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}
Wythoff symbol	2 3 4/3 \|
Symmetry group	O_h, [4,3], *432
Index references	U₂₀, C₆₇, W₉₃
Dual polyhedron	Great disdyakis dodecahedron
Vertex figure	4.6/5.8/3
Bowers acronym	Quitco

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 centered at the origin are all permutations of

(±1, ±(1−√2), ±(1−2√2)).

References

Maeder, Roman. "20: great truncated cuboctahedron". MathConsult. Archived from the original on 2020-02-17.