# Decachoron

Decachoron
Rank4
TypeUniform
Notation
Bowers style acronymDeca
Coxeter diagramo3x3x3o ()
Elements
Cells10 truncated tetrahedra
Faces20 triangles, 20 hexagons
Edges60
Vertices30
Vertex figureTetragonal disphenoid, edge lengths 1 (base) and 3 (sides)
Edge figuretut 6 tut 6 tut 3
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {2}}\approx 1.41421}$
Inradius${\displaystyle {\frac {\sqrt {10}}{4}}\approx 0.79057}$
Hypervolume${\displaystyle {\frac {115{\sqrt {5}}}{48}}\approx 5.35725}$
Dichoral anglesTut–6–tut: ${\displaystyle \arccos \left(-{\frac {1}{4}}\right)\approx 104.47751^{\circ }}$
Tut–3–tut: ${\displaystyle \arccos \left({\frac {1}{4}}\right)\approx 75.52248^{\circ }}$
Central density1
Number of external pieces10
Level of complexity3
Related polytopes
ArmyDeca
RegimentDeca
DualBidecachoron
ConjugateNone
Abstract & topological properties
Flag count720
Euler characteristic0
OrientableYes
Properties
SymmetryA4×2, order 240
Flag orbits3
ConvexYes
NatureTame

The decachoron, or deca, also commonly called the 10-cell, bitruncated 5-cell or bitruncated pentachoron, is a convex noble uniform polychoron that consists of 10 truncated tetrahedra as cells. Four cells join at each vertex. It is the medial stage of the truncation series between a regular pentachoron and its dual. Equivalently, it is also the stellation core of the compound of two dual pentachora, the stellated decachoron.

It is also the 10-3 gyrochoron.

## Vertex coordinates

The vertices of a decachoron of edge length 1 are given by the following points:

• ${\displaystyle \pm \left(0,\,{\frac {\sqrt {6}}{3}},\,{\frac {2{\sqrt {3}}}{3}},\,0\right)}$,
• ${\displaystyle \pm \left(0,\,{\frac {\sqrt {6}}{3}},\,-{\frac {\sqrt {3}}{3}},\,\pm 1\right)}$,
• ${\displaystyle \pm \left({\frac {\sqrt {10}}{4}},\,{\frac {\sqrt {6}}{12}},\,{\frac {2{\sqrt {3}}}{3}},\,0\right)}$,
• ${\displaystyle \pm \left({\frac {\sqrt {10}}{4}},\,{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {3}}{3}},\,\pm 1\right)}$,
• ${\displaystyle \pm \left({\frac {\sqrt {10}}{4}},\,{\frac {5{\sqrt {6}}}{12}},\,{\frac {\sqrt {3}}{3}},\,0\right)}$,
• ${\displaystyle \pm \left({\frac {\sqrt {10}}{4}},\,{\frac {5{\sqrt {6}}}{12}},\,-{\frac {\sqrt {3}}{6}},\,\pm {\frac {1}{2}}\right)}$,
• ${\displaystyle \pm \left({\frac {\sqrt {10}}{4}},\,-{\frac {\sqrt {6}}{4}},\,0,\,\pm 1\right)}$,
• ${\displaystyle \pm \left({\frac {\sqrt {10}}{4}},\,-{\frac {\sqrt {6}}{4}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}}\right)}$.

Simpler coordinates are given by all odd sign changes of:

• ${\displaystyle \left({\frac {3{\sqrt {2}}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {10}}{4}}\right)}$,

and all permutations of the first 3 coordinates of:

• ${\displaystyle \left(\pm {\sqrt {2}},\,0,\,0,\,0\right)}$.

Much simpler coordinates can be given in five dimensions, as all permutations of:

• ${\displaystyle \left({\sqrt {2}},\,{\sqrt {2}},\,{\frac {\sqrt {2}}{2}},\,0,\,0\right)}$.

## Representations

A decachoron has the following Coxeter diagrams:

• o3x3x3o () (full symmetry)
• oox3xux3xoo&#xt (A3 axial, cell-first)
• oxuxo ooxux3xuxoo&#xt (A2×A1 axial, triangle-first)

## Variations

The following variants of the decachoron exist:

## Related polytopes

The hexagonal faces of the decachoron form a regular polyhedron, {6,4∣3}

The tripesic hexacosichoron is a uniform polychoron compound composed of 60 decachora.

### Isogonal derivatives

Substitution by vertices of these following elements will produce these convex isogonal polychora: