Small prismatodecachoron

Small prismatodecachoron
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Spid
Coxeter diagram	x3o3o3x ()
Elements
Cells	10 tetrahedra, 20 triangular prisms
Faces	40 triangles, 30 squares
Edges	60
Vertices	20
Vertex figure	Triangular antiprism, edge lengths 1 (base) and √2 (sides)
Edge figure	tet 3 trip 4 trip 4 trip 3
Measures (edge length 1)
Circumradius	1
Hypervolume	${\frac {35{\sqrt {5}}}{48}}\approx 1.63047$
Dichoral angles	Trip–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 density	1
Number of external pieces	30
Level of complexity	4
Related polytopes
Army	Spid
Regiment	Spid
Dual	Triangular-antitegmatic icosachoron
Conjugate	None
Abstract & topological properties
Flag count	960
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A4×2, order 240
Convex	Yes
Nature	Tame

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.

Gallery

• 

  Cross-sections

• 

  Cross-section animation

• 

  Net

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:

• Small disprismatopentapentachoron - Single pen symmetry, 2 sets of each type of cell
• Triangular-prismatic 10-3 double gyrostep prism - 1 of the hulls of 10-3 step prisms, has gyrochoron symmetry

Related polytopes

The small prismatodecachoron is the colonel of a regiment with 5 uniform polytopes. 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.

The square faces of the small prismatodecachoron form a regular skew polyhedron, {4,6∣3}.

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.

Polychoron compounds

Uniform polychoron compounds composed of small prismatodecachora include:

• Compound of 6 small prismatodecachora (6)
• dodecahedronary prismatohexacosichoron (60)

Isogonal derivatives

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

• Tetrahedron (10): Bidecachoron
• Triangular prism (20): Biambodecachoron
• Square (30): Decachoron
• Triangle (40): Bitruncatodecachoron
• Edge (60): Rectified small prismatodecachoron

External links

• Bowers, Jonathan. "Category 11: Antipodiumverts" (#442).

• Bowers, Jonathan. "Pennic and Decaic Isogonals".

• Klitzing, Richard. "spid".
• Quickfur. "The Runcinated 5-cell".

• Wikipedia contributors. "Runcinated 5-cell".