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Architectonic and catoptric tessellation
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<h2 id="enumeration">Enumeration</h2> <p>The pairs of architectonic and catoptric tessellations are listed below with their <a href="/facts/Symmetry_group/CNqeseMp">symmetry group</a>. These tessellations only represent four symmetry <a href="/facts/Space_group/2XQx0J7M">space groups</a>, and also all within the <a href="/facts/Cubic_crystal_system/92Co7BKM">cubic crystal system</a>. Many of these tessellations can be defined in multiple symmetry groups, so in each case the highest symmetry is expressed. </p> <table><tbody><tr><th rowspan="2">Ref.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>indices</th><th rowspan="2">Symmetry</th><th colspan="3">Architectonic tessellation</th><th colspan="3">Catoptric tessellation</th></tr><tr><th>Name<a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagram</a>Image</th><th><a href="/facts/Vertex_figure/3u2ySK0C">Vertex figure</a>Image</th><th>Cells</th><th>Name</th><th>Cell</th><th><a href="/facts/Vertex_figure/3u2ySK0C">Vertex figures</a></th></tr><tr><th>J11,15A1W1G22δ4</th><th>nc[4,3,4]</th><td><a href="/facts/Cubic_honeycomb/eMxySFur">Cubille</a>(Cubic honeycomb)</td><td><a href="/facts/Octahedron/G38q92xW">Octahedron</a>, </td><td></td><td><a href="/facts/Cubic_honeycomb/eMxySFur">Cubille</a></td><td><a href="/facts/Cube/65lUaAqN">Cube</a>, </td><td></td></tr><tr><th>J12,32A15W14G7t1δ4</th><th>nc[4,3,4]</th><td><a href="/facts/Rectified_cubic_honeycomb/eMxySFur">Cuboctahedrille</a>(Rectified cubic honeycomb)</td><td><a href="/facts/Cuboid/eLLKX4rh">Cuboid</a>, </td><td></td><td><a href="/facts/Square_bipyramidal_honeycomb/Ae30mwnf">Oblate octahedrille</a></td><td>Isosceles <a href="/facts/Square_bipyramid/G38q92xW">square bipyramid</a></td><td> , </td></tr><tr><th>J13A14W15G8t0,1δ4</th><th>nc[4,3,4]</th><td><a href="/facts/Truncated_cubic_honeycomb/eMxySFur">Truncated cubille</a>(Truncated cubic honeycomb)</td><td>Isosceles <a href="/facts/Square_pyramid/wDetDohM">square pyramid</a></td><td></td><td><a href="/facts/Hexakis_cubic_honeycomb/Ae30mwnf">Pyramidille</a></td><td>Isosceles <a href="/facts/Square_pyramid/wDetDohM">square pyramid</a></td><td>, </td></tr><tr><th>J14A17W12G9t0,2δ4</th><th>nc[4,3,4]</th><td><a href="/facts/Cantellated_cubic_honeycomb/eMxySFur">2-RCO-trille</a>(Cantellated cubic honeycomb)</td><td>Wedge</td><td></td><td><a href="/facts/Quarter_oblate_octahedrille/eMxySFur">Quarter oblate octahedrille</a></td><td>irr. <a href="/facts/Triangular_bipyramid/fU4kIBTV">Triangular bipyramid</a></td><td>, , </td></tr><tr><th>J16A3W2G28t1,2δ4</th><th>bc[[4,3,4]]</th><td><a href="/facts/Bitruncated_cubic_honeycomb/j6Prl5h0">Truncated octahedrille</a>(Bitruncated cubic honeycomb)</td><td><a href="/facts/Tetragonal_disphenoid/7rU7jc7U">Tetragonal disphenoid</a></td><td></td><td><a href="/facts/Disphenoid_tetrahedral_honeycomb/Ae30mwnf">Oblate tetrahedrille</a></td><td><a href="/facts/Tetragonal_disphenoid/7rU7jc7U">Tetragonal disphenoid</a></td><td></td></tr><tr><th>J17A18W13G25t0,1,2δ4</th><th>nc[4,3,4]</th><td><a href="/facts/Cantitruncated_cubic_honeycomb/eMxySFur">n-tCO-trille</a>(Cantitruncated cubic honeycomb)</td><td>Mirrored <a href="/facts/Sphenoid_(geometry)/7mkSrl6j">sphenoid</a></td><td></td><td><a href="/facts/Triangular_pyramidille/eMxySFur">Triangular pyramidille</a></td><td>Mirrored <a href="/facts/Sphenoid_(geometry)/7mkSrl6j">sphenoid</a></td><td>, , </td></tr><tr><th>J18A19W19G20t0,1,3δ4</th><th>nc[4,3,4]</th><td><a href="/facts/Runcitruncated_cubic_honeycomb/eMxySFur">1-RCO-trille</a>(Runcitruncated cubic honeycomb)</td><td>Trapezoidal <a href="/facts/Pyramid/76Qx8TLD">pyramid</a></td><td></td><td><a