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List of Euclidean uniform tilings
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<h2 id="laves-tilings">Laves tilings</h2> <p>In the 1987 book, <i><a href="/facts/Tilings_and_patterns/bX261kQS">Tilings and patterns</a></i>, <a href="/facts/Branko_Gr%25C3%25BCnbaum/taVzX1MG">Branko Grünbaum</a> calls the vertex-uniform tilings <i>Archimedean</i>, in parallel to the <a href="/facts/Archimedean_solid/wYWeHDYT">Archimedean solids</a>. Their <a href="/facts/Dual_graph/1t14hhEF">dual tilings</a> are called <i>Laves tilings</i> in honor of <a href="/facts/Crystallography/9B5toeqG">crystallographer</a> <a href="/facts/Fritz_Laves/UpqMG6FJ">Fritz Laves</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> They're also called Shubnikov–Laves tilings after <a href="/facts/Alexei_Vasilievich_Shubnikov/aYknz2FY">Aleksei Shubnikov</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> <a href="/facts/John_Horton_Conway/g3PA1zoK">John Conway</a> called the uniform duals <i>Catalan tilings</i>, in parallel to the <a href="/facts/Catalan_solid/R8D2cP7D">Catalan solid</a> polyhedra. </p><p>The Laves tilings have vertices at the centers of the regular polygons, and edges connecting centers of regular polygons that share an edge. The <a href="/facts/Prototile/zh5RBHy5">tiles</a> of the Laves tilings are called <i><a href="/facts/Planigon/QFexCc9b">planigons</a></i>. This includes the 3 regular tiles (triangle, square and hexagon) and 8 irregular ones.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> Each vertex has edges evenly spaced around it. Three dimensional analogues of the <i>planigons</i> are called <a href="/facts/Stereohedron/cpE5Rqw1">stereohedrons</a>. </p><p>These dual tilings are listed by their <a href="/facts/Face_configuration/wWaBS6t0">face configuration</a>, the number of faces at each vertex of a face. For example <i>V4.8.8</i> means isosceles triangle tiles with one corner with four triangles, and two corners containing eight triangles. The orientations of the vertex planigons (up to <a href="/facts/Dihedral_group/1PUlRh4M">D12</a>) are consistent with the vertex diagrams in the below sections. </p> Eleven planigons<table><tbody><tr><th colspan="4">Triangles</th><th colspan="3">Quadrilaterals</th><th colspan="3">Pentagons</th><th>Hexagon</th></tr><tr><td>V63</td><td>V4.82</td><td>V4.6.12</td><td>V3.122</td><td>V44</td><td>V(3.6)2</td><td>V3.4.6.4</td><td>V32.4.3.4</td><td>V34.6</td><td>V33.42</td><td>V36</td></tr></tbody></table> <h2 id="convex-uniform-tilings-of-the-euclidean-plane">Convex uniform tilings of the Euclidean plane</h2> <p>All reflectional forms can be made by <a href="/facts/Wythoff_construction/8yTYjp5e">Wythoff constructions</a>, represented by <a href="/facts/Wythoff_symbol/ECD0Wl5x">Wythoff symbols</a>, or <a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter-Dynkin diagrams</a>, each operating upon one of three <a href="/facts/Schwarz_triangle/bIz52h1W">Schwarz triangle</a> (4,4,2), (6,3,2), or (3,3,3), with