# Stellated decachoron

Stellated decachoron Rank4
TypeRegular
Notation
Bowers style acronymSted
Coxeter diagramxo3oo3oo3ox (           )
Elements
Components2 pentachora
Cells10 tetrahedra
Faces20 triangles
Edges20
Vertices10
Vertex figureTetrahedron, edge length 1
Measures (edge length 1)
Circumradius${\frac {\sqrt {10}}{5}}\approx 0.63246$ Inradius${\frac {\sqrt {10}}{20}}\approx 0.15811$ Hypervolume${\frac {\sqrt {5}}{48}}\approx 0.046585$ Dichoral angle$\arccos \left({\frac {1}{4}}\right)\approx 75.52249^{\circ }$ Related polytopes
ArmyBideca
RegimentSted
DualStellated decachoron
ConjugateNone
Convex coreDecachoron
Abstract & topological properties
Schläfli type{3,3,3}
OrientableYes
Properties
SymmetryA4×2, order 240
ConvexNo
NatureTame

The stellated decachoron or sted is the simplest regular compound polychoron, and the third in an infinite series of regular bi-simplex compounds. It is a compound of two pentachora in dual orientations. It has 10 tetrahedra as cells, with 4 cells joining at a vertex. As the name suggests, it is also a stellation of the uniform decachoron.

## Vertex coordinates

The vertices of a stellated decachoron of edge length 1, centered at the origin, are given by:

• $\pm \left(\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}},\,-{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {10}}{20}}\right)$ ,
• $\pm \left(0,\,{\frac {\sqrt {3}}{3}},\,-{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {10}}{20}}\right)$ ,
• $\pm \left(0,\,0,\,{\frac {\sqrt {6}}{4}},\,-{\frac {\sqrt {10}}{20}}\right)$ ,
• $\left(0,\,0,\,0,\,\pm {\frac {\sqrt {10}}{5}}\right)$ .