Hexacosichoron

A **hexacosichoron** is a four-dimensional regular polychoron with 600 cells, each of which is a tetrahedron. There are five cells to each edge and twenty to each vertex. It is called a **hydrochoron** under the elemental system, and its Bowers name is **ex**.

## Structure and Subfacets

The hexacosichoron has 600 cells. These are joined twenty to a vertex in an icosahedral formation.

### Related polychora

The hexacosichoron, do to yielding regular icosahedra under vertices, possesses many unit-edged diminishings. Two of these are uniform polychora. The grand antiprism can be made by diminishing two orthogonal decagonal sets of vertices, while the snub icositetrachoron can be made ty diminishing 24 vertices forming an icositetrachoron. Many other diminishing have only unit edges.

### Vertex coordinate

The vertices of a hexacosichoron of edge length

2

ϕ

−
2

{\displaystyle 2 \phi ^ {-2}}

, can be represented by the 4D Cartesian coordinates:

* the permutatiions of:
	+ (
	±
	2
	,
	±
	2
	,
	0
	,
	0
	)

	{\displaystyle (\pm 2, \pm 2, 0, 0)}

	,
	+ (
	±

	5

	,
	±
	1
	,
	±
	1
	,
	±
	1
	)

	{\displaystyle (\pm \sqrt{5}, \pm 1, \pm 1, \pm 1)}

	,
	+ (
	±
	ϕ
	,
	±
	ϕ
	,
	±
	ϕ
	,
	±

	ϕ

	−
	2

	)

	{\displaystyle (\pm \phi, \pm \phi, \pm \phi, \pm \phi ^ {-2})}
	+ (
	±

	ϕ
	2

	,
	±

	ϕ

	−
	1

	,
	±

	ϕ

	−
	1

	,
	±

	ϕ

	−
	1

	)

	{\displaystyle (\pm \phi^2, \pm \phi^{-1}, \pm \phi^{-1}, \pm \phi^{-1})}
* and the even permutations of:
	+ (
	±

	ϕ
	2

	,
	±

	ϕ

	−
	2

	,
	±
	1
	,
	0
	)

	{\displaystyle (\pm \phi^2, \pm \phi^{-2}, \pm 1, 0)}
	+ (
	±

	5

	,
	±

	ϕ

	−
	1

	,
	±
	ϕ
	,
	0
	)

	{\displaystyle (\pm \sqrt{5}, \pm \phi^{-1}, \pm \phi, 0)}
	+ (
	±
	2
	,
	±
	1
	,
	±
	ϕ
	,
	±

	ϕ

	−
	1

	)

	{\displaystyle (\pm 2, \pm 1, \pm \phi, \pm \phi^{-1})}

where

# ϕ

1

5

2

{\displaystyle \phi = {1 + \sqrt{5} \over 2}}

.

## References

1. ↑ Coxeter, H.S.M. *Regular Polytopes* Dover Publications Inc., 1973, p. 157

## See also

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

{
3
,
3
,
2
}

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

{
3
,
3
,
3
}

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

{
3
,
3
,
4
}

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

{
3
,
3
,
5
}

{\displaystyle \{3,3,5\}}

{
3
,
3
,
6
}

{\displaystyle \{3,3,6\}}

{
2
,
3
,
5
}

{\displaystyle \{2,3,5\}}

{
3
,
3
,
5
}

{\displaystyle \{3,3,5\}}

{
4
,
3
,
5
}

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

{
5
,
3
,
5
}

{\displaystyle \{5,3,5\}}

{
6
,
3
,
5
}

{\displaystyle \{6,3,5\}}

t
0

{
5
,
3
,
3
}

{\displaystyle t\_0 \{5,3,3\}}

t
1

{
5
,
3
,
3
}

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

t
2

{
5
,
3
,
3
}

{\displaystyle t\_2 \{5,3,3\}}

t
3

{
5
,
3
,
3
}

{\displaystyle t\_3 \{5,3,3\}}

t

0
,
1

{
5
,
3
,
3
}

{\displaystyle t\_{0,1} \{5,3,3\}}

t

1
,
2

{
5
,
3
,
3
}

{\displaystyle t\_{1,2} \{5,3,3\}}

t

2
,
3

{
5
,
3
,
3
}

{\displaystyle t\_{2,3} \{5,3,3\}}

t

0
,
2

{
5
,
3
,
3
}

{\displaystyle t\_{0,2} \{5,3,3\}}

t

1
,
3

{
5
,
3
,
3
}

{\displaystyle t\_{1,3} \{5,3,3\}}

t

0
,
1
,
2

{
5
,
3
,
3
}

{\displaystyle t\_{0,1,2} \{5,3,3\}}

t

1
,
2
,
3

{
5
,
3
,
3
}

{\displaystyle t\_{1,2,3} \{5,3,3\}}

t

0
,
3

{
5
,
3
,
3
}

{\displaystyle t\_{0,3} \{5,3,3\}}

t

0
,
2
,
3

{
5
,
3
,
3
}

{\displaystyle t\_{0,2,3} \{5,3,3\}}

t

0
,
1
,
3

{
5
,
3
,
3
}

{\displaystyle t\_{0,1,3} \{5,3,3\}}

t

0
,
1
,
2
,
3

{
5
,
3
,
3
}

{\displaystyle t\_{0,1,2,3} \{5,3,3\}}