Truncated rhombicuboctahedron

Truncated rhombicuboctahedron
Schläfli symbol	trr{4,3} = $tr{\begin{Bmatrix}4\\3\end{Bmatrix}}$
Conway notation	taaC
Faces	50: 24 {4} 8 {6} 6+12 {8}
Edges	144
Vertices	96
Symmetry group	Oh, [4,3], (*432) order 48
Rotation group	O, [4,3]+, (432), order 24
Dual polyhedron	Disdyakis enneacontahexahedron
Properties	convex, zonohedron

The truncated rhombicuboctahedron is a polyhedron, constructed as a truncated rhombicuboctahedron. It has 50 faces, 18 octagons, 8 hexagons, and 24 squares.

Other names

• Truncated small rhombicuboctahedron
• Beveled cuboctahedron

Zonohedron

As a zonohedron, it can be constructed with all but 12 octagons as regular polygons. It is 2-uniform, with 2 sets of 48 vertices existing on two distances from its center.

It represents the Minkowski sum of a cube, a truncated octahedron, and a rhombic dodecahedron.

Excavated truncated rhombicuboctahedron

Excavated truncated rhombicuboctahedron
Faces	148: 8 {3} 24+96+6 {4} 8 {6} 6 {8}
Edges	312
Vertices	144
Euler characteristic	-20
genus	11
Symmetry group	Oh, [4,3], (*432) order 48

The truncated rhombicuboctahedron can have its 12 irregular octagonal faces removed, and a toroidal polyhedron seen as a network of 6 square cupola, 8 triangular cupola, and 12 triangular prisms. It has 148 faces (8 triangles, 126 squares, 8 hexagons, and 6 octagons), 312 edges, and 144 vertices. With Euler characteristic χ = f + v - e = -20, its genus, g = (2-χ)/2 is 11.

Without the triangular prisms, the toroidal polyhedron becomes a truncated cuboctahedron.

Excavated

Truncated rhombicuboctahedron	Truncated cuboctahedron

Related polyhedra

The truncated cuboctahedron is similar, with all regular faces, and 4.6.8 vertex figure.

The triangle and squares of the rhombicuboctahedron can be independently rectified or truncated, creating four permutations of polyhedra. The partially truncated forms can be seen as edge contractions of the truncated form.

rectified/truncated rhombicuboctahedron

Rectified	Partially truncated		Truncated
4.4.4.4 and 3.4.4.4	4.4.4.6 and 4.6.6	4.6.8 and 3.4.6.4	4.8.8 and 4.6.8

The truncated rhombicuboctahedron can be seen in sequence of rectification and truncation operations from the cuboctahedron. A further alternation step leads to the snub rhombicuboctahedron.

Name	Cuboctahedron	Rhombi- cuboctahedron	Truncated rhombi- cuboctahedron	Snub rhombi- cuboctahedron
Coxeter	CO (rC)	rCO (rrC)	trCO (trrC)	srCO (htrrC)
Conway	aC	aaC = eC	taaC = baC	saC
Image
Conway	jC	oC	maC	gaC
Dual

See also

• Expanded cuboctahedron
• Truncated rhombicosidodecahedron

References

• Eppstein, David (1996). "Zonohedra and zonotopes". Mathematica in Education and Research. 5 (4): 15–21.
• Coxeter Regular Polytopes, Third edition, (1973), Dover edition, ISBN 0-486-61480-8 (pp. 145–154 Chapter 8: Truncation)
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5

External links

• George Hart's Conway interpreter: generates polyhedra in VRML, taking Conway notation as input
• Prism Expansions [1] Toroid model