View source for Truncated 24-cells

{{redir|Truncated 24-cell|the uniform 4-polytope with cubical and cuboctahedral cells|rectified 24-cell}}
{| class=wikitable align=right width=450
|- align=center valign=top
|[[File:Schlegel wireframe 24-cell.png|150px]]<BR>[[24-cell]]<BR>{{CDD|node|3|node|4|node|3|node_1}}
|[[Image:Schlegel half-solid truncated 24-cell.png|150px]]<BR>Truncated 24-cell<BR>{{CDD|node|3|node|4|node_1|3|node_1}}
|[[File:Bitruncated 24-cell Schlegel halfsolid.png|150px]]<BR>Bitruncated 24-cell<BR>{{CDD|node|3|node_1|4|node_1|3|node}}
|- align=center
|colspan=4|Schlegel diagrams centered on one [3,4] (cells at opposite at [4,3])
|}
In [[geometry]], a '''truncated 24-cell''' is a [[uniform 4-polytope]] (4-dimensional uniform [[polytope]]) formed as the [[Truncation (geometry)|truncation]] of the regular [[24-cell]].

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

== Truncated 24-cell==

{| class="wikitable" align="right" style="margin-left:10px" width="280"
|-
!bgcolor=#e7dcc3 colspan=3|[[Image:Schlegel half-solid truncated 24-cell.png|280px]]<BR>[[Schlegel diagram]]
|-
!bgcolor=#e7dcc3 colspan=3|Truncated 24-cell
|-
|bgcolor=#e7dcc3|Type
|colspan=2|[[Uniform 4-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]s
|colspan=2|t{3,4,3}<BR>tr{3,3,4}=<math>t\left\{\begin{array}{l}3\\3,4\end{array}\right\}</math><BR>t{3<sup>1,1,1</sup>} = <math>t\left\{\begin{array}{l}3\\3\\3\end{array}\right\}</math>

|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]
|colspan=2|{{CDD|node_1|3|node_1|4|node|3|node}}<BR>{{CDD|node_1|3|node_1|3|node_1|4|node}}<BR>{{CDD|node_1|splitsplit1|branch3_11|node_1}}
|-
|bgcolor=#e7dcc3|Cells
|48
|24 [[Truncated octahedron|''4.6.6'']] [[Image:Truncated octahedron.png|20px]]<BR>24 [[cube|''4.4.4'']] [[Image:Hexahedron.png|20px]]
|-
|bgcolor=#e7dcc3|Faces
|240
|144 [[square (geometry)|{4}]]<br>96 [[hexagon|{6}]]
|-
|bgcolor=#e7dcc3|Edges
|colspan=2|384
|-
|bgcolor=#e7dcc3|Vertices
|colspan=2|192
|-
|bgcolor=#e7dcc3|[[Vertex figure]]
|colspan=2|[[Image:Truncated 24-cell verf.png|50px]]<BR>equilateral triangular pyramid
|-
|bgcolor=#e7dcc3|[[Coxeter notation|Symmetry group]]
|colspan=2|F<sub>4</sub> [3,4,3], order 1152
|-
|bgcolor=#e7dcc3|[[Rotation subgroup]]
|colspan=2|[3,4,3]<sup>+</sup>, order 576
|-
|bgcolor=#e7dcc3|[[Commutator subgroup]]
|colspan=2|[3<sup>+</sup>,4,3<sup>+</sup>], order 288
|-
|bgcolor=#e7dcc3|Properties
|colspan=2|[[Convex polytope|convex]] 
|-
|bgcolor=#e7dcc3|Uniform index
|colspan=2|''[[rectified 24-cell|23]]'' 24 ''[[cantellated 24-cell|25]]''
|}
The '''truncated 24-cell''' or '''truncated icositetrachoron''' is a uniform 4-dimensional polytope (or [[uniform 4-polytope]]), which is bounded by 48 [[cell (mathematics)|cells]]: 24 [[cube]]s, and 24 [[truncated octahedron|truncated octahedra]]. Each vertex joins three truncated octahedra and one cube, in an equilateral triangular pyramid [[vertex figure]].

===Construction===

The '''truncated 24-cell''' can be constructed from polytopes with three symmetry groups:
*F<sub>4</sub> [3,4,3]: A [[Truncation (geometry)|truncation]] of the [[24-cell]]. 
*B<sub>4</sub> [3,3,4]: A [[cantitruncation]] of the [[16-cell]], with two families of truncated octahedral cells.
*D<sub>4</sub> [3<sup>1,1,1</sup>]: An [[omnitruncation]] of the [[demihypercube|demitesseract]], with three families of truncated octahedral cells.

{| class="wikitable"
![[Coxeter group]]
!<math>{F}_4</math> = [3,4,3]
!<math>{C}_4</math> = [4,3,3]
!<math>{D}_4</math> = [3,3<sup>1,1</sup>]
|-
![[Schläfli symbol]]
!t{3,4,3}
!tr{3,3,4}
!t{3<sup>1,1,1</sup>}
|-
!Order
|1152
|384
|192
|-
!Full<BR>symmetry<BR>group
|[3,4,3]
|[4,3,3]
|<[3,3<sup>1,1</sup>]> = [4,3,3]<BR>[3[3<sup>1,1,1</sup>]] = [3,4,3]
|-
![[Coxeter diagram]]
|{{CDD|node_1|3|node_1|4|node|3|node}}
|{{CDD|node|4|node_1|3|node_1|3|node_1}}
|{{CDD|node_1|splitsplit1|branch3_11|node_1}}
|-
![[Facet (geometry)|Facets]]
|'''3:''' {{CDD|node_1|3|node_1|4|node}}<BR>'''1:''' {{CDD|node_1|4|node|3|node}}
|'''2:''' {{CDD|node_1|3|node_1|3|node_1}}<BR>'''1:''' {{CDD|node|4|node_1|3|node_1}}<BR>'''1:''' {{CDD|node|4|node_1|2|node_1}}
|'''1,1,1:''' {{CDD|node_1|3|node_1|3|node_1}}<BR>'''1:''' {{CDD|node_1|2|node_1|2|node_1}}
|-
![[Vertex figure]]
|[[File:Truncated 24-cell verf.png|80px]]
|[[File:Cantitruncated 16-cell verf.png|80px]]
|[[File:omnitruncated demitesseract verf.png|80px]]
|}

=== Zonotope===
It is also a [[zonotope]]: it can be formed as the [[Minkowski sum]] of the six line segments connecting opposite pairs among the twelve permutations of the vector (+1,&minus;1,0,0).

=== Cartesian coordinates ===
The [[Cartesian coordinate]]s of the vertices of a truncated 24-cell having edge length sqrt(2) are all coordinate permutations and sign combinations of:
: (0,1,2,3) [4!×2<sup>3</sup> = 192 vertices]

The dual configuration has coordinates at all coordinate permutation and signs of
: (1,1,1,5) [4×2<sup>4</sup> = 64 vertices]
: (1,3,3,3) [4×2<sup>4</sup> = 64 vertices]
: (2,2,2,4) [4×2<sup>4</sup> = 64 vertices]

===Structure===

The 24 cubical cells are joined via their square faces to the truncated octahedra; and the 24 truncated octahedra are joined to each other via their hexagonal faces.

===Projections===

The parallel projection of the truncated 24-cell into 3-dimensional space, truncated octahedron first, has the following layout:

* The projection envelope is a [[truncated cuboctahedron]].
* Two of the truncated octahedra project onto a truncated octahedron lying in the center of the envelope.
* Six cuboidal volumes join the square faces of this central truncated octahedron to the center of the octagonal faces of the great rhombicuboctahedron. These are the images of 12 of the cubical cells, a pair of cells to each image.
* The 12 square faces of the great rhombicuboctahedron are the images of the remaining 12 cubes.
* The 6 octagonal faces of the great rhombicuboctahedron are the images of 6 of the truncated octahedra.
* The 8 (non-uniform) truncated octahedral volumes lying between the hexagonal faces of the projection envelope and the central truncated octahedron are the images of the remaining 16 truncated octahedra, a pair of cells to each image.

