Small prismatotetracontoctachoron

Small prismatotetracontoctachoron
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Spic
Coxeter diagram	x3o4o3x ()
Elements
Cells	48 octahedra, 192 triangular prisms
Faces	384 triangles, 288 squares
Edges	576
Vertices	144
Vertex figure	Square antiprism, edge lengths 1 (base) and √2 (sides)
Edge figure	(oct 3 trip 4 trip 4 trip 3)
Measures (edge length 1)
Circumradius	${\sqrt {2+{\sqrt {2}}}}\approx 1.84776$
Hypervolume	$2(11+8{\sqrt {2}})\approx 44.62742$
Dichoral angles	Trip–4–trip: $\arccos \left(-{\frac {2{\sqrt {2}}}{3}}\right)\approx 160.52878^{\circ }$
	Oct–3–trip: 150°
Central density	1
Number of external pieces	240
Level of complexity	4
Related polytopes
Army	Spic
Regiment	Spic
Dual	Square-antitegmatic hecatontetracontatetrachoron
Conjugate	Quasiprismatotetracontoctachoron
Abstract & topological properties
Flag count	9216
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	F4×2, order 2304
Convex	Yes
Nature	Tame

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

Vertex coordinates

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

• $\left(\pm {\frac {2+{\sqrt {2}}}{2}},\,\pm {\frac {\sqrt {2}}{2}},\,0,\,0\right),$
• $\left(\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right).$

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

Representations

The small prismatotetracontoctachoron has the following Coxeter diagrams:

• x3o4o3x (full symmetry)
• oxoxoxoxo4oooxxxooo3xxwoqowxx&#xt (BC3 axial, octahedron-first)
• ox4oo3xx3qo&#zx (BC4 symmetry)
• qoo3xxx3oqo *b3ooq&#zx (D4 symmetry)
• Uwqxo oxoxo4oooxx3xxwoq&#zx (BC3×A1 symmetry)
• ooxxowq4oxoxwoq qowxoxo4qwoxxoo&#zx (BC2×BC2 symmetry)

Variations

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

Related polychora

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:

• Small prismated chirohecatonicosachoron (5)
• Small prismated dodecahedronary hexacosichoron (25)

Isogonal derivatives

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

• Octahedron (48): Bitetracontoctachoron
• Triangular prism (192): Biambotetracontoctachoron
• Square (288): Tetracontoctachoron
• Triangle (384): Bitruncatotetracontoctachoron
• Edge (576): Rectified small prismatotetracontoctachoron

External links

• Bowers, Jonathan. "Category 13: Spic and Giddic Regiments" (#512).

• Klitzing, Richard. "spic".

• Quickfur. "The Runcinated 24-cell".

• Wikipedia contributors. "Runcinated 24-cell".