Chamfered cube

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Chamfered cube
Chamfered cube
Type Conway polyhedron: cC = t4daC
Goldberg polyhedron GIV(2,0)
Faces 6 squares
12 hexagons
Edges 48 (2 types)
Vertices 32 (2 types)
Vertex configuration (24) 4.6.6
(8) 6.6.6
Symmetry Oh, [4,3], (*432)
Dual polyhedron Tetrakis cuboctahedron
Properties convex, zonohedron, equilateral-faced

The chamfered cube (also called truncated rhombic dodecahedron) is a convex polyhedron constructed from the rhombic dodecahedron by truncating the 6 (order 4) vertices.

The 6 vertices are truncated such that all edges are equal length. The original 12 rhombic faces become flattened hexagons, and the truncated vertices become squares.

The hexagonal faces are equilateral but not regular. They are formed by a truncated rhombus, have 2 internal angles of about 109.47 degrees (\cos^{-1}(-\frac{1}{3})) and 4 internal angles of about 125.26 degrees, while a regular hexagon would have all 120 degree angles.

Because all its faces have an even number of sides with 180 degree rotation symmetry, it is a zonohedron. It is also the Goldberg polyhedron GIV(2,0), containing square and hexagonal faces.

Coordinates[edit]

The chamfered cube is the Minkowski sum of a rhombic dodecahedron and a cube of side length 1 when eight vertices of the rhombic dodecahedron are at (\pm 1, \pm 1, \pm 1) and its six vertices are at the permutations of (\pm 2, 0, 0).

Related polyhedra[edit]

This polyhedron is similar to the uniform truncated octahedron:

Truncated rhombic dodecahedron
Truncated rhombic dodecahedron2.png
Truncated octahedron
Truncated octahedron.png

This polyhedron is also a part of a sequence of truncated rhombic polyhedra and tilings with [n,3] Coxeter group symmetry. The cube can be seen as a rhombic hexahedron where the rhombi are squares. The truncated forms have regular n-gons at the truncated vertices, and nonregular hexagonal faces. The sequence has two vertex figures (n.6.6) and (6,6,6). The hexagonal tiling can be considered a truncated rhombille tiling.

Polyhedra Euclidean tiling Hyperbolic tiling
[3,3] [4,3] [5,3] [6,3] [7,3] [8,3]
Hexahedron.svg
Cube
Rhombicdodecahedron.jpg
Rhombic dodecahedron
Rhombictriacontahedron.jpg
Rhombic triacontahedron
Rhombic star tiling.png
Rhombille
Order73 qreg rhombic til.png Uniform dual tiling 433-t01-yellow.png
Alternate truncated cube.png
Chamfered tetrahedron
Truncated rhombic dodecahedron2.png
Chamfered cube
Truncated rhombic triacontahedron.png
Chamfered dodecahedron
Truncated rhombille tiling.png
Hexagonal tiling

Rhombic dodecahedron[edit]

The name truncated rhombic dodecahedron is ambiguous since only 6 (order-4) vertices were truncated. A truncation on just the 3-vertices of the rhombic dodecahedron would cause an icosahedron with 12 equilateral hexagons and 20 triangles, forming 30 vertices in total; this figure's dual can be called the triakis cuboctahedron. Another alternate truncated rhombic dodecahedron can appear by truncating all 14 vertices, yielding 12 irregular octagonal faces. The dual of the full truncation is a triangular tetracontaoctahedron labeled the tritetrakis cuboctahedron, which is a complete Kleetope of the cuboctahedron. The tetrahedron is to the truncated cube as the cube is to the full truncation or a bitruncated cuboctahedron.

See also[edit]

References[edit]

  • Antoine Deza, Michel Deza, Viatcheslav Grishukhin, Fullerenes and coordination polyhedra versus half-cube embeddings, 1998 PDF [1]

External links[edit]