Truncated cube

File:Truncated cube.stl

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 Template:Math and Template:Math, where δ_S is the silver ratio, Template:Sqrt +1.

Area and volume

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

<math>\begin{align}

A &= 2\left(6+6\sqrt{2}+\sqrt{3}\right)a^2 &&\approx 32.434\,6644a^2 \\ V &= \frac{21+14\sqrt{2}}{3}a^3 &&\approx 13.599\,6633a^3. \end{align}</math>

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
Solid			File:Polyhedron truncated 6 from blue max.png	File:Polyhedron truncated 6 from red max.png	File:Polyhedron truncated 6 from yellow max.png
Wireframe	File:Cube t01 v.png	File:Cube t01 e38.png	File:Cube t01 e88.png	File:3-cube t01 B2.svg	File:3-cube t01.svg
Dual	File:Dual truncated cube t01 v.png	File:Dual truncated cube t01 e8.png	File:Dual truncated cube t01 e88.png	File:Dual truncated cube t01 B2.png	File:Dual truncated cube t01.png
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.

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

Cartesian coordinates

File:Icosidecahedron in truncated cube.png

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 2Template:Sfrac are all the permutations of

(±Template:Sfrac, ±1, ±1),

where δ_S=Template:Sqrt+1.

If we let a parameter ξ= Template:Sfrac, 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.

File: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

File:Dissected truncated cube.png

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.<ref>B. M. Stewart, Adventures Among the Toroids (1970) Template:Isbn</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

File:Excavated truncated cube.png

Vertex arrangement

It shares the vertex arrangement with three nonconvex uniform polyhedra:

File:Truncated hexahedron.png Truncated cube

File:Uniform great rhombicuboctahedron.png Nonconvex great rhombicuboctahedron

File:Great cubicuboctahedron.png Great cubicuboctahedron

File:Great rhombihexahedron.png Great rhombihexahedron

Related polyhedra

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. Template:Octahedral truncations

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. Template:Truncated figure1 small table Template:Truncated figure4 table

Alternated truncation

Template:Multiple image 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

The truncated cube, is second in a sequence of truncated hypercubes: Template:Truncated hypercube polytopes

Truncated cubical graph

Template:Infobox graph 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.<ref>Template:Citation</ref>

File:3-cube t01.svg Orthographic

See also

• Spinning truncated cube
• Cube-connected cycles, a family of graphs that includes the skeleton of the truncated cube
• Chamfered cube, obtained by replacing the edges of a cube with non-uniform hexagons

References

• Template:The Geometrical Foundation of Natural Structure (book) (Section 3-9)
• Cromwell, P. Polyhedra, CUP hbk (1997), pbk. (1999). Ch.2 p. 79-86 Archimedean solids

External links

• Template:Mathworld2
  • Template:Mathworld
• Template:KlitzingPolytopes
• Editable printable net of a truncated cube with interactive 3D view
• The Uniform Polyhedra
• Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra
  • VRML model
  • Conway Notation for Polyhedra Try: "tC"