Small prismatotetracontoctachoron

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Small prismatotetracontoctachoron
Bowers style acronymSpic
Coxeter diagramx3o4o3x ()
Cells48 octahedra, 192 triangular prisms
Faces384 triangles, 288 squares
Vertex figureSquare antiprism, edge lengths 1 (base) and 2 (sides)
Edge figure(oct 3 trip 4 trip 4 trip 3)
Measures (edge length 1)
Dichoral anglesTrip–4–trip:
 Oct–3–trip: 150°
Central density1
Number of external pieces240
Level of complexity4
Related polytopes
DualSquare-antitegmatic hecatontetracontatetrachoron
Abstract & topological properties
Flag count9216
Euler characteristic0
SymmetryF4×2, order 2304

The small prismatotetracontoctachoron, or spic, also commonly called the runcinated 24-cell, is a convex uniform polychoron that consists of 48 regular octahedra and 192 triangular prisms. 2 octahedra and 8 triangular prisms join at each vertex. It is the result of expanding the cells of an icositetrachoron outwards.

The small prismatotetracontoctachoron contains the vertices of an octagonal duoprism, small rhombated tesseract, truncated cubic prism, triangular-hexagonal prismantiprismoid and the triangular double prismantiprismoid.

Cross-sections[edit | edit source]

Card with cell counts, vertex figure, and cross-sections.

Vertex coordinates[edit | edit source]

The vertices of a small prismatotetracontoctachoron of edge length 1 are all permutations of:

  • ,
  • .

The second set of vertices are identical to the vertices of an inscribed small rhombated tesseract.

Representations[edit | edit source]

The small prismatotetracontoctachoron has the following Coxeter diagrams:

  • x3o4o3x () (full symmetry)
  • oxoxoxoxo4oooxxxooo3xxwoqowxx&#xt (B3 axial, octahedron-first)
  • ox4oo3xx3qo&#zx (B4 symmetry)
  • qoo3xxx3oqo *b3ooq&#zx (D4 symmetry)
  • Uwqxo oxoxo4oooxx3xxwoq&#zx (B3×A1 symmetry)
  • ooxxowq4oxoxwoq qowxoxo4qwoxxoo&#zx (B2×B2 symmetry)

Variations[edit | edit source]

The small prismatotetracontoctachoron has a semi-uniform variant under half symmetry called the small disprismatoicositetricositetrachoron.

Related polychora[edit | edit source]

The small prismatotetracontoctachoron is the colonel of a regiment that includes 20 members, 2 fissaries, and a compound. Among the members, 8 in total have doubled F4 symmetry, including the small prismatotetracontoctachoron itself, the small distetracontoctachoron, and the noble small retrotetracontoctachoron, while the 12 remaining members, and the fissaries and compound, have single F4 symmetry only.

The small prismatotetracontoctachoron can be diminished by removing octahedron atop small rhombicuboctahedron segmentochora. If 8 of these caps are removed, the result is the small rhombated tesseract.

Uniform polychoron compounds composed of small prismatotetracontoctachora include:

Isogonal derivatives[edit | edit source]

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

External links[edit | edit source]