# Small prismatodecachoron

Small prismatodecachoron Rank4
TypeUniform
Notation
Bowers style acronymSpid
Coxeter diagramx3o3o3x (       )
Elements
Cells10 tetrahedra, 20 triangular prisms
Faces40 triangles, 30 squares
Edges60
Vertices20
Vertex figureTriangular antiprism, edge lengths 1 (base) and 2 (sides) Edge figuretet 3 trip 4 trip 4 trip 3
Measures (edge length 1)
Hypervolume${\frac {35{\sqrt {5}}}{48}}\approx 1.63047$ Dichoral anglesTrip–4–trip: $\arccos \left(-{\frac {2}{3}}\right)\approx 131.81032^{\circ }$ Tet–3–trip: $\arccos \left(-{\frac {\sqrt {6}}{4}}\right)\approx 127.76124^{\circ }$ Central density1
Number of external pieces30
Level of complexity4
Related polytopes
ArmySpid
RegimentSpid
DualTriangular-antitegmatic icosachoron
ConjugateNone
Abstract & topological properties
Flag count960
Euler characteristic0
OrientableYes
Properties
SymmetryA4×2, order 240
ConvexYes
NatureTame

The small prismatodecachoron, or spid, also commonly called the runcinated 5-cell or runcinated pentachoron, is a convex uniform polychoron that consists of 10 regular tetrahedra and 20 triangular prisms. 2 tetrahedra and 6 triangular prisms join at each vertex. It is the result of expanding the cells of a pentachoron outwards.

The small prismatodecachoron of edge length (5+1)/2 can be vertex-inscribed into a grand antiprism, and indeed the regular hexacosichoron as well.

It can also be obtained as one of several isogonal hulls of 2 10-3 step prisms, which could be called the triangular-prismatic 10-3 double gyrostep prism.

## Vertex coordinates

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

• $\pm \left(0,\,0,\,0,\,\pm 1\right),$ • $\pm \left(0,\,0,\,\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}}\right),$ • $\pm \left(0,\,{\frac {\sqrt {6}}{3}},\,-{\frac {\sqrt {3}}{3}},\,0\right),$ • $\pm \left(0,\,{\frac {\sqrt {6}}{3}},\,{\frac {\sqrt {3}}{6}},\,\pm {\frac {1}{2}}\right),$ • $\pm \left({\frac {\sqrt {10}}{4}},\,-{\frac {\sqrt {6}}{4}},\,0,\,0\right),$ • $\pm \left({\frac {\sqrt {10}}{4}},\,{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {3}}{3}},\,0\right),$ • $\pm \left({\frac {\sqrt {10}}{4}},\,{\frac {\sqrt {6}}{12}},\,{\frac {\sqrt {3}}{6}},\,\pm {\frac {1}{2}}\right).$ Simpler coordinates are given by all even sign changes of:

• $\left({\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {10}}{4}}\right),$ and all permutations of the first 3 coordinates of:

• $\left(\pm {\frac {\sqrt {2}}{2}},\,\pm {\frac {\sqrt {2}}{2}},\,0,\,0\right).$ Much simpler coordinates can be given in five dimensions, as all permutations of:

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

A small prismatodecachoron has the following Coxeter diagrams:

• x3o3o3x (full symmetry)
• xxo3ooo3oxx&#xt (A3 axial, tetrahedron-first)
• x(ou)x x(xo)o3o(xo)x&#xt (A2×A1 axial, triangular prism-first)
• (xoxxox)(uo) (oxxoxx)(ou)&#xr (A1×A1 axial)

## Variations

The small prismatodecachoron has a few subsymmetrical isogonal variants:

## Related polychora

The small prismatodecachoron is the colonel of a 5-member regiment. Its other members include the decahemidecachoron, the prismatohemidecachoron, the prismatopentahemidecachoron, and the spinoprismatodispentachoron. The first two of these polychora have full symmetry, while the latter two have single symmetry only.

A small prismatodecachoron can be cut in half to produce two identical tetrahedron atop cuboctahedron segmentochora, with the tetrahedral bases in dual orientations. The triangular cupofastegium can also be obtained as a wedge of the small prismatodecachoron, in triangular prism-first orientation.

Uniform polychoron compounds composed of small prismatodecachora include:

### Isogonal derivatives

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