Truncated octahedron atop truncated cube

Truncated octahedron atop truncated cube
(OFF file)
Rank	4
Type	Segmentotope
Notation
Bowers style acronym	Toatic
Coxeter diagram	ox4xx3xo&#x
Elements
Cells	12 tetrahedra, 8 triangular cupolas, 6 square cupolas, 1 truncated octahedron, 1 truncated cube
Faces	8+24+24 triangles, 6+24 squares, 8 hexagons, 6 octagons
Edges	12+12+24+24+48
Vertices	24+24
Vertex figures	24 skewed rectangular pyramids, base edge lengths 1 and √2, side edge lengths 1, 1, √2+√2, √2+√2
	24 isosceles trapezoidal pyramids, base edge lengths 1, √2, √2, √2, side edge lengths 1, 1, √3, √3
Measures (edge length 1)
Circumradius	${\sqrt {\frac {11+8{\sqrt {2}}}{7}}}\approx 1.78541$
Hypervolume	${\sqrt {\frac {2686+2594{\sqrt {2}}-301{\sqrt {3}}+42{\sqrt {6}}}{72}}}\approx 9.07989$
Dichoral angles	Tet–3–tricu: $\arccos \left({\frac {1-3{\sqrt {2}}}{4}}\right)\approx 144.16048^{\circ }$
	Tet–3–squacu: $\arccos \left({\frac {2-3{\sqrt {2}}}{4}}\right)\approx 124.10147^{\circ }$
	Toe–6–tricu: $\arccos \left({\frac {2-3{\sqrt {2}}}{4}}\right)\approx 124.10147^{\circ }$
	Tic–8–squacu: $\arccos \left({\frac {{\sqrt {2}}-2}{2}}\right)\approx 107.03125^{\circ }$
	Squacu–4–tricu: $\arccos \left({\frac {{\sqrt {2}}-2}{2}}\right)\approx 107.03125^{\circ }$
	Toe–4–squacu: $\arccos \left({\frac {2-{\sqrt {2}}}{2}}\right)\approx 72.96875^{\circ }$
	Tic–3–tricu: $\arccos \left({\frac {3{\sqrt {2}}-2}{4}}\right)\approx 55.89854^{\circ }$
Height	${\frac {\sqrt {2{\sqrt {2}}-1}}{2}}\approx 0.67610$
Central density	1
Related polytopes
Army	Toatic
Regiment	Toatic
Dual	Tetrakis hexahedral-triakis octahedral tegmoid
Conjugate	Truncated octahedron atop truncated cube
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	B3×I, order 48
Convex	Yes
Nature	Tame

Truncated octahedron atop truncated cube, or toatic, is a CRF segmentochoron (designated K-4.98 on Richard Klitzing's list). As the name suggests, it consists of a truncated octahedron and a truncated cube as bases, connected by 12 tetrahedra, 8 triangular cupolas, and 6 square cupolas.

Vertex coordinates

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

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

External links

• Klitzing, Richard. "toatic".