Small rhombicuboctahedron atop truncated octahedron

Small rhombicuboctahedron atop truncated octahedron
(OFF file)
Rank	4
Type	Segmentotope
Notation
Bowers style acronym	Sircoatoe
Coxeter diagram	ox4xo3xx&#x
Elements
Cells	12 triangular prisms, 6 square antiprisms, 8 triangular cupolas, 1 small rhombicuboctahedron, 1 truncated octahedron
Faces	8+24+24 triangles, 6+6+12+24 squares, 8 hexagons
Edges	12+24+24+24+48
Vertices	24+24
Vertex figures	24 skewed wedges, edge lengths 1 (4) and √2 (5)
	24 isosceles trapezoidal pyramids, base edge lengths 1, 1, 1, √2, side edge lengths √2, √2, √3, √3
Measures (edge length 1)
Circumradius	${\sqrt {\frac {35+16{\sqrt {2}}}{23}}}\approx 1.58289$
Hypervolume	${\frac {\sqrt {18405+16124{\sqrt {2}}}}{24}}\approx 8.45821$
Dichoral angles	Tricu–4–trip: $\arccos \left({\frac {{\sqrt {6}}-4{\sqrt {3}}}{6}}\right)\approx 138.28318^{\circ }$
	Squap–3–trip: $\arccos \left(-{\frac {\sqrt {24{\sqrt {2}}}}{8}}\right)\approx 136.73885^{\circ }$
	Squap–3–tricu: $\arccos \left(-{\frac {\sqrt {9{\sqrt {2}}-8}}{4}}\right)\approx 122.92868^{\circ }$
	Sirco–4–trip: $\arccos \left({\frac {{\sqrt {6}}-2{\sqrt {3}}}{3}}\right)\approx 109.76749^{\circ }$
	Sirco–4–squap: $\arccos \left(-{\frac {\sqrt {3{\sqrt {2}}-4}}{2}}\right)\approx 104.25825^{\circ }$
	Toe–6–tricu: $\arccos \left({\frac {4-3{\sqrt {2}}}{4}}\right)\approx 93.47771^{\circ }$
	Sirco–3–tricu: $\arccos \left({\frac {3{\sqrt {2}}-4}{4}}\right)\approx 86.52229^{\circ }$
	Toe–4–squap: $\arccos \left({\frac {\sqrt {3{\sqrt {2}}-4}}{2}}\right)\approx 75.74175^{\circ }$
Height	${\frac {\sqrt {4{\sqrt {2}}-3}}{2}}\approx 0.81499$
Central density	1
Related polytopes
Army	Sircoatoe
Regiment	Sircoatoe
Dual	Deltoidal icositetrahedral-tetrakis hexahedral tegmoid
Conjugate	Small rhombicuboctahedron atop truncated octahedron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	B3×I, order 48
Convex	Yes
Nature	Tame

Small rhombicuboctahedron atop truncated octahedron, or sircoatoe, is a CRF segmentochoron (designated K-4.75 on Richard Klitzing's list). As the name suggests, it consists of a small rhombicuboctahedron and a truncated octahedron as bases, connected by 12 triangular prisms, 6 square antiprisms, and 8 triangular cupolas.

Vertex coordinates

The vertices of a small rhombicuboctahedron atop truncated octahedron segmentochoron of edge length 1 are given by:

• $\left(\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,{\frac {\sqrt {4{\sqrt {2}}-3}}{2}}\right)$ and all permutations of the first three coordinates,
• $\left(\pm {\sqrt {2}},\,\pm {\frac {\sqrt {2}}{2}},\,0,\,0\right)$ and all permutations of the first three coordinates.

External links

• Klitzing, Richard. "sircoatoe".