symmetry represented by <a href="/facts/Coxeter_group/bgzHFm1Q">Coxeter groups</a>: [4,4], [6,3], or [3[3]]. <a href="/facts/Alternation_(geometry)/H2nVs4xj">Alternated</a> forms such as the snub can also be represented by special markups within each system. Only one uniform tiling can't be constructed by a Wythoff process, but can be made by an <a href="/facts/Elongation_(geometry)/8SR7CyNz">elongation</a> of the triangular tiling. An orthogonal mirror construction [∞,2,∞] also exists, seen as two sets of parallel mirrors making a rectangular fundamental domain. If the domain is square, this symmetry can be doubled by a diagonal mirror into the [4,4] family. </p><p>Families: </p> <ul><li>(4,4,2), B C ~ 2 {\displaystyle {\tilde {BC}}_{2}} , [4,4] – Symmetry of the regular <a href="/facts/Square_tiling/3ym86mvv">square tiling</a> <ul><li> I ~ 1 2 {\displaystyle {\tilde {I}}_{1}^{2}} , [∞,2,∞]</li></ul></li> <li>(6,3,2), G ~ 2 {\displaystyle {\tilde {G}}_{2}} , [6,3] – Symmetry of the regular <a href="/facts/Hexagonal_tiling/HwG5F0YH">hexagonal tiling</a> and <a href="/facts/Triangular_tiling/Jx8f8Krh">triangular tiling</a>. <ul><li>(3,3,3), A ~ 2 {\displaystyle {\tilde {A}}_{2}} , [3[3]]</li></ul></li></ul> <h3> The [4,4] group family</h3> <table><tbody><tr><th>Uniform tilings(Platonic and Archimedean)</th><th><a href="/facts/Vertex_figure/3u2ySK0C">Vertex figure</a> and dual face<a href="/facts/Wythoff_symbol/ECD0Wl5x">Wythoff symbol(s)</a><a href="/facts/Symmetry_group/CNqeseMp">Symmetry group</a><a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagram</a>(s)</th><th><a href="/facts/Dual_polyhedron/rkIHbmAf">Dual</a>-uniform tilings(called Laves or Catalan tilings)</th></tr><tr><td><a href="/facts/Square_tiling/3ym86mvv">Square tiling</a> (quadrille)</td><td>4.4.4.4 (or 44)4 | 2 4<a href="/facts/442_symmetry/oqpMEMBw">p4m</a>, [4,4], (*442)</td><td><a href="/facts/Self-dual/8jGuyQIU">self-dual</a> (quadrille)</td></tr><tr><td><a href="/facts/Truncated_square_tiling/Mbj1fVzm">Truncated square tiling</a> (truncated quadrille)</td><td>4.8.82 | 4 44 4 2 |<a href="/facts/442_symmetry/oqpMEMBw">p4m</a>, [4,4], (*442) or </td><td><a href="/facts/Tetrakis_square_tiling/oqpMEMBw">Tetrakis square tiling</a> (kisquadrille)</td></tr><tr><td><a href="/facts/Snub_square_tiling/hNu3KtEA">Snub square tiling</a> (snub quadrille)</td><td>3.3.4.3.4| 4 4 2<a href="/facts/442_symmetry/oqpMEMBw">p4g</a>, [4+,4], (4*2) or </td><td><a href="/facts/Cairo_pentagonal_tiling/IYXIjHs0">Cairo pentagonal tiling</a> (4-fold pentille)</td></tr></tbody></table> <h3> The [6,3] group family</h3> <table><tbody><tr><th>Platonic and Archimedean tilings</th><th><a href="/facts/Vertex_figure/3u2ySK0C">Vertex figure</a> and dual face<a href="/facts/Wythoff_symbol/ECD0Wl5x">Wythoff symbol(s)</a><a href="/facts/Symmetry_group/CNqeseMp">Symmetry group</a><a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagram</a>(s)</th><th><a href="/facts/Dual_polyhedron/rkIHbmAf">Dual</a> Laves tilings</th></tr><tr><td><a href="/facts/Hexagonal_tiling/HwG5F0YH">Hexagonal