrt{3}},\ \pm3\right)</math>
:<math>\left(\sqrt{\frac{2}{5}},\  \sqrt{\frac{2}{3}},\    \frac{-4}{\sqrt{3}},\ \pm2\right)</math>
:<math>\left(\sqrt{\frac{2}{5}},\  -\sqrt{\frac{2}{3}},\   \frac{4}{\sqrt{3}},\  \pm2\right)</math>
:<math>\left(\sqrt{\frac{2}{5}},\  -\sqrt{\frac{2}{3}},\   \frac{1}{\sqrt{3}},\  \pm3\right)</math>
:<math>\left(\sqrt{\frac{2}{5}},\  -\sqrt{\frac{2}{3}},\   \frac{-5}{\sqrt{3}},\ \pm1\right)</math>
:<math>\left(\frac{-3}{\sqrt{10}},\ \frac{5}{\sqrt{6}},\   \frac{2}{\sqrt{3}},\  \pm2\right)</math>
:<math>\left(\frac{-3}{\sqrt{10}},\ \frac{5}{\sqrt{6}},\   \frac{-4}{\sqrt{3}},\ 0\right)</math>
:<math>\left(\frac{-3}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\   \frac{4}{\sqrt{3}},\  \pm2\right)</math>
:<math>\left(\frac{-3}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\   \frac{1}{\sqrt{3}},\  \pm3\right)</math>
|
:<math>\left(\frac{-3}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\   \frac{-5}{\sqrt{3}},\ \pm1\right)</math>
:<math>\left(\frac{-3}{\sqrt{10}},\ \frac{-7}{\sqrt{6}},\  \frac{2}{\sqrt{3}},\  0\right)</math>
:<math>\left(\frac{-3}{\sqrt{10}},\ \frac{-7}{\sqrt{6}},\  \frac{-1}{\sqrt{3}},\ \pm1\right)</math>
:<math>\left(-4\sqrt{\frac{2}{5}},\ 2\sqrt{\frac{2}{3}},\  \frac{1}{\sqrt{3}},\  \pm1\right)</math>
:<math>\left(-4\sqrt{\frac{2}{5}},\ 2\sqrt{\frac{2}{3}},\  \frac{-2}{\sqrt{3}},\ 0\right)</math>
:<math>\left(-4\sqrt{\frac{2}{5}},\ 0,\                    \pm\sqrt{3},\         \pm1\right)</math>
:<math>\left(-4\sqrt{\frac{2}{5}},\ 0,\                    0,\                   \pm2\right)</math>
:<math>\left(-4\sqrt{\frac{2}{5}},\ -2\sqrt{\frac{2}{3}},\ \frac{2}{\sqrt{3}},\  0\right)</math>
:<math>\left(-4\sqrt{\frac{2}{5}},\ -2\sqrt{\frac{2}{3}},\ \frac{-1}{\sqrt{3}},\ 1\right)</math>
|}

The vertices can be more simply constructed on a [[hyperplane]] in 5-space, as the [[permutation]]s of:
: (0,1,1,2,3)

This construction is from the positive [[orthant]] [[facet (geometry)|facet]] of the [[runcitruncated 5-orthoplex]].

