View source for Truncated 5-cell

{| class=wikitable align=right width=450
|- align=center valign=top
|[[File:Schlegel wireframe 5-cell.png|150px]]<BR>[[5-cell]]<BR>{{CDD|node|3|node|3|node|3|node_1}}
|[[Image:Schlegel half-solid truncated pentachoron.png|150px]]<BR>Truncated 5-cell<BR>{{CDD|node|3|node|3|node_1|3|node_1}}
|[[File:Schlegel half-solid bitruncated 5-cell.png|150px]]<BR>Bitruncated 5-cell<BR>{{CDD|node|3|node_1|3|node_1|3|node}}
|- align=center
|colspan=4|Schlegel diagrams centered on [3,3] (cells at opposite at [3,3])
|}
In [[geometry]], a '''truncated 5-cell''' is a [[uniform 4-polytope]] (4-dimensional uniform [[polytope]]) formed as the [[Truncation (geometry)|truncation]] of the regular [[5-cell]].

There are two degrees of truncations, including a [[bitruncation]].

== Truncated 5-cell==

{| class="wikitable" align="right" style="margin-left:10px" width="250"
|-
!bgcolor=#e7dcc3 colspan=3|Truncated 5-cell
|-
|bgcolor=#ffffff align=center colspan=3|[[Image:Schlegel half-solid truncated pentachoron.png|250px]]<BR>[[Schlegel diagram]]<BR>([[tetrahedron]] cells visible)
|-
|bgcolor=#e7dcc3|Type
|colspan=2|[[Uniform 4-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]
|colspan=2|t<sub>0,1</sub>{3,3,3}<BR>t{3,3,3}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]
|colspan=2|{{CDD|node_1|3|node_1|3|node|3|node}}
|-
|bgcolor=#e7dcc3|Cells
|10
|5 [[Tetrahedron|(''3.3.3'')]] [[Image:Tetrahedron.png|20px]]<BR>5 [[truncated tetrahedron|(''3.6.6'')]] [[Image:Truncated tetrahedron.png|20px]]
|-
|bgcolor=#e7dcc3|Faces
|30
|20 {3}<BR>10 {6}
|-
|bgcolor=#e7dcc3|Edges
|colspan=2|40
|-
|bgcolor=#e7dcc3|Vertices
|colspan=2|20
|-
|bgcolor=#e7dcc3|[[Vertex figure]]
|colspan=2|[[Image:Truncated 5-cell verf.png|80px]]<BR>Equilateral-triangular pyramid
|-
|bgcolor=#e7dcc3|[[Coxeter group|Symmetry 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|''[[Rectified 5-cell|2]]'' 3 ''[[Cantellated 5-cell|4]]''
|}
The '''truncated 5-cell''', '''truncated pentachoron''' or '''truncated 4-simplex''' is bounded by 10 [[cell (geometry)|cells]]: 5 [[tetrahedron|tetrahedra]], and 5 [[truncated tetrahedron|truncated tetrahedra]]. Each vertex is surrounded by 3 truncated tetrahedra and one tetrahedron; the [[vertex figure]] is an elongated tetrahedron.

===Construction===

The truncated 5-cell may be constructed from the [[5-cell]] by [[Truncation (geometry)|truncating]] its vertices at 1/3 of its edge length. This transforms the 5 tetrahedral cells into truncated tetrahedra, and introduces 5 new tetrahedral cells positioned near the original vertices.

===Structure===

The truncated tetrahedra are joined to each other at their hexagonal faces, and to the tetrahedra at their triangular faces.