=== Images ===
{{24-cell_4-cube_Coxeter_plane_graphs|t01|150|t23}}

{| class="wikitable" width=480
|[[Image:Schlegel half-solid truncated 24-cell.png|240px]]<BR>[[Schlegel diagram]]<BR>([[cube|cubic]] cells visible)
|[[Image:Schlegel half-solid cantitruncated 16-cell.png|240px]]<BR>[[Schlegel diagram]]<BR>8 of 24 truncated octahedral cells visible
|-
|colspan=2 align=center|[[Image:Truncated xylotetron stereographic oblique.png|240px]]<BR>[[Stereographic projection]]<BR>Centered on [[truncated tetrahedron]]
|}

{| class=wikitable
|+ [[Net (polyhedron)|Nets]]
|[[Image:Truncated 24-cell net.png|200px]]<BR> Truncated 24-cell
|[[File:Dual_tico_net.png|200px]]<BR>Dual to truncated 24-cell
|}

=== Related polytopes ===
The convex hull of the truncated 24-cell and its dual (assuming that they are congruent) is a nonuniform polychoron composed of 480 cells: 48 [[cube]]s, 144 [[square antiprism]]s, 288 [[tetrahedron|tetrahedra]] (as tetragonal disphenoids), and 384 vertices. Its vertex figure is a hexakis [[triangular cupola]].

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

== Bitruncated 24-cell==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
|-
!bgcolor=#e7dcc3 colspan=3|Bitruncated 24-cell
|-
|bgcolor=#ffffff align=center colspan=3|[[File:Bitruncated 24-cell Schlegel halfsolid.png|220px]]<BR>[[Schlegel diagram]], centered on truncated cube, with alternate cells hidden
|-
|bgcolor=#e7dcc3|Type
|colspan=2|[[Uniform 4-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]
|colspan=2|2t{3,4,3}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]
|colspan=2|{{CDD|node|3|node_1|4|node_1|3|node}}
|-
|bgcolor=#e7dcc3|Cells
|colspan=2|48 [[Truncated cube|(''3.8.8'')]] [[Image:Truncated hexahedron.png|20px]]
|-
|bgcolor=#e7dcc3|Faces
|336
|192 [[triangle|''{3}'']]<BR>144 [[octagon|''{8}'']]
|-
|bgcolor=#e7dcc3|Edges
|colspan=2|576
|-
|bgcolor=#e7dcc3|Vertices
|colspan=2|288
|-
|bgcolor=#e7dcc3|[[Edge figure]]
|colspan=2| 3.8.8
|-
|bgcolor=#e7dcc3|[[Vertex figure]]
|colspan=2|[[Image:Bitruncated 24-cell vertex figure.png|80px]]<BR>[[tetragonal disphenoid]]
|-
|bgcolor=#e7dcc3|[[dual polytope]]
|colspan=2|[[Truncated 24-cells#Disphenoidal 288-cell|Disphenoidal 288-cell]]
|-
|bgcolor=#e7dcc3|[[Coxeter notation|Symmetry group]]
|colspan=2|[[Automorphism|Aut]](F<sub>4</sub>), <nowiki>[[</nowiki>3,4,3]], order 2304
|-
|bgcolor=#e7dcc3|Properties
|colspan=2|[[Convex polytope|convex]], [[Isogonal figure|isogonal]], [[Isotoxal figure|isotoxal]], [[Isochoric figure|isochoric]]
|-
|bgcolor=#e7dcc3|Uniform index
|colspan=2|''[[Runcinated 24-cell|26]]'' 27 ''[[Cantitruncated 24-cell|28]]''
|}
[[File:Tetracontoctachoron net.png|thumb|right|[[Net (polyhedron)|Net]]]]
The '''bitruncated 24-cell'''. '''48-cell''', or '''tetracontoctachoron''' is a 4-dimensional uniform [[polytope]] (or [[uniform 4-polytope]]) derived from the [[24-cell]]. 

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

It is constructed by [[bitruncation (geometry)|bitruncating]] the 24-cell (truncating at halfway to the depth which would yield the [[Dual polytope|dual]] 24-cell).

Being a uniform 4-polytope, it is [[vertex-transitive]]. In addition, it is [[cell-transitive]], consisting of 48 [[truncated cube]]s, and also [[edge-transitive]], with 3 [[truncated cube]]s cells per edge and with one triangle and two octagons around each edge.

The 48 cells of the bitruncated 24-cell correspond with the 24 cells and 24 vertices of the 24-cell. As such, the centers of the 48 cells form the [[root system]] of type [[F4 (mathematics)|F<sub>4</sub>]].

Its vertex figure is a ''[[tetragonal disphenoid]]'', a tetrahedron with 2 opposite edges length 1 and all 4 lateral edges length √(2+√2).

===Alternative names===
* Bitruncated 24-cell ([[Norman Johnson (mathematician)|Norman W. Johnson]]) 
* 48-cell as a [[Isochoric figure|cell-transitive]] 4-polytope
* Bitruncated icositetrachoron 
* Bitruncated polyoctahedron 
* Tetracontaoctachoron (Cont) (Jonathan Bowers)

===Structure===
The truncated cubes are joined to each other via their octagonal faces in ''anti'' orientation; i.&nbsp;e., two adjoining truncated cubes are rotated 45&nbsp;degrees relative to each other so that no two triangular faces share an edge.

The sequence of truncated cubes joined to each other via opposite octagonal faces form a cycle of 8. Each truncated cube belongs to 3 such cycles. On the other hand, the sequence of truncated cubes joined to each other via opposite triangular faces form a cycle of 6. Each truncated cube belongs to 4 such cycles.

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. Edges exist at 4 symmetry positions. Squares exist at 3 positions, hexagons 2 positions, and octagons one. Finally the 4 types of cells exist centered on the 4 corners of the fundamental simplex.<ref>{{KlitzingPolytopes|../incmats/cont.htm|o3x4x3o - cont}}</ref>
{| class=wikitable
!F<sub>4</sub>||{{CDD|node|3|node_1|4|node_1|3|node}}
! [[k-face|''k''-face]]||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>||[[vertex figure|''k''-figure]]
!Notes
|- align=right
|rowspan=3|A<sub>1</sub>A<sub>1</sub> ||{{CDD|node|2|node_x|2|node_x|2|node}}||( ) 
!f<sub>0</sub>
|BGCOLOR="#ffe0e0"|'''288'''||2||2||1||4||1||2||2||[[tetragonal disphenoid|s{2,4}]] || rowspan=3|F<sub>4</sub>/A<sub>1</sub>A<sub>1</sub> = 288
|- align=right
|{{CDD|node_x|2|node_1|2|node_x|2|node}}||rowspan=2|{ } 
!rowspan=2|f<sub>1</sub>
||2||BGCOLOR="#ffffe0"|'''288'''||BGCOLOR="#ffffe0"|*||1||2||0||2||1|| rowspan=2|[[Isosceles triangle|{ }v( )]]
|- align=right
|{{CDD|node|2|node_x|2|node_1|2|node_x}}
||2||BGCOLOR="#ffffe0"|*||BGCOLOR="#ffffe0"|'''288'''||0||2||1||1||2
|- align=right
|A<sub>2</sub>A<sub>1</sub> || {{CDD|node|3|node_1|2|node_x|2|node}}|| [[triangle|{3}]]
!rowspan=3|f<sub>2</sub>
||3||3||0||BGCOLOR="#e0ffe0"|'''96'''||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|*||2||0||rowspan=3|{ } || F<sub>4</sub>/A<sub>2</sub>A<sub>1</sub> = 1152/6/2 = 96
|- align=right
|B<sub>2</sub> || {{CDD|node_x|2|node_1|4|node_1|2|node_x}}|| [[octagon|t{4}]]
||8||4||4||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|'''144'''||BGCOLOR="#e0ffe0"|*||1||1 || F<sub>4</sub>/B<sub>2</sub> = 1152/8 = 144
|- align=right
|A<sub>2</sub>A<sub>1</sub> || {{CDD|node|2|node_x|2|node_1|3|node}}|| [[triangle|{3}]]
||3||0||3||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|'''96'''||0||2 || F<sub>4</sub>/A<sub>2</sub>A<sub>1</sub> = 1152/6/2 = 96
|- align=right
|rowspan=2|B<sub>3</sub> || {{CDD|node|3|node_1|4|node_1|2|node_x}}|| rowspan=2|[[Truncated cube|t{4,3}]]
!rowspan=2|f<sub>3</sub>
||24||24||12||8||6||0||BGCOLOR="#e0ffff"|'''24'''||BGCOLOR="#e0ffff"|*||rowspan=2|( ) ||  rowspan=2|F<sub>4</sub>/B<sub>3</sub> = 1152/48 = 24
|- align=right
|| {{CDD|node_x|2|node_1|4|node_1|3|node}}
||24||12||24||0||6||8||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|'''24'''
|}

