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Truncated cube

Archimedean solid with 14 regular faces From Wikipedia, the free encyclopedia

Truncated cube
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In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces (6 octagonal and 8 triangular), 36 edges, and 24 vertices.

Truncated cube
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(Click here for rotating model)
TypeArchimedean solid
Uniform polyhedron
ElementsF = 14, E = 36, V = 24 (χ = 2)
Faces by sides8{3}+6{8}
Conway notationtC
Schläfli symbolst{4,3}
t0,1{4,3}
Wythoff symbol2 3 | 4
Coxeter diagram
Symmetry groupOh, B3, [4,3], (*432), order 48
Rotation groupO, [4,3]+, (432), order 24
Dihedral angle3-8: 125°15′51″
8-8: 90°
ReferencesU09, C21, W8
PropertiesSemiregular convex
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Colored faces
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3.8.8
(Vertex figure)
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Triakis octahedron
(dual polyhedron)
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Net
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3D model of a truncated cube

If the truncated cube has unit edge length, its dual triakis octahedron has edges of lengths 2 and δS +1, where δS is the silver ratio, 2 +1.

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Area and volume

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Perspective

The area A and the volume V of a truncated cube of edge length a are:

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Orthogonal projections

The truncated cube has five special orthogonal projections, centered, on a vertex, on two types of edges, and two types of faces: triangles, and octagons. The last two correspond to the B2 and A2 Coxeter planes.

More information Centered by, Vertex ...
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Spherical tiling

The truncated cube can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.

More information Orthographic projection, Stereographic projections ...

Cartesian coordinates

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A truncated cube with its octagonal faces pyritohedrally dissected with a central vertex into triangles and pentagons, creating a topological icosidodecahedron

Cartesian coordinates for the vertices of a truncated hexahedron centered at the origin with edge length 21/δS are all the permutations of

1/δS, ±1, ±1),

where δS=2+1.

If we let a parameter ξ= 1/δS, in the case of a Regular Truncated Cube, then the parameter ξ can be varied between ±1. A value of 1 produces a cube, 0 produces a cuboctahedron, and negative values produces self-intersecting octagrammic faces.

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If the self-intersected portions of the octagrams are removed, leaving squares, and truncating the triangles into hexagons, truncated octahedra are produced, and the sequence ends with the central squares being reduced to a point, and creating an octahedron.

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Dissection

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Dissected truncated cube, with elements expanded apart

The truncated cube can be dissected into a central cube, with six square cupolae around each of the cube's faces, and 8 regular tetrahedra in the corners. This dissection can also be seen within the runcic cubic honeycomb, with cube, tetrahedron, and rhombicuboctahedron cells.

This dissection can be used to create a Stewart toroid with all regular faces by removing two square cupolae and the central cube. This excavated cube has 16 triangles, 12 squares, and 4 octagons.[1][2]

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Vertex arrangement

It shares the vertex arrangement with three nonconvex uniform polyhedra:

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Truncated cube
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Nonconvex great rhombicuboctahedron
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Great cubicuboctahedron
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Great rhombihexahedron
Summarize
Perspective

The truncated cube is related to other polyhedra and tilings in symmetry.

The truncated cube is one of a family of uniform polyhedra related to the cube and regular octahedron.

More information Uniform octahedral polyhedra, Symmetry: [4,3], (*432) ...

Symmetry mutations

This polyhedron is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (3.2n.2n), and [n,3] Coxeter group symmetry, and a series of polyhedra and tilings n.8.8.

More information *n32 symmetry mutation of truncated spherical tilings: t{n,3}, Symmetry*n32 [n,3] ...
More information Symmetry*n42 [n,4], Spherical ...

Alternated truncation

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Tetrahedron, its edge truncation, and the truncated cube

Truncating alternating vertices of the cube gives the chamfered tetrahedron, i.e. the edge truncation of the tetrahedron.

The truncated triangular trapezohedron is another polyhedron which can be formed from cube edge truncation.

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The truncated cube, is second in a sequence of truncated hypercubes:

Truncated cubical graph

Quick Facts Vertices, Edges ...

In the mathematical field of graph theory, a truncated cubical graph is the graph of vertices and edges of the truncated cube, one of the Archimedean solids. It has 24 vertices and 36 edges, and is a cubic Archimedean graph.[3]

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Orthographic

See also

References

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