Nonconvex great rhombicuboctahedron

Nonconvex great rhombicuboctahedron
Uniform great rhombicuboctahedron.png
Type Uniform star polyhedron
Elements F = 26, E = 48
V = 24 (χ = 2)
Faces by sides 8{3}+(6+12){4}
Wythoff symbol 3/2 4 | 2
3 4/3 | 2
Symmetry group Oh, [4,3], *432
Index references U17, C59, W85
Dual polyhedron Great deltoidal icositetrahedron
Vertex figure Uniform great rhombicuboctahedron vertfig.png
4.4.4.3/2
Bowers acronym Querco

In geometry, the nonconvex great rhombicuboctahedron is a nonconvex uniform polyhedron, indexed as U17. It has 26 faces (8 triangles and 18 squares), 48 edges, and 24 vertices.[1] It is represented by Schläfli symbol t0,2{4,​32} and Coxeter-Dynkin diagram of CDel node 1.pngCDel 4.pngCDel node.pngCDel d3.pngCDel rat.pngCDel d2.pngCDel node 1.png. Its vertex figure is a crossed quadrilateral.

3D model of a nonconvex great rhombicuboctahedron

This model shares the name with the convex great rhombicuboctahedron, also called the truncated cuboctahedron.

An alternate name for this figure is quasirhombicuboctahedron. From that derives its Bowers acronym: querco.

Orthogonal projectionsEdit

 

Cartesian coordinatesEdit

Cartesian coordinates for the vertices of a nonconvex great rhombicuboctahedron centered at the origin with edge length 1 are all the permutations of

ξ, ±1, ±1),

where ξ = 2 − 1.

Related polyhedraEdit

It shares the vertex arrangement with the convex truncated cube. It additionally shares its edge arrangement with the great cubicuboctahedron (having the triangular faces and 6 square faces in common), and with the great rhombihexahedron (having 12 square faces in common). It has the same vertex figure as the pseudo great rhombicuboctahedron, which is not a uniform polyhedron.

 
Truncated cube
 
Great rhombicuboctahedron
 
Great cubicuboctahedron
 
Great rhombihexahedron
 
Pseudo great rhombicuboctahedron

Great deltoidal icositetrahedronEdit

Great deltoidal icositetrahedron
 
Type Star polyhedron
Face  
Elements F = 24, E = 48
V = 26 (χ = 2)
Symmetry group Oh, [4,3], *432
Index references DU17
dual polyhedron Nonconvex great rhombicuboctahedron
 
3D model of a great deltoidal icositetrahedron

The great deltoidal icositetrahedron is the dual of the nonconvex great rhombicuboctahedron.

ReferencesEdit

  1. ^ Maeder, Roman. "17: great rhombicuboctahedron". MathConsult.

External linksEdit

Weisstein, Eric W. "Great Deltoidal Icositetrahedron". MathWorld.