href="/facts/Square_quarter_pyramidille/eMxySFur">Square quarter pyramidille</a></td><td>Irr. <a href="/facts/Pyramid/76Qx8TLD">pyramid</a></td><td>, , , </td></tr><tr><th>J19A22W18G27t0,1,2,3δ4</th><th>bc[[4,3,4]]</th><td><a href="/facts/Omnitruncated_cubic_honeycomb/eMxySFur">b-tCO-trille</a>(Omnitruncated cubic honeycomb)</td><td><a href="/facts/Phyllic_disphenoid/7mkSrl6j">Phyllic disphenoid</a></td><td></td><td><a href="/facts/Phyllic_disphenoidal_honeycomb/Ae30mwnf">Eighth pyramidille</a></td><td><a href="/facts/Phyllic_disphenoid/7mkSrl6j">Phyllic disphenoid</a></td><td>, </td></tr><tr><th>J21,31,51A2W9G1hδ4</th><th>fc[4,31,1]</th><td><a href="/facts/Tetrahedral-octahedral_honeycomb/mX4CxSvp">Tetroctahedrille</a>(Tetrahedral-octahedral honeycomb) or </td><td><a href="/facts/Cuboctahedron/hD7p0Mem">Cuboctahedron</a>, </td><td></td><td><a href="/facts/Rhombic_dodecahedral_honeycomb/1wjLc5oW">Dodecahedrille</a> or </td><td><a href="/facts/Rhombic_dodecahedron/X3npAlmr">Rhombic dodecahedron</a>, </td><td>, </td></tr><tr><th>J22,34A21W17G10h2δ4</th><th>fc[4,31,1]</th><td><a href="/facts/Cantic_cubic_honeycomb/mX4CxSvp">truncated tetraoctahedrille</a>(Truncated tetrahedral-octahedral honeycomb) or </td><td>Rectangular pyramid</td><td></td><td><a href="/facts/Rhombic_pyramidal_honeycomb/1wjLc5oW">Half oblate octahedrille</a> or </td><td>rhombic <a href="/facts/Pyramid/76Qx8TLD">pyramid</a></td><td>, , </td></tr><tr><th>J23A16W11G5h3δ4</th><th>fc[4,31,1]</th><td><a href="/facts/Runcic_cubic_honeycomb/mX4CxSvp">3-RCO-trille</a>(Cantellated tetrahedral-octahedral honeycomb) or </td><td>Truncated triangular pyramid</td><td></td><td><a href="/facts/Quarter_cubille/mX4CxSvp">Quarter cubille</a></td><td> irr. <a href="/facts/Triangular_bipyramid/fU4kIBTV">triangular bipyramid</a></td><td></td></tr><tr><th>J24A20W16G21h2,3δ4</th><th>fc[4,31,1]</th><td><a href="/facts/Runcicantic_cubic_honeycomb/mX4CxSvp">f-tCO-trille</a>(Cantitruncated tetrahedral-octahedral honeycomb) or </td><td><a href="/facts/Sphenoid_(geometry)/7mkSrl6j">Mirrored sphenoid</a></td><td></td><td><a href="/facts/Half_pyramidille/mX4CxSvp">Half pyramidille</a></td><td> <a href="/facts/Sphenoid_(geometry)/7mkSrl6j">Mirrored sphenoid</a></td><td></td></tr><tr><th>J25,33A13W10G6qδ4</th><th>d[[3[4]]]</th><td><a href="/facts/Quarter_cubic_honeycomb/22qQazbw">Truncated tetrahedrille</a>(Cyclotruncated tetrahedral-octahedral honeycomb) or </td><td>Isosceles <a href="/facts/Triangular_prism/Yjp4Kw9M">triangular prism</a></td><td></td><td><a href="/facts/Trigonal_trapezohedral_honeycomb/sW4PrKn6">Oblate cubille</a></td><td><a href="/facts/Trigonal_trapezohedron/w5qPRnR3">Trigonal trapezohedron</a></td><td></td></tr></tbody></table> <h2 id="vertex-figures">Vertex Figures</h2> <p>The vertex figures of all architectonic honeycombs, and the dual cells of all catoptric honeycombs are shown below, at the same scale and the same orientation: </p> <h2 id="symmetry">Symmetry</h2> <p>These four symmetry groups are labeled as: </p> <table><tbody><tr><th>Label</th><th>Description</th><th><a href="/facts/Space_group/2XQx0J7M">space group</a>Intl symbol</th><th>Geometricnotation<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a></th><th><a href="/facts/Coxeter_notation/ArUw6goe">Coxeternotation</a></th><th><a href="/facts/Fibrifold_notation/2Eb13dNI">Fibrifoldnotation</a></th></tr><tr><th>bc</th><td>bicubic symmetryor extended cubic symmetry</td><td>(221) Im3m</td><td>I43</td><td>[[4,3,4]]</td><td>8°:2</td></tr><tr><th>nc</th><td>normal cubic symmetry</td><td>(229) Pm3m</td><td>P43</td><td>[4,3,4]</td><td>4−:2</td></tr><tr><th>fc</th><td>half-cubic symmetry</td><td>(225) Fm3m</td><td>F43</td><td>[4,31,1] = [4,3,4,1+]</td><td>2−:2</td></tr><tr><th>d</th><td><i>diamond symmetry</i>or extended quarter-cubic symmetry</td><td>(227) Fd3m</td><td>Fd4n3</td><td>[[3[4]]] = [[1+,4,3,4,1+]]</td><td>2+:2</td></tr></tbody></table> <ul><li><a href="https://books.google.com/books?id=nVx-tu596twC&q=space-filling+packings&pg=PA54">Crystallography of Quasicrystals: Concepts, Methods and Structures</a> by Walter Steurer, Sofia Deloudi (2009), p. 54-55. 