Seen in a [[configuration (polytope)|configuration matrix]], all incidence counts between elements are shown. The diagonal [[f-vector]] numbers are derived through the [[Wythoff construction]], dividing the full group order of a subgroup order by removing one mirror at a time.<ref>{{KlitzingPolytopes|../incmats/cont.htm|x3x4o3o - tip}}</ref>
{| class=wikitable
!A<sub>4</sub>||{{CDD|node_1|3|node_1|3|node|3|node}}
! [[k-face|''k''-face]]||f<sub>k</sub>||f<sub>0</sub>||colspan=2|f<sub>1</sub>||colspan=2|f<sub>2</sub>||colspan=2|f<sub>3</sub>||[[vertex figure|''k''-figure]]
!Notes
|- align=right
|A<sub>2</sub> ||{{CDD|node_x|2|node_x|2|node|3|node}}|| ( ) 
!f<sub>0</sub>
|BGCOLOR="#ffe0e0"|'''20'''||1||3||3||3||3||1||[[Triangular pyramid|{3}v( )]] || A<sub>4</sub>/A<sub>2</sub> = 5!/3! = 20
|- align=right
|A<sub>2</sub>A<sub>1</sub> ||{{CDD|node_1|2|node_x|2|node|3|node}}|| rowspan=2|{ }
!rowspan=2|f<sub>1</sub>
||2||BGCOLOR="#ffffe0"|'''10'''||BGCOLOR="#ffffe0"|*||3||0||3||0||[[triangle|{3}]] || A<sub>4</sub>/A<sub>2</sub>A<sub>1</sub> = 5!/3!/2 = 10
|- align=right
|A<sub>1</sub>A<sub>1</sub> ||{{CDD|node_x|2|node_1|2|node_x|2|node}}
||2||BGCOLOR="#ffffe0"|*||BGCOLOR="#ffffe0"|'''30'''||1||2||2||1||[[Isosceles triangle|{ }v( )]] || A<sub>4</sub>/A<sub>1</sub>A<sub>1</sub> = 5!/2/2 = 30
|- align=right
|A<sub>2</sub>A<sub>1</sub> ||{{CDD|node_1|3|node_1|2|node_x|2|node}}|| [[hexagon|t{3}]]
!rowspan=2|f<sub>2</sub>
||6||3||3||BGCOLOR="#e0ffe0"|'''10'''||BGCOLOR="#e0ffe0"|*||2||0||rowspan=2|{ } || A<sub>4</sub>/A<sub>2</sub>A<sub>1</sub> = 5!/3!/2 = 10
|- align=right
|A<sub>2</sub> ||{{CDD|node_x|2|node_1|3|node|2|node_x}}|| [[triangle|{3}]]
||3||0||3||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|'''20'''||1||1|| A<sub>4</sub>/A<sub>2</sub> = 5!/3! = 20
|- align=right
|rowspan=2|A<sub>3</sub> ||{{CDD|node_1|3|node_1|3|node|2|node_x}}|| [[truncated tetrahedron|t{3,3}]]
!rowspan=2|f<sub>3</sub>
 ||12||6||12||4||4||BGCOLOR="#e0ffff"|'''5'''||BGCOLOR="#e0ffff"|*|| rowspan=2|( )||rowspan=2| A<sub>4</sub>/A<sub>3</sub> = 5!/4! = 5
|- align=right
||{{CDD|node_x|2|node_1|3|node|3|node}}|| [[Tetrahedron|{3,3}]]
||4||0||6||0||4||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|'''5'''
|}

===Projections===

The truncated tetrahedron-first Schlegel diagram projection of the truncated 5-cell into 3-dimensional space has the following structure:

* The projection envelope is a [[truncated tetrahedron]].
* One of the truncated tetrahedral cells project onto the entire envelope.
* One of the tetrahedral cells project onto a tetrahedron lying at the center of the envelope.
* Four flattened tetrahedra are joined to the triangular faces of the envelope, and connected to the central tetrahedron via 4 radial edges. These are the images of the remaining 4 tetrahedral cells.
* Between the central tetrahedron and the 4 hexagonal faces of the envelope are 4 irregular truncated tetrahedral volumes, which are the images of the 4 remaining truncated tetrahedral cells.

This layout of cells in projection is analogous to the layout of faces in the face-first projection of the truncated tetrahedron into 2-dimensional space. The truncated 5-cell is the 4-dimensional analogue of the truncated tetrahedron.

=== Images ===
{{4-simplex Coxeter plane graphs|t01|150}}

<gallery>
Image:Truncated pentachoron net.png|[[Net (polytope)|net]]
Image:Truncated simplex stereographic.png|[[stereographic projection]]<BR>(centered on [[truncated tetrahedron]])
</gallery>

=== Alternate names ===
* Truncated pentatope 
* Truncated [[Simplex|4-simplex]]
* Truncated pentachoron (Acronym: tip) (Jonathan Bowers)

===Coordinates===

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

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

More simply, the vertices of the ''truncated 5-cell'' can be constructed on a [[hyperplane]] in 5-space as permutations of (0,0,0,1,2) ''or'' of (0,1,2,2,2). These coordinates come from positive [[orthant]] facets of the [[truncated pentacross]] and [[bitruncated penteract]] respectively.