{{-}}
== Omnitruncated 5-cell==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
|-
|bgcolor=#e7dcc3 align=center colspan=3|'''Omnitruncated 5-cell'''
|-
|bgcolor=#ffffff align=center colspan=3|[[File:Schlegel half-solid omnitruncated 5-cell.png|220px]]<BR>[[Schlegel diagram]] with half of the truncated octahedral cells shown.
|-
|bgcolor=#e7dcc3|Type
|colspan=2|[[Uniform 4-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]
|colspan=2| t<sub>0,1,2,3</sub>{3,3,3}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]
|colspan=2|{{CDD|node_1|3|node_1|3|node_1|3|node_1}}<BR>or {{CDD|branch_11|3ab|nodes_11}} or {{CDD|branch_11|3ab-cross|nodes_11}}
|-
|bgcolor=#e7dcc3|Cells
|30
|10 [[File:truncated octahedron.png|20px]][[truncated octahedron|(4.6.6)]]<BR>20 [[File:hexagonal prism.png|20px]][[hexagonal prism|(4.4.6)]]
|-
|bgcolor=#e7dcc3|Faces
|150
|90{4}<BR>60{6}
|-
|bgcolor=#e7dcc3|Edges
|colspan=2|240
|-
|bgcolor=#e7dcc3|Vertices
|colspan=2|120
|-
|bgcolor=#e7dcc3|[[Vertex figure]]
|colspan=2|[[File:Omnitruncated 5-cell vertex figure.png|80px]]<BR>[[Phyllic disphenoid]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]
|colspan=2|[[Automorphism|Aut]](A<sub>4</sub>), {{brackets|3,3,3}}, order 240
|-
|bgcolor=#e7dcc3|Properties
|colspan=2|[[Convex polytope|convex]], [[isogonal figure|isogonal]], [[zonotope]]
|-
|bgcolor=#e7dcc3|Uniform index
|colspan=2|''[[Runcitruncated 5-cell|8]]'' 9 ''[[Tesseract|10]]''
|}
The '''omnitruncated 5-cell''' or '''great prismatodecachoron''' is composed of 120 vertices, 240 edges, 150 faces (90 [[Square (geometry)|squares]] and 60 [[hexagon]]s), and 30 cells. The cells are: 10 [[truncated octahedron|truncated octahedra]], and 20 [[hexagonal prism]]s. Each vertex is surrounded by four cells: two truncated octahedra, and two hexagonal prisms, arranged in two phyllic disphenoidal [[vertex figure]]s.

[[Coxeter]] calls this '''Hinton's polytope''' after [[Charles Howard Hinton|C. H. Hinton]],  who described it in his book ''The Fourth Dimension'' in 1906.  It forms a [[#Tessellations|uniform honeycomb]] which Coxeter calls [[Hinton's honeycomb]].<ref name=cox>''The Beauty of Geometry: Twelve Essays'' (1999), Dover Publications, {{LCCN|99035678}}, {{isbn|0-486-40919-8}} (The classification of Zonohededra, page 73)</ref>

=== Alternative names ===
* Omnitruncated [[5-cell]]
* Omnitruncated [[pentachoron]]
* Omnitruncated [[Simplex|4-simplex]]
* Great prismatodecachoron (Acronym: gippid) (Jonathan Bowers)
* Hinton's polytope ([[Coxeter]])

=== Images ===
{{4-simplex2 Coxeter plane graphs|t0123|100}}

{| class=wikitable
|+ [[Net (polyhedron)|Net]]
|[[File:Great prismatodecachoron net.png|200px]]<BR> Omnitruncated 5-cell
|[[File:Dual gippid net.png|200px]]<BR>Dual to omnitruncated 5-cell
|}

=== Perspective projections ===
{| class=wikitable width=600
|+ 
|- align=center
|[[File:Omnitruncated 5-cell.png|300px]]<BR>Perspective [[Schlegel diagram]]<BR>Centered on [[truncated octahedron]]
|[[File:Omnitruncated simplex stereographic.png|300px]]<BR>[[Stereographic projection]]
|}

=== Permutohedron ===

Just as the [[truncated octahedron]] is the [[permutohedron]] of order 4, the omnitruncated 5-cell is the permutohedron of order 5.<ref>[http://www.math.ubc.ca/~holroyd/cayley/s5.html The permutahedron of order 5]</ref>
The omnitruncated 5-cell is a [[zonotope]], the [[Minkowski sum]] of five line segments parallel to the five lines through the origin and the five vertices of the 5-cell.
[[File:Omnitruncated 5Cell as Permutohedron.svg|thumb|[[Orthogonal projection]] as a [[permutohedron]]]]

=== Tessellations ===

The [[omnitruncated 5-cell honeycomb]] can tessellate 4-dimensional space by translational copies of this cell, each with 3 hypercells around each face. This honeycomb's [[Coxeter diagram]] is {{CDD||branch_11|3ab|nodes_11|split2|node_1}}.<ref>George Olshevsky, ''Uniform Panoploid Tetracombs'', manuscript (2006): Lists the tessellation as [140 of 143] ''Great-prismatodecachoric tetracomb'' (Omnitruncated pentachoric 4d honeycomb)</ref> Unlike the analogous honeycomb in three dimensions, the [[bitruncated cubic honeycomb]] which has three different [[Coxeter group]] [[Wythoff construction]]s, this honeycomb has only one such construction.<ref name=cox />

