# Runcinated 5-simplex

Runcinated 5-simplex
Rank5
TypeUniform
Notation
Bowers style acronymSpix
Coxeter diagramx3o3o3x3o ()
Elements
Tera20 triangular duoprisms, 15 octahedral prisms, 6 rectified pentachora, 6 small prismatodecachora
Cells15+30 tetrahedra, 60+60+60 triangular prisms, 30 octahedra
Faces60+60+120 triangles, 180 squares
Edges90+180
Vertices60
Vertex figureTriangular antifastegium, edge lengths 1 (base triangular prism and top triangle) and 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {51}}{6}}\approx 1.19024}$
Hypervolume${\displaystyle {\frac {309{\sqrt {3}}}{160}}\approx 3.34502}$
Diteral anglesOpe–trip–triddip: 135°
Rap–oct–ope: ${\displaystyle \arccos \left(-{\frac {\sqrt {10}}{5}}\right)\approx 129.23152^{\circ }}$
Spid–trip–triddip: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }}$
Spid–trip–ope: ${\displaystyle \arccos \left(-{\frac {\sqrt {10}}{10}}\right)\approx 108.43495^{\circ }}$
Spid–tet–rap: ${\displaystyle \arccos \left(-{\frac {1}{5}}\right)\approx 101.53696^{\circ }}$
Spid–tet–spid: ${\displaystyle \arccos \left({\frac {1}{5}}\right)\approx 78.46304^{\circ }}$
Central density1
Number of external pieces47
Level of complexity25
Related polytopes
ArmySpix
RegimentSpix
DualAntitegmatonotched hexecontateron
ConjugateNone
Abstract & topological properties
Flag count18000
Euler characteristic2
OrientableYes
Properties
SymmetryA5, order 720
ConvexYes
NatureTame

The runcinated 5-simplex, also called the small prismated hexateron or spix, is a convex uniform 5-polytope. It consists of 20 triangular duoprisms, 15 octahedral prisms, 6 rectified pentachora, and 6 small prismatodecachora. One rectified pentachoron, 2 small prismatodecachora, 3 octahedral prisms, and 3 triangular duoprisms join at each vertex. As the name suggests, it is the runcination of the 5-simplex.

## Vertex coordinates

The vertices of a runcinated 5-simplex of edge length 1 can be given in 6 dimensions as all permutations of:

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

## Representations

A runcinated 5-simplex has the following Coxeter diagrams:

• x3o3o3x3o () (full symmetry)
• oox3xxo3ooo3oxx&#xt (A4 axial, rectified pentachoron-first)

## Related polytopes

The regiment of the runcinated 5-simplex includes a total of 37 members plus one fissary member. This makes it the largest regiment of uniform polytera with A5 symmetry.