=== Related polytopes ===
The convex hull of the truncated 5-cell and its dual (assuming that they are congruent) is a nonuniform polychoron composed of 60 cells: 10 [[tetrahedron|tetrahedra]], 20 [[octahedron|octahedra]] (as triangular antiprisms), 30 [[tetrahedron|tetrahedra]] (as tetragonal disphenoids), and 40 vertices. Its vertex figure is a hexakis [[triangular cupola]].

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

== Bitruncated 5-cell==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
|-
!bgcolor=#e7dcc3 colspan=3|Bitruncated 5-cell
|-
|bgcolor=#ffffff align=center colspan=3|[[Image:Schlegel half-solid bitruncated 5-cell.png|220px]]<BR>[[Schlegel diagram]] with alternate cells hidden.
|-
|bgcolor=#e7dcc3|Type
|colspan=2|[[Uniform 4-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]
|colspan=2| t<sub>1,2</sub>{3,3,3}<BR>2t{3,3,3}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]
|colspan=2|{{CDD|node|3|node_1|3|node_1|3|node}}<BR>or {{CDD|branch_11|3ab|nodes}} or {{CDD|branch_11|3ab-cross|nodes}}
|-
|bgcolor=#e7dcc3|Cells
|colspan=2|10 ([[Truncated tetrahedron|''3.6.6'']]) [[Image:Truncated tetrahedron.png|20px]]
|-
|bgcolor=#e7dcc3|Faces
|40
|20 {3}<br/>20 {6}
|-
|bgcolor=#e7dcc3|Edges
|colspan=2|60
|-
|bgcolor=#e7dcc3|Vertices
|colspan=2|30
|-
|bgcolor=#e7dcc3 valign=center|[[Vertex figure]]
|colspan=2|[[Image:Bitruncated 5-cell vertex figure.png|80px]]<BR>([[Tetragonal disphenoid|{ }v{ }]])
|-
|bgcolor=#e7dcc3|[[dual polytope]]
|colspan=2|[[#Disphenoidal 30-cell|Disphenoidal 30-cell]]
|-
|bgcolor=#e7dcc3|[[Coxeter group|Symmetry group]]
|colspan=2|[[Automorphism|Aut]](A<sub>4</sub>), <nowiki>[[3,3,3]]</nowiki>, order 240
|-
|bgcolor=#e7dcc3|Properties
|colspan=2|[[Convex polytope|convex]], [[Isogonal figure|isogonal]], [[Isotoxal figure|isotoxal]], [[Isochoric figure|isochoric]]
|-
|bgcolor=#e7dcc3|Uniform index
|colspan=2|''[[Runcinated pentachoron|5]]'' 6 ''[[Cantitruncated 5-cell|7]]''
|}
The '''[[bitruncation (geometry)|bitruncated]] [[5-cell]]''' (also called a '''bitruncated pentachoron''', '''decachoron''' and '''10-cell''') is a 4-dimensional [[polytope]], or [[4-polytope]], composed of 10 [[cell (mathematics)|cells]] in the shape of [[truncated tetrahedron|truncated tetrahedra]].

Topologically, under its highest symmetry, <nowiki>[[3,3,3]]</nowiki>, there is only one geometrical form, containing 10 uniform truncated tetrahedra. The hexagons are always regular because of the polychoron's inversion symmetry, of which the regular hexagon is the only such case among ditrigons (an isogonal hexagon with 3-fold symmetry).

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

Each hexagonal face of the truncated tetrahedra is joined in complementary orientation to the neighboring truncated tetrahedron. Each edge is shared by two hexagons and one triangle. Each vertex is surrounded by 4 truncated tetrahedral cells in a [[tetragonal disphenoid]] [[vertex figure]].

The bitruncated 5-cell is the [[intersection (set theory)|intersection]] of two [[Pentachoron|pentachora]] in dual configuration.  As such, it is also the intersection of a [[penteract]] with the hyperplane that bisects the penteract's long diagonal orthogonally.  In this sense it is a 4-dimensional analog of the [[Octahedron|regular octahedron]] (intersection of regular tetrahedra in dual configuration / [[tesseract]] bisection on long diagonal) and the regular hexagon (equilateral triangles / cube).  The 5-dimensional analog is the [[birectified 5-simplex]], and the <math>n</math>-dimensional analog is the polytope whose [[Coxeter diagram|Coxeter–Dynkin diagram]] is linear with rings on the middle one or two nodes.

