Truncated cube

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search
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
Polyhedron truncated 6 max.png
Colored faces
Polyhedron truncated 6 vertfig.svg
3.8.8
(Vertex figure)
Polyhedron truncated 6 dual.png
Triakis octahedron
(dual polyhedron)
Polyhedron truncated 6 net.svg
Net
3D model of a truncated cube

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
Solid
Polyhedron truncated 6 from blue max.png
Polyhedron truncated 6 from red max.png Polyhedron truncated 6 from yellow max.png
Wireframe Cube t01 v.png Cube t01 e38.png Cube t01 e88.png 3-cube t01 B2.svg 3-cube t01.svg
Dual 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]

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 2ξ are all the permutations of

ξ, ±1, ±1),

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 octahedra 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 tilings in symmetry.

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

Uniform octahedral polyhedra
Symmetry: [4,3], (*432) [4,3]+
(432)
[1+,4,3] = [3,3]
(*332)
[3+,4]
(3*2)
{4,3} t{4,3} r{4,3}
r{31,1}
t{3,4}
t{31,1}
{3,4}
{31,1}
rr{4,3}
s2{3,4}
tr{4,3} sr{4,3} h{4,3}
{3,3}
h2{4,3}
t{3,3}
s{3,4}
s{31,1}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png
CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
= CDel nodes 11.pngCDel split2.pngCDel node.png
CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
= CDel nodes 11.pngCDel split2.pngCDel node 1.png
CDel node h0.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
= CDel nodes.pngCDel split2.pngCDel node 1.png
CDel node 1.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png =
CDel nodes 10ru.pngCDel split2.pngCDel node.png or CDel nodes 01rd.pngCDel split2.pngCDel node.png
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png =
CDel nodes 10ru.pngCDel split2.pngCDel node 1.png or CDel nodes 01rd.pngCDel split2.pngCDel node 1.png
CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node h0.png =
CDel node h.pngCDel split1.pngCDel nodes hh.png
Uniform polyhedron-43-t0.svg Uniform polyhedron-43-t01.svg Uniform polyhedron-43-t1.svg
Uniform polyhedron-33-t02.png
Uniform polyhedron-43-t12.svg
Uniform polyhedron-33-t012.png
Uniform polyhedron-43-t2.svg
Uniform polyhedron-33-t1.png
Uniform polyhedron-43-t02.png
Rhombicuboctahedron uniform edge coloring.png
Uniform polyhedron-43-t012.png Uniform polyhedron-43-s012.png Uniform polyhedron-33-t0.pngUniform polyhedron-33-t2.png Uniform polyhedron-33-t01.pngUniform polyhedron-33-t12.png Uniform polyhedron-43-h01.svg
Uniform polyhedron-33-s012.svg
Duals to uniform polyhedra
V43 V3.82 V(3.4)2 V4.62 V34 V3.43 V4.6.8 V34.4 V33 V3.62 V35
CDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 4.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node fh.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 3.pngCDel node fh.pngCDel 4.pngCDel node.png
CDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 4.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 3.pngCDel node fh.pngCDel 3.pngCDel node fh.png
Octahedron.svg Triakisoctahedron.jpg Rhombicdodecahedron.jpg Tetrakishexahedron.jpg Hexahedron.svg Deltoidalicositetrahedron.jpg Disdyakisdodecahedron.jpg Pentagonalicositetrahedronccw.jpg Tetrahedron.svg Triakistetrahedron.jpg Dodecahedron.svg

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.

*n32 symmetry mutation of truncated spherical tilings: t{n,3}
Symmetry
*n32
[n,3]
Spherical Euclid. Compact hyperb. Paraco.
*232
[2,3]
*332
[3,3]
*432
[4,3]
*532
[5,3]
*632
[6,3]
*732
[7,3]
*832
[8,3]...
*∞32
[∞,3]
Truncated
figures
Spherical triangular prism.png Uniform tiling 332-t01-1-.png Uniform tiling 432-t01.png Uniform tiling 532-t01.png Uniform tiling 63-t01.svg Truncated heptagonal tiling.svg H2-8-3-trunc-dual.svg H2 tiling 23i-3.png
Symbol t{2,3} t{3,3} t{4,3} t{5,3} t{6,3} t{7,3} t{8,3} t{∞,3}
Triakis
figures
Spherical trigonal bipyramid.png Spherical triakis tetrahedron.png Spherical triakis octahedron.png Spherical triakis icosahedron.png Tiling Dual Semiregular V3-12-12 Triakis Triangular.svg Order-7 triakis triangular tiling.svg H2-8-3-kis-primal.svg Ord-infin triakis triang til.png
Config. V3.4.4 V3.6.6 V3.8.8 V3.10.10 V3.12.12 V3.14.14 V3.16.16 V3.∞.∞
*n42 symmetry mutation of truncated tilings: n.8.8
Symmetry
*n42
[n,4]
Spherical Euclidean Compact hyperbolic Paracompact
*242
[2,4]
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]
Truncated
figures
Octagonal dihedron.svg Uniform tiling 432-t01.png Uniform tiling 44-t12.svg H2-5-4-trunc-primal.svg H2 tiling 246-6.png H2 tiling 247-6.png H2 tiling 248-6.png H2 tiling 24i-6.png
Config. 2.8.8 3.8.8 4.8.8 5.8.8 6.8.8 7.8.8 8.8.8 ∞.8.8
n-kis
figures
Spherical octagonal hosohedron.png Spherical triakis octahedron.png 1-uniform 2 dual.svg H2-5-4-kis-dual.svg Order4 hexakis hexagonal til.png Order4 heptakis heptagonal til.png H2-8-3-primal.svg Ord4 apeirokis apeirogonal til.png
Config. V2.8.8 V3.8.8 V4.8.8 V5.8.8 V6.8.8 V7.8.8 V8.8.8 V∞.8.8

Alternated truncation[edit]

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.

Related polytopes[edit]

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

Truncated hypercubes
Image 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 ...
Name Octagon Truncated cube Truncated tesseract Truncated 5-cube Truncated 6-cube Truncated 7-cube Truncated 8-cube
Coxeter diagram 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
Vertex figure ( )v( ) Truncated cube vertfig.png
( )v{ }
Truncated 8-cell verf.png
( )v{3}
Truncated 5-cube verf.png
( )v{3,3}
( )v{3,3,3} ( )v{3,3,3,3} ( )v{3,3,3,3,3}

Truncated cubical graph[edit]

Truncated cubical graph
Truncated cubic graph.png
4-fold symmetry Schlegel diagram
Vertices24
Edges36
Automorphisms48
Chromatic number3
PropertiesCubic, Hamiltonian, regular, zero-symmetric
Table of graphs and parameters

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]