Cuboctahedron atop truncated cube

Cuboctahedron atop truncated cube
(OFF file)
Rank	4
Type	Segmentotope
Notation
Bowers style acronym	Coatic
Coxeter diagram	ox4xx3oo&#x
Elements
Cells	8 triangular prisms, 6 square cupolas, 1 cuboctahedron, 1 truncated cube
Faces	8+8+12 triangles, 6+24 squares, 6 octagons
Edges	12+24+24+24
Vertices	12+24
Vertex figures	12 wedges, edge lengths 1 (two edges of base and top edge) and √2 (rest of edges)
	24 sphenoids, edge lengths 1, √2, and √2+√2 (2 each)
Measures (edge length 1)
Circumradius	${\sqrt {4+2{\sqrt {2}}}}\approx 2.61313$
Hypervolume	${\frac {39+32{\sqrt {2}}}{4}}\approx 3.51062$
Dichoral angles	Co–3–trip: 150°
	Squacu–4–trip: $\arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.735611^{\circ }$
	Co–4–squacu: 135°
	Squacu–3–squacu: 120°
	Tic–8–squacu: 45°
	Tic–3–trip: 30°
Height	${\frac {1}{2}}=0.5$
Central density	1
Related polytopes
Army	Coatic
Regiment	Coatic
Dual	Rhombic dodecahedral-triakis octahedral tegmoid
Conjugate	Cuboctahedron atop quasitruncated hexahedron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	B3×I, order 48
Convex	Yes
Nature	Tame

Cuboctahedron atop truncated cube, or coatic, is a CRF segmentochoron (designated K-4.129 on Richard Klitzing's list). As the name suggests, it consists of a cuboctahedron and a truncated cube as bases, connected by 8 triangular prisms and 6 square cupolas.

It can be obtained as a cap of the small rhombated icositetrachoron, which can be obtained by attaching 8 of these segmentochora to the truncated cubic cells of the prismatorhombated hexadecachoron.

Vertex coordinates

The vertices of a cuboctahedron atop truncated cube segmentochoron of edge length 1 are given by:

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

External links

• Klitzing, Richard. "coatic".
• Quickfur. "Cuboctahedron atop Truncated Cube".