The bitruncated 5-cell is one of the two non-regular convex [[uniform 4-polytope]]s which are [[cell-transitive]]. The other is the [[bitruncated 24-cell]], which is composed of 48 truncated cubes.

=== Symmetry ===

This 4-polytope has a higher extended pentachoric symmetry (2×A<sub>4</sub>, <nowiki>[[3,3,3]]</nowiki>), doubled to order 240, because the element corresponding to any element of the underlying 5-cell can be exchanged with one of those corresponding to an element of its dual.

=== Alternative names ===
* Bitruncated 5-cell ([[Norman Johnson (mathematician)|Norman W. Johnson]])
* 10-cell as a [[Isochoric figure|cell-transitive]] 4-polytope
* Bitruncated pentachoron 
* Bitruncated pentatope
* Bitruncated [[Simplex|4-simplex]] 
* Decachoron (Acronym: deca) (Jonathan Bowers)

=== Images ===
{{4-simplex2 Coxeter plane graphs|t12|150}}

{| class=wikitable width=440
|[[File:Decachoron stereographic (hexagon).png|220px]]<BR>stereographic projection of spherical 4-polytope <br>(centred on a hexagon face)
|[[File:Bitruncated 5-cell net.png|220px]]<BR>[[Net (polytope)]]
|}

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

{| class="wikitable collapsible collapsed"
! colspan=2|Coordinates
|-
|
:<math>\pm\left(\sqrt{\frac{5}{2}},\ \frac{5}{\sqrt{6}},\  \frac{2}{\sqrt{3}},\  0\right)</math>
:<math>\pm\left(\sqrt{\frac{5}{2}},\ \frac{5}{\sqrt{6}},\  \frac{-1}{\sqrt{3}},\ \pm1\right)</math>
:<math>\pm\left(\sqrt{\frac{5}{2}},\ \frac{1}{\sqrt{6}},\  \frac{4}{\sqrt{3}},\  0\right)</math>
:<math>\pm\left(\sqrt{\frac{5}{2}},\ \frac{1}{\sqrt{6}},\  \frac{-2}{\sqrt{3}},\ \pm2\right)</math>
|
:<math>\pm\left(\sqrt{\frac{5}{2}},\ -\sqrt{\frac{3}{2}},\ \pm\sqrt{3},\         \pm1\right)</math>
:<math>\pm\left(\sqrt{\frac{5}{2}},\ -\sqrt{\frac{3}{2}},\ 0,\                   \pm2\right)</math>
:<math>\pm\left(0,\                  2\sqrt{\frac{2}{3}},\ \frac{4}{\sqrt{3}},\  0\right)</math>
:<math>\pm\left(0,\                  2\sqrt{\frac{2}{3}},\ \frac{-2}{\sqrt{3}},\ \pm2\right)</math>
|}
More simply, the vertices of the bitruncated 5-cell can be constructed on a [[hyperplane]] in 5-space as permutations of (0,0,1,2,2). These represent positive [[orthant]] facets of the [[bitruncated pentacross]]. Another 5-space construction, centered on the origin are all 20 permutations of (-1,-1,0,1,1).

==== Related polytopes====

The [[bitruncated 5-cell]] can be seen as the intersection of two regular [[5-cell]]s in dual positions. {{CDD|branch_11|3ab|nodes}} = {{CDD|branch|3ab|nodes_10l}} ∩ {{CDD|branch|3ab|nodes_01l}}.

{{Isotopic uniform simplex polytopes}}

=== Configuration===
Seen in a [[configuration (polytope)|configuration matrix]], all incidence counts between elements are shown. The diagonal [[f-vector]] numbers are derived through the [[Wythoff construction]], dividing the full group order of a subgroup order by removing one mirror at a time.<ref>{{KlitzingPolytopes|../incmats/srip.htm|x3o4x3o - srip}}</ref>

