Truncated cube

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Truncated cube
Truncatedhexahedron.jpg
(Click here for rotating model)
Type Archimedean solid
Uniform polyhedron
Elements F = 14, E = 36, V = 24 (χ = 2)
Faces by sides 8{3}+6{8}
Conway notation tC
Schläfli symbols t{4,3}
t0,1{4,3}
Wythoff symbol 2 3 | 4
Coxeter diagram CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
Symmetry group Oh, B3, [4,3], (*432), order 48
Rotation group O, [4,3]+, (432), order 24
Dihedral Angle 3-8: 125°15′51″
8-8: 90°
References U09, C21, W8
Properties Semiregular convex
Truncated hexahedron.png
Colored faces
Truncated cube vertfig.png
3.8.8
(Vertex figure)
Triakisoctahedron.jpg
Triakis octahedron
(dual polyhedron)
Truncated hexahedron flat.svg
Net

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.

If the truncated cube has unit edge length, its dual triakis octahedron has edges of lengths 2 and 2 + 2.

Area and volume[edit]

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

Orthogonal projections[edit]

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.

Orthogonal projections
Centered by Vertex Edge
3-8
Edge
8-8
Face
Octagon
Face
Triangle
Truncated
cube
Cube t01 v.png Cube t01 e38.png Cube t01 e88.png 3-cube t01 B2.svg 3-cube t01.svg
Triakis
octahedron
Dual truncated cube t01 v.png Dual truncated cube t01 e8.png Dual truncated cube t01 e88.png Dual truncated cube t01 B2.png Dual truncated cube t01.png
Projective
symmetry
[2] [2] [2] [4] [6]

Spherical tiling[edit]

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.

Uniform tiling 432-t01.png Truncated cube stereographic projection octagon.png
octagon-centered
Truncated cube stereographic projection triangle.png
triangle-centered
Orthographic projection Stereographic projections

Cartesian coordinates[edit]

The following Cartesian coordinates define the vertices of a truncated hexahedron centered at the origin with edge length 2ξ:

ξ, ±1, ±1),
(±1, ±ξ, ±1),
(±1, ±1, ±ξ)

where ξ = 2 − 1.

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.

Truncated cube sequence.png

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

Dissection[edit]

Dissected truncated cube, with elements expanded apart

The truncated cube can be dissected into a central cube, with six square cupola around each of the cube's faces, and 8 regular tetrahedral 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 cupola and the central cube. This excavated cube has 16 triangles, 12 squares, and 4 octagons.[1][2]

Excavated truncated cube.png

Vertex arrangement[edit]

It shares the vertex arrangement with three nonconvex uniform polyhedra:

Truncated hexahedron.png
Truncated cube
Uniform great rhombicuboctahedron.png
Nonconvex great rhombicuboctahedron
Great cubicuboctahedron.png
Great cubicuboctahedron
Great rhombihexahedron.png
Great rhombihexahedron

Related polyhedra[edit]

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

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

Symmetry mutations[edit]

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.

Alternated truncation[edit]

A cube can be alternately truncated producing tetrahedral symmetry, with six hexagonal faces, and four triangles at the truncated vertices. It is one of a sequence of alternate truncations of polyhedra and tiling.

Alternate truncated cube.png

Related polytopes[edit]

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

Truncated hypercubes
Regular polygon 8 annotated.svg 3-cube t01.svgTruncated hexahedron.png 4-cube t01.svgSchlegel half-solid truncated tesseract.png 5-cube t01.svg5-cube t01 A3.svg 6-cube t01.svg6-cube t01 A5.svg 7-cube t01.svg7-cube t01 A5.svg 8-cube t01.svg8-cube t01 A7.svg ...
Octagon Truncated cube Truncated tesseract Truncated 5-cube Truncated 6-cube Truncated 7-cube Truncated 8-cube
CDel node 1.pngCDel 4.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

Truncated cubical graph[edit]

Truncated cubical graph
Truncated cubic graph.png
4-fold symmetry schlegel diagram
Vertices 24
Edges 36
Automorphisms 48
Chromatic number 3
Properties Cubic, Hamiltonian, regular, zero-symmetric

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]

3-cube t01.svg
Orthographic

See also[edit]

References[edit]

  1. ^ B. M. Stewart, Adventures Among the Toroids (1970) ISBN 978-0-686-11936-4
  2. ^ http://www.doskey.com/polyhedra/Stewart05.html
  3. ^ Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269 
  • Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X.  (Section 3-9)
  • Cromwell, P. Polyhedra, CUP hbk (1997), pbk. (1999). Ch.2 p.79-86 Archimedean solids

External links[edit]