# Decachoron

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.

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

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: