Icositetrachoron

The **icositetrachoron** is a four-dimensional regular polytope composed of 24 octahedral cells, three of which join at each edge. It is also called a **24-cell** because of this. Its Bowers acronym is "**ico**", and it can also be called a **xylochoron**. It is one of the three regular polychora that tile 4-dimensional space, leading to the icositetrachoric tetracomb.

## Construction

One way to construct the icositetrachoron is to augment cubic pyramids onto the cells of the tesseract. If the sides of the pyramids are exactly the same edge length as the tesseract, the pyramidal cells merge into octahedra.

Another way is via the rectification of the hexadecachoron. The tetrahedral cells will rectify into octahedra, while the vertices become further octahedral cells. This can be represented as

r
{
3
,
3
,
4
}

{\displaystyle r\{3,3,4\}}

.

## Structure and Sections

### Structure

The icositetrachoron is composed of octahedra joining six to a vertex. Seen from a cell, we first have one, then the eight joining to it. The next layer is six octahedra sharing only a vertex with the top, then another eight, and finally the opposite cell.

### Subfacets

* 24 points (0D)
* 96 line segments (1D)
* 96 triangles (2D)
* 24 octahedra (3D)
* 1 icositetrachoron

### Hypervolumes

* vertex count =

24

{\displaystyle 24}
* edge length =

96
l

{\displaystyle 96l}
* surface area =

24

3

l
2

{\displaystyle 24\sqrt{3}l^2}
* surcell volume =

8

2

l
3

{\displaystyle 8\sqrt{2}l^3}
* surteron bulk =

2

l
4

{\displaystyle 2l^4}

### Vertex Coordinates

The vertices of an icositetrachoron, of edge length

2

{\displaystyle \sqrt{2}}

, can be denoted on the 4D Euclidean flune by the permutations of (±1, ±1, 0, 0). It can also be seen as the superposition of a tesseract and a hexadecachoron, giving the vertices (±1, ±1, ±1, ±1), along with the rotations of (±2, 0, 0, 0).

## References

1. ↑ Coxeter, H.S.M. *Regular Polytopes*, Dover Publications, p. 157

## See also

**Self-intersecting regular polychora:** fix · gohi · gahi · sishi · gaghi · gishi · gashi · gofix · gax · gogishi
**Tesseractihemioctachoron:** tho

**Have Full Hexadecachoric symmetry:** oh · shahoh · huhoh · hohoh · ohuhoh · ghahoh
**Have Full Demitesseractic symmetry:**
ratho · odho ·
hodho · hoh honho · dod honho · doh honho

{
3
,
4
,
2
}

{\displaystyle \{3,4,2\}}

{
3
,
4
,
3
}

{\displaystyle \{3,4,3\}}

{
3
,
4
,
4
}

{\displaystyle \{3,4,4\}}

Template:Ariant Nav X 4 3

t
0

{
3
,
4
,
3
}

{\displaystyle t\_0 \{3,4,3\}}

t
1

{
3
,
4
,
3
}

{\displaystyle t\_1 \{3,4,3\}}

t
2

{
3
,
4
,
3
}

{\displaystyle t\_2 \{3,4,3\}}

t
3

{
3
,
4
,
3
}

{\displaystyle t\_3 \{3,4,3\}}

t

0
,
1

{
3
,
4
,
3
}

{\displaystyle t\_{0,1} \{3,4,3\}}

t

1
,
2

{
3
,
4
,
3
}

{\displaystyle t\_{1,2} \{3,4,3\}}

t

2
,
3

{
3
,
4
,
3
}

{\displaystyle t\_{2,3} \{3,4,3\}}

t

0
,
2

{
3
,
4
,
3
}

{\displaystyle t\_{0,2} \{3,4,3\}}

t

1
,
3

{
3
,
4
,
3
}

{\displaystyle t\_{1,3} \{3,4,3\}}

t

0
,
1
,
2

{
3
,
4
,
3
}

{\displaystyle t\_{0,1,2} \{3,4,3\}}

t

1
,
2
,
3

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3
,
4
,
3
}

{\displaystyle t\_{1,2,3} \{3,4,3\}}

t

0
,
3

{
3
,
4
,
3
}

{\displaystyle t\_{0,3} \{3,4,3\}}

t

0
,
2
,
3

{
3
,
4
,
3
}

{\displaystyle t\_{0,2,3} \{3,4,3\}}

t

0
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1
,
3

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3
,
4
,
3
}

{\displaystyle t\_{0,1,3} \{3,4,3\}}

t

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,
1
,
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{
3
,
4
,
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}

{\displaystyle t\_{0,1,2,3} \{3,4,3\}}

t
0

{
4
,
3
,
3
}

{\displaystyle t\_0 \{4,3,3\}}

t
1

{
4
,
3
,
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}

{\displaystyle t\_1 \{4,3,3\}}

t
2

{
4
,
3
,
3
}

{\displaystyle t\_2 \{4,3,3\}}

t
3

{
4
,
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,
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}

{\displaystyle t\_3 \{4,3,3\}}

t

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3
,
3
}

{\displaystyle t\_{0,1} \{4,3,3\}}

t

1
,
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4
,
3
,
3
}

{\displaystyle t\_{1,2} \{4,3,3\}}

t

2
,
3

{
4
,
3
,
3
}

{\displaystyle t\_{2,3} \{4,3,3\}}

t

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,
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{
4
,
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,
3
}

{\displaystyle t\_{0,2} \{4,3,3\}}

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,
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}

{\displaystyle t\_{1,3} \{4,3,3\}}

t

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4
,
3
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}

{\displaystyle t\_{0,1,2} \{4,3,3\}}

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,
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4
,
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}

{\displaystyle t\_{1,2,3} \{4,3,3\}}

t

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4
,
3
,
3
}

{\displaystyle t\_{0,3} \{4,3,3\}}

t

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,
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,
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}

{\displaystyle t\_{0,2,3} \{4,3,3\}}

t

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,
3
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}

{\displaystyle t\_{0,1,3} \{4,3,3\}}

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{\displaystyle t\_{0,1,2,3} \{4,3,3\}}

t
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{\displaystyle t\_0 \{3,3,b3\}}

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{\displaystyle t\_1 \{3,3,b3\}}

t
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{\displaystyle t\_2 \{3,3,b3\}}

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,
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}

{\displaystyle t\_3 \{3,3,b3\}}

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}

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,
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}

{\displaystyle t\_{1,2} \{3,3,b3\}}

t

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,
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}

{\displaystyle t\_{2,3} \{3,3,b3\}}

t

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}

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t

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t

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{\displaystyle t\_{1,2,3} \{3,3,b3\}}

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t

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{
2
,
3
,
4
}

{\displaystyle t\_1 \{2,3,4\}}

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1

{
3
,
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,
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}

{\displaystyle t\_1 \{3,3,4\}}

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1

{
4
,
3
,
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}

{\displaystyle t\_1 \{4,3,4\}}

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1

{
5
,
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,
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}

{\displaystyle t\_1 \{5,3,4\}}

t
1

{
6
,
3
,
4
}

{\displaystyle t\_1 \{6,3,4\}}

t
1

{
3
,
3
,
2
}

{\displaystyle t\_1 \{3,3,2\}}

t
1

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,
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,
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}

{\displaystyle t\_1 \{3,3,3\}}

t
1

{
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,
3
,
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}

{\displaystyle t\_1 \{3,3,4\}}

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1

{
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,
3
,
5
}

{\displaystyle t\_1 \{3,3,5\}}

t
1

{
3
,
3
,
6
}

{\displaystyle t\_1 \{3,3,6\}}