=== Symmetry ===

The '''omnitruncated 5-cell''' has extended pentachoric symmetry, {{brackets|3,3,3}}, order 240. The [[vertex figure]] of the '''omnitruncated 5-cell''' represents the [[Goursat tetrahedron#3-sphere .28finite.29 solutions|Goursat tetrahedron]] of the [3,3,3] [[Coxeter group]]. The extended symmetry comes from a 2-fold rotation across the middle order-3 branch, and is represented more explicitly as [2<sup>+</sup>[3,3,3]].
:[[File:Omnitruncated 5-cell vertex figure.png|120px]]

=== Coordinates ===

The [[Cartesian coordinate]]s of the vertices of an origin-centered omnitruncated 5-cell having edge length&nbsp;2 are:

{| class=wikitable
|
:<math>\left(\pm\sqrt{10},\ \pm\sqrt{6},\ \pm\sqrt{3},\ \pm1\right)</math>
:<math>\left(\pm\sqrt{10},\ \pm\sqrt{6},\ 0,\ \pm2\right)</math>
:<math>\pm\left(\pm\sqrt{10},\ \sqrt{\frac{2}{3}},\ \frac{5}{\sqrt{3}},\ \pm1\right)</math>
:<math>\pm\left(\pm\sqrt{10},\ \sqrt{\frac{2}{3}},\ \frac{-1}{\sqrt{3}},\ \pm3\right)</math>
:<math>\pm\left(\pm\sqrt{10},\ \sqrt{\frac{2}{3}},\ \frac{-4}{\sqrt{3}},\ \pm2\right)</math>
:<math>\left(\sqrt{\frac{5}{2}},\ 3\sqrt{\frac{3}{2}},\ \pm\sqrt{3},\ \pm1\right)</math>
:<math>\left(-\sqrt{\frac{5}{2}},\ -3\sqrt{\frac{3}{2}},\ \pm\sqrt{3},\ \pm1\right)</math>
:<math>\left(\sqrt{\frac{5}{2}},\ 3\sqrt{\frac{3}{2}},\ 0,\ \pm2\right)</math>
:<math>\left(-\sqrt{\frac{5}{2}},\ -3\sqrt{\frac{3}{2}},\ 0,\ \pm2\right)</math>
|
:<math>\pm\left(\sqrt{\frac{5}{2}},\ \frac{1}{\sqrt{6}},\ \frac{7}{\sqrt{3}},\ \pm1\right)</math>
:<math>\pm\left(\sqrt{\frac{5}{2}},\ \frac{1}{\sqrt{6}},\ \frac{-2}{\sqrt{3}},\ \pm4\right)</math>
:<math>\pm\left(\sqrt{\frac{5}{2}},\ \frac{1}{\sqrt{6}},\ \frac{-5}{\sqrt{3}},\ \pm3\right)</math>
:<math>\pm\left(\sqrt{\frac{5}{2}},\ -\sqrt{\frac{3}{2}},\ \pm2\sqrt{3},\ \pm2\right)</math>
:<math>\pm\left(\sqrt{\frac{5}{2}},\ -\sqrt{\frac{3}{2}},\ 0,\ \pm4\right)</math>
:<math>\pm\left(\sqrt{\frac{5}{2}},\ \frac{-7}{\sqrt{6}},\ \frac{5}{\sqrt{3}},\ \pm1\right)</math>
:<math>\pm\left(\sqrt{\frac{5}{2}},\ \frac{-7}{\sqrt{6}},\ \frac{-1}{\sqrt{3}},\ \pm3\right)</math>
|
:<math>\pm\left(\sqrt{\frac{5}{2}},\ \frac{-7}{\sqrt{6}},\ \frac{-4}{\sqrt{3}},\ \pm2\right)</math>
:<math>\pm\left(0,\ 4\sqrt{\frac{2}{3}},\ \frac{5}{\sqrt{3}},\ \pm1\right)</math>
:<math>\pm\left(0,\ 4\sqrt{\frac{2}{3}},\ \frac{-1}{\sqrt{3}},\ \pm3\right)</math>
:<math>\pm\left(0,\ 4\sqrt{\frac{2}{3}},\ \frac{-4}{\sqrt{3}},\ \pm2\right)</math>
:<math>\pm\left(0,\ 2\sqrt{\frac{2}{3}},\ \frac{7}{\sqrt{3}},\ \pm1\right)</math>
:<math>\pm\left(0,\ 2\sqrt{\frac{2}{3}},\ \frac{-2}{\sqrt{3}},\ \pm4\right)</math>
:<math>\pm\left(0,\ 2\sqrt{\frac{2}{3}},\ \frac{-5}{\sqrt{3}},\ \pm3\right)</math>
|}