===Coordinates===
The [[Cartesian coordinate]]s of a bitruncated 24-cell having edge length&nbsp;2 are all permutations of coordinates and sign of:

:(0, 2+√2, 2+√2, 2+2√2)
:(1, 1+√2, 1+√2, 3+2√2)

===Projections===

====Projection to 2 dimensions====

{{24-cell_Coxeter_plane_graphs2|t12|160}}

====Projection to 3 dimensions====

{| class=wikitable width=800
!Orthographic
!Perspective
|-
|valign=top width=500|The following animation shows the orthographic projection of the bitruncated 24-cell into 3 dimensions. The animation itself is a perspective projection from the static 3D image into 2D, with rotation added to make its structure more apparent.<BR>[[Image:Bitruncated-24cell-parallelproj-01.gif]]<BR>The images of the 48 truncated cubes are laid out as follows:
* The central truncated cube is the cell closest to the 4D viewpoint, highlighted to make it easier to see. To reduce visual clutter, the vertices and edges that lie on this central truncated cube have been omitted.
* Surrounding this central truncated cube are 6 truncated cubes attached via the octagonal faces, and 8 truncated cubes attached via the triangular faces. These cells have been made transparent so that the central cell is visible.
* The 6 outer square faces of the projection envelope are the images of another 6 truncated cubes, and the 12 oblong octagonal faces of the projection envelope are the images of yet another 12 truncated cubes.
* The remaining cells have been culled because they lie on the far side the bitruncated 24-cell, and are obscured from the 4D viewpoint. These include the antipodal truncated cube, which would have projected to the same volume as the highlighted truncated cube, with 6 other truncated cubes surrounding it attached via octagonal faces, and 8 other truncated cubes surrounding it attached via triangular faces.
|valign=top width=300|The following animation shows the cell-first perspective projection of the bitruncated 24-cell into 3 dimensions. Its structure is the same as the previous animation, except that there is some [[foreshortening]] due to the perspective projection.<BR>
[[Image:Bitruncated 24cell perspective 04.gif]]
|}

{| class=wikitable
|+[[Stereographic projection]]
|[[Image:Bitruncated xylotetron stereographic close-up.png|220px]]
|}

=== Related regular skew polyhedron ===

The [[regular skew polyhedron]], {8,4|3}, exists in 4-space with 4 octagonal around each vertex, in a zig-zagging nonplanar vertex figure. These octagonal faces can be seen on the bitruncated 24-cell, using all 576 edges and 288 vertices. The 192 triangular faces of the bitruncated 24-cell can be seen as removed. The dual regular skew polyhedron, {4,8|3}, is similarly related to the square faces of the [[runcinated 24-cell]].

=== Disphenoidal 288-cell ===
{| class="wikitable" align="right" style="margin-left:20px" width="262"
|-
!bgcolor=#e7dcc3 colspan=3|Disphenoidal 288-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>F<sub>4</sub><ref name="Gevay" /><BR> (1,0,0,0)<sub>F<sub>4</sub></sub> ⊕ (0,0,0,1)<sub>F<sub>4</sub></sub><ref name="Koca">[https://arxiv.org/abs/1203.4574 Quaternionic Construction of the W(F4) Polytopes with Their Dual Polytopes and Branching under the Subgroups W(B4) and W(B3) × W(A1)] Mehmet Koca 1, Mudhahir Al-Ajmi 2 and Nazife Ozdes Koca 3 Department of Physics, College of Science, Sultan Qaboos University P. O. Box 36, Al-Khoud 123, Muscat, Sultanate of Oman, p.18. ''5.7 Dual polytope of the polytope (0, 1, 1, 0)F<sub>4</sub> = W(F<sub>4</sub>)(&omega;<sub>2</sub>+&omega;<sub>3</sub>)''</ref>
|-
|bgcolor=#e7dcc3|Coxeter
|colspan=2|{{CDD|node|3|node_f1|4|node_f1|3|node}}
|-
|bgcolor=#e7dcc3|Cells
|colspan=2|[[File:Disphenoid tetrahedron.png|80px]]<BR>288 congruent [[tetragonal disphenoid]]s
|-
|bgcolor=#e7dcc3|Faces
|colspan=2|576 congruent [[triangle|isosceles]]<BR> &nbsp; (2 short edges)
|-
|bgcolor=#e7dcc3|Edges
|336
| style="width:30em;" | 192 of length <math>\scriptstyle 1</math><BR>144 of length <math>\scriptstyle \sqrt{2-\sqrt2}</math>
|-
|bgcolor=#e7dcc3|Vertices
|colspan=2|48
|-
|bgcolor=#e7dcc3|[[Vertex figure]]
|colspan=2|[[File:Disphenoidal 288-cell vertex figure.png|80px]]<BR>([[Triakis octahedron]])
|-
|bgcolor=#e7dcc3|Dual
|colspan=2| [[Truncated 24-cells#Bitruncated 24-cell|Bitruncated 24-cell]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]
|colspan=2| [[Automorphism|Aut]](F<sub>4</sub>), <nowiki>[[3,4,3]]</nowiki>, order 2304
|-
|bgcolor=#e7dcc3|Orbit vector
|colspan=2| (1, 2, 1, 1)
|-
|bgcolor=#e7dcc3|Properties
|colspan=2|[[Convex polytope|convex]], [[Isochoric figure|isochoric]]
|}
The '''disphenoidal 288-cell''' is the [[Dual polytope|dual]] of the [[#bitruncated 24-cell|bitruncated 24-cell]]. It is a 4-dimensional [[polytope]] (or [[polychoron]]) derived from the [[24-cell]]. It is constructed by doubling and rotating the 24-cell, then constructing the [[convex hull]].