12 packings of 2 or more uniform polyhedra with cubic symmetry</li></ul> <h2 id="further-reading">Further reading</h2> <ul><li><a href="/facts/John_Horton_Conway/g3PA1zoK">Conway, John H.</a>; Burgiel, Heidi; Goodman-Strauss, Chaim (2008). "21. Naming Archimedean and Catalan Polyhedra and Tilings". <i>The Symmetries of Things</i>. A K Peters, Ltd. pp. 292–298. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-56881-220-5.</li> <li>Inchbald, Guy (July 1997). "The Archimedean honeycomb duals". <i>The Mathematical Gazette</i>. 81 (491). Leicester: The Mathematical Association: 213–219. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F3619198">10.2307/3619198</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/3619198">3619198</a>. <a href="http://www.steelpillow.com/polyhedra/AHD/AHD.htm">[1]</a></li> <li><a href="/facts/Branko_Gr%C3%BCnbaum/taVzX1MG">Branko Grünbaum</a>, (1994) Uniform tilings of 3-space. <a href="/facts/Geombinatorics/5BgWJzA2">Geombinatorics</a> 4, 49 - 56.</li> <li><a href="/facts/Norman_Johnson_(mathematician)/bQtL9w56">Norman Johnson</a> (1991) <i>Uniform Polytopes</i>, Manuscript</li> <li><a href="/facts/Alfredo_Andreini/GkW29WaF">A. Andreini</a>, (1905) <i>Sulle reti di poliedri regolari e semiregolari e sulle corrispondenti reti correlative</i> (On the regular and semiregular nets of polyhedra and on the corresponding correlative nets), Mem. Società Italiana della Scienze, Ser.3, 14 75–129. <a href="/facts/PDF/C9Mn1vYL">PDF</a> <a href="https://web.archive.org/web/20140429195143/http://media.accademiaxl.it/memorie/Serie3_T14.pdf">[2]</a></li> <li>George Olshevsky, (2006) <i>Uniform Panoploid Tetracombs</i>, Manuscript <a href="/facts/PDF/C9Mn1vYL">PDF</a> <a href="http://bendwavy.org/4HONEYS.pdf">[3]</a></li> <li>Pearce, Peter (1980). <a href="https://books.google.com/books?id=sfc2OEuE8oQC&q=%22the+tetrahedron+and+octahedron+space+filling%22"><i>Structure in Nature is a Strategy for Design</i></a>. The MIT Press. pp. 41–47. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780262660457.</li> <li>Kaleidoscopes: Selected Writings of <a href="/facts/Harold_Scott_MacDonald_Coxeter/D2l5jWGG">H. S. M. Coxeter</a>, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-471-01003-6 <a href="http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html">[4]</a> <ul><li>(Paper 24) H.S.M. Coxeter, <i>Regular and Semi-Regular Polytopes III</i>, [Math. Zeit. 200 (1988) 3-45] See p318 <a href="https://books.google.com/books?id=fUm5Mwfx8rAC&dq=%22quarter+cubic+honeycomb%22+q%7B4%2C3%2C4%7D&pg=PA318">[5]</a></li></ul></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>For cross-referencing of Architectonic solids, they are given with list indices from Andreini (1-22), Williams(1-2,9-19), Johnson (11-19, 21-25, 31-34, 41-49, 51-52, 61-65), and Grünbaum(1-28). Coxeters names are based on δ4 as a cubic honeycomb, hδ4 as an alternated cubic honeycomb, and qδ4 as a quarter cubic honeycomb. <a href="/wiki/Cubic_honeycomb" target="_blank">/wiki/Cubic_honeycomb</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Hestenes, David; Holt, Jeremy (February 27, 2007). "Crystallographic space groups in geometric algebra" (PDF). Journal of Mathematical Physics. 48 (2). AIP Publishing LLC: 023514. doi:10.1063/1.2426416. ISSN 1089-7658. <a href="https://davidhestenes.net/geocalc/pdf/CrystalGA.pdf" target="_blank">https://davidhestenes.net/geocalc/pdf/CrystalGA.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> </ol>