Truncated cube

Truncated cube

(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, BC3, [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

Colored faces
3.8.8
(Vertex figure)

Triakis octahedron
(dual polyhedron)
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 \scriptstyle {2+\sqrt{2}}.

Area and volume

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

A = 2\left(6+6\sqrt{2}+\sqrt{3}\right)a^2 \approx 32.4346644a^2
V = \frac{1}{3}\left(21+14\sqrt{2}\right)a^3 \approx 13.5996633a^3.

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.

Orthogonal projections
Centered by Vertex Edge
3-8		 Edge
8-8		 Face
Octagon		 Face
Triangle
Truncated
cube
Triakis
octahedron
Projective
symmetry		 [2] [2] [2] [4] [6]

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.


octagon-centered
triangle-centered
Orthographic projection Stereographic projections

Cartesian coordinates

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 = \scriptstyle {\sqrt2 - 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.

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

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]

Vertex arrangement

It shares the vertex arrangement with three nonconvex uniform polyhedra:


Truncated cube
Nonconvex great rhombicuboctahedron
Great cubicuboctahedron
Great rhombihexahedron

Related polyhedra

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

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

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.

Related polytopes

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

Truncated hypercubes
...
Octagon Truncated cube Truncated tesseract Truncated 5-cube Truncated 6-cube Truncated 7-cube Truncated 8-cube

Truncated cubical graph

Truncated cubical graph

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]


Orthographic

See also

References

  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

External links

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