Being the dual of a uniform polychoron, it is [[cell-transitive]], consisting of 288 congruent [[disphenoid|tetragonal disphenoids]]. In addition, it is [[vertex-transitive]] under the group Aut(F<sub>4</sub>).<ref name="Koca" />

==== Images ====
{| class=wikitable
|+ Orthogonal projections 
|-
![[Coxeter plane]]s
!B<sub>2</sub>
!B<sub>3</sub>
!F<sub>4</sub>
|-
!Disphenoidal<BR>288-cell
|colspan=2|[[File:Dual bitruncated 24-cell B2-3.png|240px]]
|[[File:F4 roots by 24-cell duals.svg|120px]]
|-
!Bitruncated<BR>24-cell
|[[File:24-cell t12 B2.svg|120px]]
|[[File:24-cell t12 B3.svg|120px]]
|[[File:24-cell t12 F4.svg|120px]]
|}

==== Geometry ====
The vertices of the 288-cell are precisely the 24 [[Hurwitz quaternion|Hurwitz unit quaternions]] with norm squared 1, united with the 24 vertices of the dual 24-cell with norm squared 2, projected to the unit [[3-sphere]]. These 48 vertices correspond to the [[Binary octahedral group|binary octahedral group <big>2O</big>]] or <2,3,4>, order 48.

Thus, the 288-cell is the only non-regular 4-polytope which is the convex hull of a quaternionic group, disregarding the infinitely many [[dicyclic group|dicyclic]] (same as binary dihedral) groups; the regular ones are the [[24-cell]] (≘ [[binary tetrahedral group|<big>2T</big>]] or <2,3,3>, order 24) and the [[600-cell]] (≘ [[binary icosahedral group|<big>2I</big>]] or <2,3,5>, order 120). (The [[16-cell]] corresponds to the [[binary dihedral group]] <big>2D</big><sub>2</sub> or <2,2,2>, order 16.)

The inscribed 3-sphere has radius 1/2+{{radic|2}}/4 ≈ 0.853553 and touches the 288-cell at the centers of the 288 tetrahedra which are the vertices of the dual bitruncated 24-cell.

The vertices can be [[F4 (mathematics)#Roots of F4|coloured in 2 colours]], say red and yellow, with the 24 Hurwitz units in red and the 24 duals in yellow, the yellow [[24-cell]] being congruent to the red one. Thus the product of 2 equally coloured quaternions is red and the product of 2 in mixed colours is yellow.

