Stellated decachoron

Stellated decachoron
(OFF file)
Rank	4
Type	Regular
Notation
Bowers style acronym	Sted
Coxeter diagram	xo3oo3oo3ox ()
Elements
Components	2 pentachora
Cells	10 tetrahedra
Faces	20 triangles
Edges	20
Vertices	10
Vertex figure	Tetrahedron, 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
Army	Bideca
Regiment	Sted
Dual	Stellated decachoron
Conjugate	None
Convex core	Decachoron
Abstract & topological properties
Schläfli type	{3,3,3}
Orientable	Yes
Properties
Symmetry	A4×2, order 240
Convex	No
Nature	Tame

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:

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

External links

• Klitzing, Richard. "sted".