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Truncated cube
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<h2 id="area-and-volume">Area and volume</h2> <p>The area <i>A</i> and the <a href="/facts/Volume/aeAXCBUd">volume</a> <i>V</i> of a truncated cube of edge length <i>a</i> are: </p> A = 2 ( 6 + 6 2 + 3 ) a 2 ≈ 32.434 6644 a 2 V = 21 + 14 2 3 a 3 ≈ 13.599 6633 a 3 . {\displaystyle {\begin{aligned}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{aligned}}} <h2 id="orthogonal-projections">Orthogonal projections</h2> <p>The <i>truncated cube</i> has five special <a href="/facts/Orthogonal_projection/sElXGkxD">orthogonal projections</a>, 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 <a href="/facts/Coxeter_plane/4B80wH8V">Coxeter planes</a>. </p> Orthogonal projections<table><tbody><tr><th>Centered by</th><th>Vertex</th><th>Edge3-8</th><th>Edge8-8</th><th>FaceOctagon</th><th>FaceTriangle</th></tr><tr><th>Solid</th><td></td><td></td><td></td><td></td><td></td></tr><tr><th>Wireframe</th><td></td><td></td><td></td><td></td><td></td></tr><tr><th>Dual</th><td></td><td></td><td></td><td></td><td></td></tr><tr><th>Projectivesymmetry</th><td>[2]</td><td>[2]</td><td>[2]</td><td>[4]</td><td>[6]</td></tr></tbody></table> <h2 id="spherical-tiling">Spherical tiling</h2> <p>The truncated cube can also be represented as a <a href="/facts/Spherical_tiling/jaglysFR">spherical tiling</a>, and projected onto the plane via a <a href="/facts/Stereographic_projection/oVyMrWqR">stereographic projection</a>. This projection is <a href="/facts/Conformal_map/cEpQnMad">conformal</a>, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane. </p> <table><tbody><tr><td></td><td><a href="/facts/Octagon/H5PuP3wP">octagon</a>-centered</td><td><a href="/facts/Triangle/cRXUwsMn">triangle</a>-centered</td></tr><tr><th><a href="/facts/Orthographic_projection/L3Cv9kYD">Orthographic projection</a></th><th colspan="2"><a href="/facts/Stereographic_projection/oVyMrWqR">Stereographic projections</a></th></tr></tbody></table> <h2 id="cartesian-coordinates">Cartesian coordinates</h2> <p><a href="/facts/Cartesian_coordinates/lkTqvMmB">Cartesian coordinates</a> for the vertices of a <a href="/facts/Truncation_(geometry)/DYb589Je">truncated</a> <a href="/facts/Hexahedron/UzruYxzd">hexahedron</a> centered at the origin with edge length 21/<i>δS</i> are all the permutations of </p> (±1/<i>δS</i>, ±1, ±1), <p>where δS=√2+1. </p><p>If we let a parameter <i>ξ</i>= 1/<i>δS</i>, in the case of a Regular Truncated Cube, then the parameter <i>ξ</i> can be varied between ±1. A value of 1 produces a <a href="/facts/Cube/65lUaAqN">cube</a>, 0 produces a <a href="/facts/Cuboctahedron/hD7p0Mem">cuboctahedron</a>, and negative values produces self-intersecting <a href="/facts/Octagram/4mfC9WqR">octagrammic</a> faces. </p> <p>If the self-intersected portions of the octagrams are removed, leaving squares, and truncating the triangles into hexagons, <a href="/facts/Truncated_octahedron/GA98b4TH">truncated octahedra</a> are produced, and the sequence ends with the central squares being reduced to a point, and creating an <a href="/facts/Octahedron/G38q92xW">octahedron</a>. </p> <h2 id="dissection">Dissection</h2> <p>The truncated cube can be dissected into a central <a href="/facts/Cube/65lUaAqN">cube</a>, with six <a href="/facts/Square_cupola/KY7dxf3w">square cupolae</a> around each of the cube's faces, and 8 regular tetrahedra in the corners. This dissection can also be seen within the <a href="/facts/Runcic_cubic_honeycomb/mX4CxSvp">runcic cubic honeycomb</a>, with <a href="/facts/Cube/65lUaAqN">cube</a>, <a href="/facts/Tetrahedron/7mkSrl6j">tetrahedron</a>, and <a href="/facts/Rhombicuboctahedron/TmxUmEnl">rhombicuboctahedron</a> cells. </p><p>This dissection can be used to create a <a href="/facts/Stewart_toroid/g6SS84k2">Stewart toroid</a> with all regular faces by removing two square cupolae and the central cube. This excavated cube has 16 <a href="/facts/Triangle/cRXUwsMn">triangles</a>, 12 <a href="/facts/Square/JHJeMvmL">squares</a>, and 4 <a href="/facts/Octagon/H5PuP3wP">octagons</a>.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a><a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p> <h2 id="vertex-arrangement">Vertex arrangement</h2> <p>It shares the <a href="/facts/Vertex_arrangement/DFgCsP0n">vertex arrangement</a> with three <a href="/facts/Nonconvex_uniform_polyhedra/rSv9XpHQ">nonconvex uniform polyhedra</a>: </p> <table><tbody><tr><td>Truncated cube</td><td><a href="/facts/Nonconvex_great_rhombicuboctahedron/XSwFaVVL">Nonconvex great rhombicuboctahedron</a></td><td><a href="/facts/Great_cubicuboctahedron/xRh9NtND">Great cubicuboctahedron</a></td><td><a href="/facts/Great_rhombihexahedron/PDA51lpE">Great rhombihexahedron</a></td></tr></tbody></table> <h2 id="related-polyhedra">Related polyhedra</h2> <p>The truncated cube is related to other polyhedra and tilings in symmetry. </p><p>The truncated cube is one of a family of uniform polyhedra related to the cube and regular octahedron. </p> <table><tbody><tr><th colspan="11"><a href="/facts/Uniform_polyhedron/PHnprqRC">Uniform octahedral polyhedra</a></th></tr><tr><th colspan="7"><a href="/facts/List_of_spherical_symmetry_groups/3X3YcHoQ">Symmetry</a>: [4,3], <a href="/facts/Octahedral_symmetry/qLqhYxgm">(*432)</a></th><th>[4,3]+(432)</th><th colspan="2">[1+,4,3] = [3,3]<a href="/facts/Tetrahedral_symmetry/hIVlf9pu">(*332)</a></th><th>[3+,4]<a href="/facts/Pyritohedral_symmetry/hIVlf9pu">(3*2)</a></th></tr><tr><td><a href="/facts/Cube/65lUaAqN">{4,3}</a></td><td>t{4,3}</td><td><a href="/facts/Cuboctahedron/hD7p0Mem">r{4,3}</a>r{31,1}</td><td><a href="/facts/Truncated_octahedron/GA98b4TH">t{3,4}</a>t{31,1}</td><td><a href="/facts/Octahedron/G38q92xW">{3,4}</a>{31,1}</td><td><a href="/facts/Rhombicuboctahedron/TmxUmEnl">rr{4,3}</a>s2{3,4}</td><td><a href="/facts/Truncated_cuboctahedron/nd2ltxcm">tr{4,3}</a></td><td><a href="/facts/Snub_cube/pbnr40gk">sr{4,3}</a></td><td><a href="/facts/Tetrahedron/7mkSrl6j">h{4,3}</a>{3,3}</td><td><a href="/facts/Truncated_tetrahedron/K9ItvmUK">h2{4,3}</a>t{3,3}</td><td><a href="/facts/Pseudoicosahedron/wWbpnDYp">s{3,4}</a>s{31,1}</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td>= </td><td>= </td><td>= </td><td></td><td></td><td></td><td> = or </td><td> = or </td><td> =</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th colspan="11">Duals to uniform polyhedra</th></tr><tr><td><a href="/facts/Octahedron/G38q92xW">V43</a></td><td><a href="/facts/Triakis_octahedron/mJ3s18DL">V3.82</a></td><td><a href="/facts/Rhombic_dodecahedron/X3npAlmr">V(3.4)2</a></td><td><a href="/facts/Tetrakis_hexahedron/RjfM6K49">V4.62</a></td><td><a href="/facts/Cube/65lUaAqN">V34</a></td><td><a href="/facts/Deltoidal_icositetrahedron/bu6NBY9X">V3.43</a></td><td><a href="/facts/Disdyakis_dodecahedron/wwAbxHWL">V4.6.8</a></td><td><a href="/facts/Pentagonal_icositetrahedron/2v6EVUPX">V34.4</a></td><td><a href="/facts/Tetrahedron/7mkSrl6j">V33</a></td><td><a href="/facts/Triakis_tetrahedron/qz6NNDMU">V3.62</a></td><td><a href="/facts/Pyritohedron/7r519eCl">V35</a></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table> <h3>Symmetry mutations</h3> <p>This polyhedron is topologically related as a part of sequence of uniform <a href="/facts/Truncation_(geometry)/DYb589Je">truncated</a> polyhedra with <a href="/facts/Vertex_configuration/wWaBS6t0">vertex configurations</a> (3.2<i>n</i>.2<i>n</i>), and [<i>n</i>,3] <a href="/facts/Coxeter_group/bgzHFm1Q">Coxeter group</a> symmetry, and a series of polyhedra and tilings <i>n</i>.8.8. </p> <table><tbody><tr><th colspan="12">*<i>n</i>32 symmetry mutation of truncated spherical tilings: t{<i>n</i>,3}</th></tr><tr><th rowspan="2">Symmetry*<i>n</i>32[n,3]</th><th colspan="4"><a href="/facts/List_of_spherical_symmetry_groups/3X3YcHoQ">Spherical</a></th><th><a href="/facts/List_of_planar_symmetry_groups/3buKYIft">Euclid.</a></th><th colspan="2">Compact hyperb.</th><th>Paraco.</th></tr><tr><th>*232[2,3]</th><th>*332[3,3]</th><th>*432[4,3]</th><th>*532[5,3]</th><th>*632[6,3]</th><th>*732[7,3]</th><th>*832[8,3]...</th><th>*∞32[∞,3]</th></tr><tr><th>Truncatedfigures</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th><a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Symbol</a></th><th><a href="/facts/Triangular_prism/Yjp4Kw9M">t{2,3}</a></th><th><a href="/facts/Truncated_tetrahedron/K9ItvmUK">t{3,3}</a></th><th>t{4,3}</th><th><a href="/facts/Truncated_dodecahedron/PVTAsUWQ">t{5,3}</a></th><th><a href="/facts/Truncated_hexagonal_tiling/c9dRoGFS">t{6,3}</a></th><th><a href="/facts/Truncated_heptagonal_tiling/TnhsfrGr">t{7,3}</a></th><th><a href="/facts/Truncated_octagonal_tiling/wxTpyggX">t{8,3}</a></th><th><a href="/facts/Truncated_order-3_apeirogonal_tiling/x98K6sIB">t{∞,3}</a></th></tr><tr><th>Triakisfigures</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th><a href="/facts/Face_configuration/wWaBS6t0">Config.</a></th><td><a href="/facts/Triangular_dipyramid/fU4kIBTV">V3.4.4</a></td><td><a href="/facts/Triakis_tetrahedron/qz6NNDMU">V3.6.6</a></td><td><a href="/facts/Triakis_octahedron/mJ3s18DL">V3.8.8</a></td><td><a href="/facts/Triakis_icosahedron/VoVKELcj">V3.10.10</a></td><td><a href="/facts/Triakis_triangular_tiling/c9dRoGFS">V3.12.12</a></td><td><a href="/facts/Truncated_heptagonal_tiling/TnhsfrGr">V3.14.14</a></td><td>V3.16.16</td><td>V3.∞.∞</td></tr></tbody></table> <table><tbody><tr><th colspan="12">*<i>n</i>42 symmetry mutation of truncated tilings: <i>n</i>.8.8 <ul><li>v</li><li>t</li><li>e</li></ul></th></tr><tr><th rowspan="2">Symmetry<a href="/facts/Orbifold_notation/HCbBoa2J">*<i>n</i>42</a>[n,4]</th><th colspan="2"><a href="/facts/List_of_spherical_symmetry_groups/3X3YcHoQ">Spherical</a></th><th><a href="/facts/List_of_planar_symmetry_groups/3buKYIft">Euclidean</a></th><th colspan="4">Compact hyperbolic</th><th>Paracompact</th></tr><tr><th>*242[2,4]</th><th>*342[3,4]</th><th>*442[4,4]</th><th>*542[5,4]</th><th>*642[6,4]</th><th>*742[7,4]</th><th>*842[8,4]...</th><th>*∞42[∞,4]</th></tr><tr><th>Truncatedfigures</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th><a href="/facts/Vertex_configuration/wWaBS6t0">Config.</a></th><th><a href="/facts/Dihedron/YkNgovlv">2.8.8</a></th><th>3.8.8</th><th><a href="/facts/Truncated_square_tiling/Mbj1fVzm">4.8.8</a></th><th><a href="/facts/Truncated_order-5_square_tiling/3NHcvnvY">5.8.8</a></th><th><a href="/facts/Truncated_order-6_square_tiling/qUNkn9yb">6.8.8</a></th><th><a href="/facts/Truncated_order-7_square_tiling/0283KFfz">7.8.8</a></th><th><a href="/facts/Octagonal_tiling/Fjj2z6ED">8.8.8</a></th><th><a href="/facts/Truncated_infinite-order_square_tiling/TvIdh5tN">∞.8.8</a></th></tr><tr><th>n-kisfigures</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th><a href="/facts/Face_configuration/wWaBS6t0">Config.</a></th><td><a href="/facts/Hosohedron/t2dyV0bW">V2.8.8</a></td><td><a href="/facts/Triakis_octahedron/mJ3s18DL">V3.8.8</a></td><td><a href="/facts/Tetrakis_square_tiling/oqpMEMBw">V4.8.8</a></td><td>V5.8.8</td><td>V6.8.8</td><td>V7.8.8</td><td>V8.8.8</td><td>V∞.8.8</td></tr></tbody></table> <h3>Alternated truncation</h3> <p>Truncating alternating vertices of the cube gives the <a href="/facts/Chamfered_tetrahedron/hQEGYQpl">chamfered tetrahedron</a>, i.e. the edge truncation of the tetrahedron. </p><p>The <a href="/facts/Truncated_triangular_trapezohedron/Sq8oAeIl">truncated triangular trapezohedron</a> is another polyhedron which can be formed from cube edge truncation. </p> <h2 id="related-polytopes">Related polytopes</h2> <p>The <i><a href="/facts/Truncation_(geometry)/DYb589Je">truncated</a> <a href="/facts/Cube/65lUaAqN">cube</a></i>, is second in a sequence of truncated <a href="/facts/Hypercube/q2mRPbIz">hypercubes</a>: </p> Truncated hypercubes<table><tbody><tr><th>Image</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td rowspan="3">...</td></tr><tr><th>Name</th><td><a href="/facts/Octagon/H5PuP3wP">Octagon</a></td><td>Truncated cube</td><td><a href="/facts/Truncated_tesseract/ShQzMKoa">Truncated tesseract</a></td><td><a href="/facts/Truncated_5-cube/kLk4Dwvl">Truncated 5-cube</a></td><td><a href="/facts/Truncated_6-cube/uDq8zaRk">Truncated 6-cube</a></td><td><a href="/facts/Truncated_7-cube/M0SeeGkR">Truncated 7-cube</a></td><td><a href="/facts/Truncated_8-cube/8Bsbj6Nw">Truncated 8-cube</a></td></tr><tr><th><a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagram</a></th><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th><a href="/facts/Vertex_figure/3u2ySK0C">Vertex figure</a></th><td>( )v( )</td><td><a href="/facts/Isosceles_triangle/rvLe3yqK">( )v{ }</a></td><td><a href="/facts/Triangular_pyramid/7mkSrl6j">( )v{3}</a></td><td><a href="/facts/5-cell/Rh6lwDcc">( )v{3,3}</a></td><td><a href="/facts/5-simplex/KIfBFfhl">( )v{3,3,3}</a></td><td><a href="/facts/6-simplex/mInfSIeH">( )v{3,3,3,3}</a></td><td><a href="/facts/7-simplex/6DwtPplH">( )v{3,3,3,3,3}</a></td></tr></tbody></table> <h2 id="truncated-cubical-graph">Truncated cubical graph</h2> <p>In the <a href="/facts/Mathematics/pxTouaz4">mathematical</a> field