{| class="wikitable" style="text-align:right; margin-right:10px;" width="260" align="left" 
|-
! style="text-align:left" | Region ||Layer||Latitude||colspan=2|red||colspan=2|yellow
|-
|rowspan=3 align="left" | Northern hemisphere || ''3'' || 1 || 1 || <math>1 </math> || 0 ||
|-
| ''2'' || {{radic|2}}/2 || 0 || || 6 || <math>\tfrac{\sqrt{2}}{2}\{ \;\;\; 1\pm\mathrm{i}, \;\;\; 1\pm\mathrm{j}, \;\;\; 1\pm\mathrm{k}\} </math>
|-
| ''1'' || 1/2 || 8 || <math>\tfrac{1}{2}\{1\pm\mathrm{i}\pm\mathrm{j}\pm\mathrm{k}\} </math> || 0 ||
|-
| align="left" | Equator || ''0'' || 0 || 6 || <math>\{ \pm\mathrm{i},\pm\mathrm{j},\pm\mathrm{k} \}</math> || 12 || <math>\tfrac{\sqrt{2}}{2}\{\,\pm\mathrm{i}\pm\mathrm{j},\,\pm\mathrm{i}\pm\mathrm{k},\,\pm\mathrm{j}\pm\mathrm{k}\} </math>
|-
|rowspan=3 align="left" | Southern hemisphere || ''–1'' || –1/2 || 8 || <math>-\tfrac{1}{2}\{1\pm\mathrm{i}\pm\mathrm{j}\pm\mathrm{k}\} </math> || 0 ||
|-
| ''–2'' || –{{radic|2}}/2 || 0 || || 6 || <math>\tfrac{\sqrt{2}}{2}\{-1\pm\mathrm{i},-1\pm\mathrm{j},-1\pm\mathrm{k}\} </math>

|-
| ''–3'' || –1 || 1 || <math>-1 </math> || 0 ||
|-
!colspan=3 style="text-align:left"| Total || 24 || || 24 ||
|}
{{-}}
Placing a fixed red vertex at the north pole (1,0,0,0), there are 6 yellow vertices in the next deeper “latitude” at ({{radic|2}}/2,x,y,z), followed by 8 red vertices in the latitude at (1/2,x,y,z). The complete coordinates are given as linear combinations of the quaternionic units {{nowrap|<math>1, \mathrm{i}, \mathrm{j}, \mathrm{k}</math>,}} which at the same time can be taken as the elements of the group {{nowrap|<big>2O</big>.}} The next deeper latitude is the equator hyperplane intersecting the 3-sphere in a 2-sphere which is populated by 6 red and 12 yellow vertices.

Layer ''2'' is a 2-sphere circumscribing a regular octahedron whose edges have length 1. A tetrahedron with vertex north pole has 1 of these edges as long edge whose 2 vertices are connected by short edges to the north pole. Another long edge runs from the north pole into layer ''1'' and 2 short edges from there into layer ''2''.

There are 192 long edges with length 1 connecting equal colours and 144 short edges with length  {{radic|2–{{radic|2}}}} ≈ 0.765367  connecting mixed colours. 192*2/48 = 8 long and 144*2/48 = 6 short, that is together 14 edges meet at any vertex.

The 576 faces are [[Triangle|isosceles]] with 1 long and 2 short edges, all congruent. The angles at the base are arccos({{radic|4+{{radic|8}}}}/4) ≈ 49.210°. 576*3/48 = 36 faces meet at a vertex, 576*1/192 = 3 at a long edge, and 576*2/144 = 8 at a short one.

The 288 cells are tetrahedra with 4 short edges and 2 antipodal and perpendicular long edges, one of which connects 2 red and the other 2 yellow vertices. All the cells are congruent. 288*4/48 = 24 cells meet at a vertex. 288*2/192 = 3 cells meet at a long edge, 288*4/144 = 8 at a short one. 288*4/576 = 2 cells meet at a triangle.

== Related polytopes==
{{Demitesseract family}}

'''B<sub>4</sub> family of uniform polytopes:'''
{{Tesseract family}}

'''F<sub>4</sub> family of uniform polytopes:'''
{{24-cell_family}}

== References==
{{reflist}}
* [[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]:
** '''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. (1966)
* {{KlitzingPolytopes|polychora.htm|4D|uniform polytopes (polychora)}} x3x4o3o=x3x3x4o - tico, o3x4x3o - cont
* {{PolyCell | urlname = section3.html| title = 3. Convex uniform polychora based on the icositetrachoron (24-cell) - Model 24, 27}}

[[Category:Uniform 4-polytopes]]