Small prismatodecachoron
Small prismatodecachoron  

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  
Dichoral angles  Trip–4–trip: 
Tet–3–trip:  
Central density  1 
Number of external pieces  30 
Level of complexity  4 
Related polytopes  
Army  Spid 
Regiment  Spid 
Dual  Triangularantitegmatic icosachoron 
Conjugate  None 
Abstract & topological properties  
Flag count  960 
Euler characteristic  0 
Orientable  Yes 
Properties  
Symmetry  A_{4}×2, order 240 
Flag orbits  4 
Convex  Yes 
Nature  Tame 
The small prismatodecachoron, or spid, also commonly called the runcinated 5cell 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 vertexinscribed into a grand antiprism, and indeed the regular hexacosichoron as well, because it also possesses gyrochoric symmetry (see below).
It can also be obtained as one of several isogonal hulls of 2 103 step prisms, which could be called the triangularprismatic 103 double gyrostep prism.
Gallery[edit  edit source]

Crosssection animation

Net
Vertex coordinates[edit  edit source]
The vertices of a small prismatodecachoron of edge length 1 are given by the following points:
 ,
 ,
 ,
 ,
 ,
 ,
 .
Simpler coordinates are given by all even sign changes of:
 ,
and all permutations of the first 3 coordinates of:
 .
Much simpler coordinates can be given in five dimensions, as all permutations of:
 .
Representations[edit  edit source]
A small prismatodecachoron has the following Coxeter diagrams:
 x3o3o3x () (full symmetry)
 xxo3ooo3oxx&#xt (A_{3} axial, tetrahedronfirst)
 x(ou)x x(xo)o3o(xo)x&#xt (A_{2}×A_{1} axial, triangular prismfirst)
 (xoxxox)(uo) (oxxoxx)(ou)&#xr (A_{1}×A_{1} axial)
Variations[edit  edit source]
The small prismatodecachoron has a few subsymmetrical isogonal variants:
 Small disprismatopentapentachoron  Single pen symmetry, 2 sets of each type of cell
 Triangularprismatic 103 double gyrostep prism  1 of the hulls of 103 step prisms, has gyrochoron symmetry
Related polytopes[edit  edit source]
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 prismfirst orientation.
Polychoron compounds[edit  edit source]
Uniform polychoron compounds composed of small prismatodecachora include:
Isogonal derivatives[edit  edit source]
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[edit  edit source]
 Bowers, Jonathan. "Category 11: Antipodiumverts" (#442).
 Bowers, Jonathan. "Pennic and Decaic Isogonals".
 Klitzing, Richard. "spid".
 Quickfur. "The Runcinated 5cell".
 Wikipedia contributors. "Runcinated 5cell".