of <a href="/facts/Graph_theory/VVPCd4TX">graph theory</a>, a truncated cubical graph is the <a href="/facts/1-skeleton/EAt2F7tC">graph of vertices and edges</a> of the <i>truncated cube</i>, one of the <a href="/facts/Archimedean_solid/wYWeHDYT">Archimedean solids</a>. It has 24 <a href="/facts/Vertex_(graph_theory)/WhlWmL3o">vertices</a> and 36 edges, and is a <a href="/facts/Cubic_graph/6M32WUIJ">cubic</a> <a href="/facts/Archimedean_graph/V3eecXxk">Archimedean graph</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> <table><tbody><tr><td>Orthographic</td></tr></tbody></table> <h2 id="see-also">See also</h2> <ul><li>Spinning truncated cube</li> <li><a href="/facts/Cube-connected_cycles/KBfIJYuu">Cube-connected cycles</a>, a family of graphs that includes the <a href="/facts/Skeleton_(topology)/EAt2F7tC">skeleton</a> of the truncated cube</li> <li><a href="/facts/Chamfered_cube/hQEGYQpl">Chamfered cube</a>, obtained by replacing the edges of a cube with non-uniform hexagons</li></ul> <ul><li><a href="/facts/Robert_Williams_(geometer)/1FJDMSJ2">Williams, Robert</a> (1979). <i>The Geometrical Foundation of Natural Structure: A Source Book of Design</i>. Dover Publications, Inc. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-23729-X. (Section 3-9)</li> <li>Cromwell, P. <i>Polyhedra</i>, CUP hbk (1997), pbk. (1999). Ch.2 p. 79-86 <i>Archimedean solids</i></li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a>, "<a href="https://mathworld.wolfram.com/TruncatedCube.html">Truncated cube</a>" ("<a href="http://mathworld.wolfram.com/ArchimedeanSolid.html">Archimedean solid</a>") at <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>. <ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/TruncatedCubicalGraph.html">"Truncated cubical graph"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li></ul></li> <li>Klitzing, Richard. <a href="https://bendwavy.org/klitzing/dimensions/polyhedra.htm">"3D convex uniform polyhedra o3x4x - tic"</a>.</li> <li><a href="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</a></li> <li><a href="http://www.mathconsult.ch/showroom/unipoly/">The Uniform Polyhedra</a></li> <li><a href="http://www.georgehart.com/virtual-polyhedra/vp.html">Virtual Reality Polyhedra</a> www.georgehart.com: The Encyclopedia of Polyhedra <ul><li><a href="/facts/VRML/EkNXun7G">VRML</a> <a href="http://www.georgehart.com/virtual-polyhedra/vrml/truncated_cube.wrl">model</a></li> <li><a href="http://www.georgehart.com/virtual-polyhedra/conway_notation.html">Conway Notation for Polyhedra</a> Try: "tC"</li></ul></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>B. M. Stewart, Adventures Among the Toroids (1970) ISBN 978-0-686-11936-4 <a href="/wiki/Adventures_Among_the_Toroids" target="_blank">/wiki/Adventures_Among_the_Toroids</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>"Adventures Among the Toroids - Chapter 5 - Simplest (R)(A)(Q)(T) Toroids of genus p=1". <a href="http://www.doskey.com/polyhedra/Stewart05.html" target="_blank">http://www.doskey.com/polyhedra/Stewart05.html</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269 <a href="/wiki/Oxford_University_Press" target="_blank">/wiki/Oxford_University_Press</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> </ol>