{| class=wikitable
|-
!Element||f<sub>''k''</sub>
!|f<sub>0</sub>
! colspan=2|f<sub>1</sub>
! colspan=3|f<sub>2</sub>
! colspan=2|f<sub>3</sub>
|- align=right
|align=left bgcolor=#ffffe0 |<!--. . . .-->{{CDD|node_x|2|node_x|2|node_x|2|node_x}}
|rowspan=1|f<sub>0</sub>
|bgcolor=#e0ffe0|30
|  bgcolor=#e0e0e0|2
|  bgcolor=#e0e0e0|2
|  bgcolor=#ffffff|1
|  bgcolor=#ffffff|4
|  bgcolor=#ffffff|1
|  bgcolor=#e0e0e0|2
|  bgcolor=#e0e0e0|2
|- align=right
|align=left bgcolor=#ffffe0 |<!--. x . .-->{{CDD|node_x|2|node_1|2|node_x|2|node_x}}
|rowspan=2|f<sub>1</sub>
|  bgcolor=#e0e0e0|2
|bgcolor=#e0ffe0|30
|  bgcolor=#ffffff|*
|  bgcolor=#e0e0e0|1
|  bgcolor=#e0e0e0|2
|  bgcolor=#e0e0e0|0
|  bgcolor=#ffffff|2
|  bgcolor=#ffffff|1
|- align=right
|align=left bgcolor=#ffffe0 |<!--. . x .-->{{CDD|node_x|2|node_x|2|node_1|2|node_x}}
|  bgcolor=#e0e0e0|2
|  bgcolor=#ffffff|*
|bgcolor=#e0ffe0|30
|  bgcolor=#e0e0e0|0
|  bgcolor=#e0e0e0|2
|  bgcolor=#e0e0e0|1
|  bgcolor=#ffffff|1
|  bgcolor=#ffffff|2
|- align=right
|align=left bgcolor=#ffffe0 |<!--o3x . .-->{{CDD|node|3|node_1|2|node_x|2|node_x}}
|rowspan=3|f<sub>2</sub>
|  bgcolor=#ffffff|3
|  bgcolor=#e0e0e0|3
|  bgcolor=#e0e0e0|0
|bgcolor=#e0ffe0|10
|  bgcolor=#ffffff|*
|  bgcolor=#ffffff|*
|  bgcolor=#e0e0e0|2
|  bgcolor=#e0e0e0|0
|- align=right
|align=left bgcolor=#ffffe0 |<!--. x3x .-->{{CDD|node_x|2|node_1|3|node_1|2|node_x}}
|  bgcolor=#ffffff|6
|  bgcolor=#e0e0e0|3
|  bgcolor=#e0e0e0|3
|  bgcolor=#ffffff|*
|bgcolor=#e0ffe0|20
|  bgcolor=#ffffff|*
|  bgcolor=#e0e0e0|1
|  bgcolor=#e0e0e0|1
|- align=right
|align=left bgcolor=#ffffe0 |<!--. . x3o-->{{CDD|node_x|2|node_x|2|node_1|3|node}}
|  bgcolor=#ffffff|3
|  bgcolor=#e0e0e0|0
|  bgcolor=#e0e0e0|3
|  bgcolor=#ffffff|*
|  bgcolor=#ffffff|*
|bgcolor=#e0ffe0|10
|  bgcolor=#e0e0e0|0
|  bgcolor=#e0e0e0|2
|- align=right
|align=left bgcolor=#ffffe0 |<!--o3x3x .-->{{CDD|node|3|node_1|3|node_1|2|node_x}}
|rowspan=2|f<sub>3</sub>
|  bgcolor=#e0e0e0|12
|  bgcolor=#ffffff|12
|  bgcolor=#ffffff|6
|  bgcolor=#e0e0e0|4
|  bgcolor=#e0e0e0|4
|  bgcolor=#e0e0e0|0
|bgcolor=#e0ffe0|5
|  bgcolor=#ffffff|*
|- align=right
|align=left bgcolor=#ffffe0 |<!--. x3x3o-->{{CDD|node_x|2|node_1|3|node_1|3|node}}
|  bgcolor=#e0e0e0|12
|  bgcolor=#ffffff|6
|  bgcolor=#ffffff|12
|  bgcolor=#e0e0e0|0
|  bgcolor=#e0e0e0|4
|  bgcolor=#e0e0e0|4
|  bgcolor=#ffffff|*
|bgcolor=#e0ffe0|5
|}

