Truncated rhombicuboctahedron

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Truncated rhombicuboctahedron
Truncated rhombicuboctahedron2.png
Schläfli symbol trr{4,3} =
Conway notation taaC
24 {4}
8 {6}
6+12 {8}
Symmetry group Oh, [4,3], (*432) order 48
Rotation group O, [4,3]+, (432), order 24
Dual polyhedron Disdyakis icositetrahedron
Disdyakis enneacontahexahedron.png
Propertiesconvex, zonohedron

The truncated rhombicuboctahedron is a polyhedron, constructed as a truncation of the rhombicuboctahedron. It has 50 faces consisting of 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


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

Excavated truncated rhombicuboctahedron
8 {3}
24+96+6 {4}
8 {6}
6 {8}
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

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 truncatedTruncated
Conway aC aaC=eC taC=bC aaaC=eaC dXCdXdC 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 and
3.4.4d.4 and
4.6i.8 and
4.8.8p and
4.6.8p and

See also

Related Research Articles

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In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids and excluding the prisms and antiprisms. They differ from the Johnson solids, whose regular polygonal faces do not meet in identical vertices.

Cuboctahedron polyhedron with 8 triangular faces and 6 square faces

A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is the only radially equilateral convex polyhedron.

Cube A geometric 3-dimensional object with 6 square faces

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Rhombicuboctahedron Archimedean solid with eight triangular and eighteen square faces

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Truncated cube

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Truncated cuboctahedron Archimedean solid in geometry

In geometry, the truncated cuboctahedron is an Archimedean solid, named by Kepler as a truncation of a cuboctahedron. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices, and 72 edges. Since each of its faces has point symmetry, the truncated cuboctahedron is a zonohedron. The truncated cuboctahedron can tessellate with the octagonal prism.

Rectification (geometry) Operation in Euclidean geometry

In Euclidean geometry, rectification, also known as critical truncation or complete-truncation is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points. The resulting polytope will be bounded by vertex figure facets and the rectified facets of the original polytope.

Runcinated tesseracts

In four-dimensional geometry, a runcinated tesseract is a convex uniform 4-polytope, being a runcination of the regular tesseract.

Cantellated tesseract

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Cubic honeycomb

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Conway polyhedron notation

In geometry, Conway polyhedron notation, invented by John Horton Conway and promoted by George W. Hart, is used to describe polyhedra based on a seed polyhedron modified by various prefix operations.

Tetradecahedron Polyhedron with 14 faces

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Crossed square cupola

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Expanded cuboctahedron

The expanded cuboctahedron is a polyhedron, constructed as an expanded cuboctahedron. It has 50 faces: 8 triangles, 30 squares, and 12 rhombs. The 48 vertices exist at two sets of 24, with a slightly different distance from its center.

Expanded icosidodecahedron

The expanded icosidodecahedron is a polyhedron, constructed as an expanded icosidodecahedron. It has 122 faces: 20 triangles, 60 squares, 12 pentagons, and 30 rhombs. The 120 vertices exist at two sets of 60, with a slightly different distance from its center.

Chamfer (geometry)

In geometry, chamfering or edge-truncation is a topological operator that modifies one polyhedron into another. It is similar to expansion, moving faces apart and outward, but also maintains the original vertices. For polyhedra, this operation adds a new hexagonal face in place of each original edge.


  1. "Prism Expansions".