These vertices can be more simply obtained in 5-space as the 120 [[permutation]]s of (0,1,2,3,4).
This construction is from the positive [[orthant]] [[facet (geometry)|facet]] of the [[runcicantitruncated 5-orthoplex]], t<sub>0,1,2,3</sub>{3,3,3,4}, {{CDD|node|4|node_1|3|node_1|3|node_1|3|node_1}}.

=== Related polytopes ===
Nonuniform variants with [3,3,3] symmetry and two types of truncated octahedra can be doubled by placing the two types of truncated octahedra on each other to produce a nonuniform polychoron with 10 [[truncated octahedron|truncated octahedra]], two types of 40 [[hexagonal prism]]s (20 ditrigonal prisms and 20 ditrigonal trapezoprisms), two kinds of 90 [[cube|rectangular trapezoprism]]s (30 with ''D<sub>2d</sub>'' symmetry and 60 with ''C<sub>2v</sub>'' symmetry), and 240 vertices. Its vertex figure is an irregular [[triangular bipyramid]].

[[File:Biomnitruncatodecachoron vertex figure.png|120px]]<BR>[[Vertex figure]]'''

This polychoron can then be alternated to produce another nonuniform polychoron with 10 [[icosahedron|icosahedra]], two types of 40 [[octahedron|octahedra]] (20 with ''S<sub>6</sub>'' symmetry and 20 with ''D<sub>3</sub>'' symmetry), three kinds of 210 [[tetrahedron|tetrahedra]] (30 tetragonal disphenoids, 60 phyllic disphenoids, and 120 irregular tetrahedra), and 120 vertices. It has a symmetry of [[3,3,3]<sup>+</sup>], order 120.

[[File:Alternated biomnitruncatodecachoron vertex figure.png|120px]]<BR>[[Vertex figure]]'''

=== Full snub 5-cell ===
[[File:Snub 5-cell verf.png|80px|thumb|Vertex figure for the '''omnisnub 5-cell''']]
The '''full snub 5-cell''' or '''omnisnub 5-cell''', defined as an [[Alternation (geometry)|alternation]] of the omnitruncated 5-cell, cannot be made uniform, but it can be given Coxeter diagram {{CDD|branch_hh|3ab|nodes_hh}}, and [[Coxeter notation|symmetry]] {{brackets|3,3,3}}<sup>+</sup>, order 120, and constructed from 90 cells: 10 [[icosahedron]]s, 20 [[octahedron]]s, and 60 [[tetrahedron]]s filling the gaps at the deleted vertices. It has 300 faces (triangles), 270 edges, and 60 vertices.

Topologically, under its highest symmetry, {{brackets|3,3,3}}<sup>+</sup>, the 10 icosahedra have ''T'' (chiral tetrahedral) symmetry, while the 20 octahedra have ''D<sub>3</sub>'' symmetry and the 60 tetrahedra have ''C<sub>2</sub>'' symmetry<ref>http://www.bendwavy.org/klitzing/incmats/s3s3s3s.htm</ref>.

== Related polytopes ==

These polytopes are a part of a family of 9 [[Uniform 4-polytope]] constructed from the [3,3,3] [[Coxeter group]].
{{Pentachoron family}}

== Notes==
{{reflist}}

== References ==
* [[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]:
** H.S.M. Coxeter, ''Regular Polytopes'', 3rd Edition, Dover New York, 1973
** '''Kaleidoscopes: Selected Writings of H.S.M. Coxeter''', edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, {{isbn|978-0-471-01003-6}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
*** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380-407, MR 2,10]
*** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591]
*** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45]
* [[Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D.
* {{PolyCell | urlname = section1.html| title = 1. Convex uniform polychora based on the pentachoron – Model 5, 8, and 9}}
* {{KlitzingPolytopes|polychora.htm|4D|uniform polytopes (polychora)}} o3x3x3o - spid, x3x3o3x - prip, x3x3x3x - gippid

{{Polytopes}}

[[Category:Polychora]]