Great truncated cuboctahedron

Great truncated cuboctahedron
Type	Uniform star polyhedron
Elements	F = 26, E = 72 V = 48 (χ = 2)
Faces by sides	12{4}+8{6}+6{8/3}
Coxeter diagram
Wythoff symbol	2 3 4/3 |
Symmetry group	Oh, [4,3], *432
Index references	U20, C67, W93
Dual polyhedron	Great disdyakis dodecahedron
Vertex figure	4.6/5.8/3
Bowers acronym	Quitco

In geometry, the great truncated cuboctahedron (or quasitruncated cuboctahedron or stellatruncated cuboctahedron) is a nonconvex uniform polyhedron, indexed as U₂₀. It has 26 faces (12 squares, 8 hexagons and 6 octagrams), 72 edges, and 48 vertices.^[1] It is represented by the Schläfli symbol tr{⁴/₃,3}, and Coxeter-Dynkin diagram . It is sometimes called the quasitruncated cuboctahedron because it is related to the truncated cuboctahedron, , except that the octagonal faces are replaced by {⁸/₃} octagrams.

Its convex hull is a nonuniform truncated cuboctahedron. The truncated cuboctahedron and the great truncated cuboctahedron form isomorphic graphs despite their different geometric structure.

Convex hull

Great truncated cuboctahedron

Cartesian coordinates for the vertices of a great truncated cuboctahedron with side length 2 centered at the origin are all permutations of $${\displaystyle {\Bigl (}\pm 1,\ \pm \left[1-{\sqrt {2}}\right],\ \pm \left[1-2{\sqrt {2}}\right]{\Bigr )}.}$$

• List of uniform polyhedra

• Weisstein, Eric W. "Great truncated cuboctahedron". MathWorld.

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