Truncated cube

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

:$\backslash begin\{align\}$

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}

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.

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:Icosidecahedron_in_truncated_cube.png 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 2{{sfrac|1|δ_S}} are all the permutations of

:(±{{sfrac|1|δ_S}}, ±1, ±1),

where δ_S={{sqrt|2}}+1.

If we let a parameter ξ= {{sfrac|1|δ_S}}, 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.

File:Dissected_truncated_cube.png

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.B. M. Stewart, Adventures Among the Toroids (1970) {{isbn|978-0-686-11936-4}}{{Cite web|url=http://www.doskey.com/polyhedra/Stewart05.html|title = Adventures Among the Toroids - Chapter 5 - Simplest (R)(A)(Q)(T) Toroids of genus p=1}}

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It shares the vertex arrangement with three nonconvex uniform 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.

{{Octahedral truncations}}

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.

{{Truncated figure1 small table}}

{{Truncated figure4 table}}

{{multiple image

| align = right | total_width = 400

| image1 = Polyhedron 4a.png

| image2 = Polyhedron chamfered 4a.png

| image3 = Polyhedron truncated 6.png

| footer = 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.

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

{{Truncated hypercube polytopes}}

{{Infobox graph

| name = Truncated cubical graph

| image = 240px

| image_caption = 4-fold symmetry Schlegel diagram

| namesake =

| vertices = 24

| edges = 36

| automorphisms = 48

| radius =

| diameter =

| girth =

| chromatic_number = 3

| chromatic_index =

| fractional_chromatic_index =

| 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.{{citation|last1=Read|first1=R. C.|last2=Wilson|first2=R. J.|title=An Atlas of Graphs|publisher=Oxford University Press|year= 1998|page=269}}

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• 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

{{reflist}}

• {{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

• {{mathworld2 |urlname=TruncatedCube |title=Truncated cube |urlname2=ArchimedeanSolid |title2=Archimedean solid}}
• {{mathworld |urlname=TruncatedCubicalGraph |title=Truncated cubical graph}}
• {{KlitzingPolytopes|polyhedra.htm|3D convex uniform polyhedra|o3x4x - tic}}
• [http://www.dr-mikes-math-games-for-kids.com/polyhedral-nets.html?net=621wh65c7Ey8v4cRpEVhGs0pPxZ5raM9uNf8HcBUgOyrp6acSwZGvkvEcL6m06RDKxmSAduYsvTvoCvEDokvHrjyVEqlGVdIH8WamnxFO1qnGpUtgt7K0ZD57RlX&name=Truncated+Cube#applet Editable printable net of a truncated cube with interactive 3D view]
• [http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
• [http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] www.georgehart.com: The Encyclopedia of Polyhedra
• VRML [http://www.georgehart.com/virtual-polyhedra/vrml/truncated_cube.wrl model]
• [http://www.georgehart.com/virtual-polyhedra/conway_notation.html Conway Notation for Polyhedra] Try: "tC"

{{Archimedean solids}}

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Category:Uniform polyhedra

Category:Archimedean solids

Category:Truncated tilings