=== Related regular skew polyhedron ===
[[File:Bitruncated_5-cell_half_net.png|thumb|A 3D net for {6,4{{pipe}}3}, with pairs of yellow triangles folded together into 4D and removed]]
The [[regular skew polyhedron]], {6,4|3}, exists in 4-space with 4 hexagonal around each vertex, in a zig-zagging nonplanar vertex figure. These hexagonal faces can be seen on the bitruncated 5-cell, using all 60 edges and 30 vertices. The 20 triangular faces of the bitruncated 5-cell can be seen as removed. The dual regular skew polyhedron, {4,6|3}, is similarly related to the square faces of the [[runcinated 5-cell#Runcinated 5-cell|runcinated 5-cell]].
{{-}}

=== Disphenoidal 30-cell ===
{| class="wikitable" align="right" style="margin-left:20px" width="262"
|-
!bgcolor=#e7dcc3 colspan=3|Disphenoidal 30-cell
|-
|bgcolor=#e7dcc3|Type
|colspan=2|perfect<ref name="Gevay">[http://www.emis.de/journals/BAG/vol.43/no.1/b43h1gev.pdf On Perfect 4-Polytopes Gabor Gévay] Contributions to Algebra and Geometry Volume 43 (2002), No. 1, 243-259
] Table 2, page 252</ref> polychoron
|-
|bgcolor=#e7dcc3|Symbol
|colspan=2|f<sub>1,2</sub>A<sub>4</sub><ref name="Gevay" />
|-
|bgcolor=#e7dcc3|Coxeter
|colspan=2|{{CDD|node|3|node_f1|3|node_f1|3|node}}
|-
|bgcolor=#e7dcc3|Cells
|colspan=2|30 congruent [[tetragonal disphenoid]]s
|-
|bgcolor=#e7dcc3|Faces
|colspan=2|60 congruent [[isosceles triangle]]s (2 short edges)
|-
|bgcolor=#e7dcc3|Edges
|40
| style="width:30em;" | 20 of length <math>\scriptstyle 1</math><BR>20 of length <math>\scriptstyle \sqrt{3/5}</math>
|-
|bgcolor=#e7dcc3|Vertices
|colspan=2|10
|-
|bgcolor=#e7dcc3|[[Vertex figure]]
|colspan=2|[[File:Disphenoidal 30-cell vertex figure.png|80px]]<BR>([[Triakis tetrahedron]])
|-
|bgcolor=#e7dcc3|Dual
|colspan=2| [[#Bitruncated 5-cell|Bitruncated 5-cell]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]
|colspan=2|[[Automorphism|Aut]](A<sub>4</sub>), <nowiki>[[3,3,3]]</nowiki>, order 240
|-
|bgcolor=#e7dcc3|Orbit vector
|colspan=2| (1, 2, 1, 1)
|-
|bgcolor=#e7dcc3|Properties
|colspan=2|[[Convex polytope|convex]], [[Isochoric figure|isochoric]]
|}
The '''disphenoidal 30-cell''' is the [[Dual polytope|dual]] of the [[#Bitruncated 5-cell|bitruncated 5-cell]]. It is a 4-dimensional [[polytope]] (or [[polychoron]]) derived from the [[5-cell]]. It is the convex hull of two [[5-cell]]s in opposite orientations.

Being the dual of a uniform polychoron, it is [[cell-transitive]], consisting of 30 congruent [[disphenoid|tetragonal disphenoids]]. In addition, it is [[vertex-transitive]] under the group Aut(A<sub>4</sub>).

{{clear}}

== Related polytopes ==

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

== 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]
*[[Coxeter]], ''The Beauty of Geometry: Twelve Essays'', Dover Publications, 1999, {{isbn|0-486-40919-8}} p.&nbsp;88 (Chapter 5: Regular Skew Polyhedra in three and four dimensions and their topological analogues, Proceedings of the London Mathematics Society, Ser. 2, Vol 43, 1937.) 
**Coxeter, H. S. M. ''Regular Skew Polyhedra in Three and Four Dimensions.'' Proc. London Math. Soc. 43, 33-62, 1937.
* [[Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. (1966)
* {{PolyCell | urlname = section1.html| title = 1. Convex uniform polychora based on the pentachoron - Model 3}}
* {{KlitzingPolytopes|polychora.htm|4D|uniform polytopes (polychora)}} x3x3o3o - tip, o3x3x3o - deca

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{{Polytopes}}

[[Category:Uniform 4-polytopes]]