Great truncated cuboctahedron

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.

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

3D model of a great truncated cuboctahedron

Convex hull

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

Orthographic projections

Cartesian coordinates

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

See also

• List of uniform polyhedra