Truncated rhombicuboctahedron

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Truncated rhombicuboctahedron
Truncated rhombicuboctahedron2.png
Schläfli symbol trr{4,3} =
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 icositetrahedron
Disdyakis enneacontahexahedron.png
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. It can fill space with the truncated cube, truncated tetrahedron and triangular prism as a truncated runcic cubic honeycomb.

Other names[edit]

  • Truncated small rhombicuboctahedron
  • Beveled cuboctahedron


As a zonohedron, it can be constructed with all but 12 octagons as regular polygons. It has two 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[edit]

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 excavated truncated rhombicuboctahedron is a toroidal polyhedron, constructed from a truncated rhombicuboctahedron with its 12 irregular octagonal faces removed. It comprises a network of 6 square cupolae, 8 triangular cupolae, and 24 triangular prisms. [1] 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.png Excavated truncated cuboctahedron.png
Truncated rhombicuboctahedron Truncated cuboctahedron

Related polyhedra[edit]

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.

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.

related polyhedra
Name r{4,3} rr{4,3} tr{4,3} Rectified
Partially truncated Truncated
Conway aC aaC=eC taC=bC aaaC=eaC dXC dXdC taaC=baC saC
Image Uniform polyhedron-43-t1.svg Uniform polyhedron-43-t02.png Uniform polyhedron-43-t012.png Expanded dual cuboctahedron.png Truncated rhombicuboctahedron2b.png Truncated rhombicuboctahedron2a.png Truncated rhombicuboctahedron2.png Snub rhombicuboctahedron2.png
VertFigs 4.6.8 and
3.4.4d.4 and
4.6i.8 and
4.8.8p and
4.6.8p and

See also[edit]


  • 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[edit]