tiling</a> (hextille)</td><td>6.6.6 (or 63)3 | 6 22 6 | 33 3 3 |<a href="/facts/632_symmetry/egxEaC8q">p6m</a>, [6,3], (*632)</td><td><a href="/facts/Triangular_tiling/Jx8f8Krh">Triangular tiling</a> (deltille)</td></tr><tr><td><a href="/facts/Trihexagonal_tiling/spAKhDvd">Trihexagonal tiling</a> (hexadeltille)</td><td>(3.6)22 | 6 33 3 | 3<a href="/facts/632_symmetry/egxEaC8q">p6m</a>, [6,3], (*632) = </td><td><a href="/facts/Rhombille_tiling/vKwFthpW">Rhombille tiling</a> (rhombille)</td></tr><tr><td><a href="/facts/Truncated_hexagonal_tiling/c9dRoGFS">Truncated hexagonal tiling</a> (truncated hextille)</td><td>3.12.122 3 | 6<a href="/facts/632_symmetry/egxEaC8q">p6m</a>, [6,3], (*632)</td><td><a href="/facts/Triakis_triangular_tiling/c9dRoGFS">Triakis triangular tiling</a> (kisdeltille)</td></tr><tr><td><a href="/facts/Triangular_tiling/Jx8f8Krh">Triangular tiling</a> (deltille)</td><td>3.3.3.3.3.3 (or 36)6 | 3 23 | 3 3| 3 3 3<a href="/facts/632_symmetry/egxEaC8q">p6m</a>, [6,3], (*632) = </td><td><a href="/facts/Hexagonal_tiling/HwG5F0YH">Hexagonal tiling</a> (hextille)</td></tr><tr><td><a href="/facts/Rhombitrihexagonal_tiling/RJPMacIY">Rhombitrihexagonal tiling</a> (rhombihexadeltille)</td><td>3.4.6.43 | 6 2<a href="/facts/632_symmetry/egxEaC8q">p6m</a>, [6,3], (*632)</td><td><a href="/facts/Deltoidal_trihexagonal_tiling/RJPMacIY">Deltoidal trihexagonal tiling</a> (tetrille)</td></tr><tr><td><a href="/facts/Truncated_trihexagonal_tiling/qMT9oGKC">Truncated trihexagonal tiling</a> (truncated hexadeltille)</td><td>4.6.122 6 3 |<a href="/facts/632_symmetry/egxEaC8q">p6m</a>, [6,3], (*632)</td><td><a href="/facts/Kisrhombille_tiling/qMT9oGKC">Kisrhombille tiling</a> (kisrhombille)</td></tr><tr><td><a href="/facts/Snub_trihexagonal_tiling/0Km9nP9l">Snub trihexagonal tiling</a> (snub hextille)</td><td>3.3.3.3.6| 6 3 2<a href="/facts/632_symmetry/egxEaC8q">p6</a>, [6,3]+, (632)</td><td><a href="/facts/Floret_pentagonal_tiling/0Km9nP9l">Floret pentagonal tiling</a> (6-fold pentille)</td></tr></tbody></table> <h3>Non-Wythoffian uniform tiling</h3> <table><tbody><tr><th>Platonic and Archimedean tilings</th><th><a href="/facts/Vertex_figure/3u2ySK0C">Vertex figure</a> and dual face<a href="/facts/Wythoff_symbol/ECD0Wl5x">Wythoff symbol(s)</a><a href="/facts/Symmetry_group/CNqeseMp">Symmetry group</a><a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagram</a></th><th><a href="/facts/Dual_polyhedron/rkIHbmAf">Dual</a> Laves tilings</th></tr><tr><td><a href="/facts/Elongated_triangular_tiling/mH6cUVHj">Elongated triangular tiling</a> (isosnub quadrille)</td><td>3.3.3.4.42 | 2 (2 2)<a href="/facts/Wallpaper_group/egxEaC8q">cmm</a>, [∞,2+,∞], (2*22)</td><td><a href="/facts/Prismatic_pentagonal_tiling/mH6cUVHj">Prismatic pentagonal tiling</a> (iso(4-)pentille)</td></tr></tbody></table> <h3>Uniform colorings</h3> <p>There are a total of 32 uniform colorings of the 11 uniform tilings: </p> <ol><li><a href="/facts/Triangular_tiling/Jx8f8Krh">Triangular tiling</a> – 9 uniform colorings, 4 wythoffian, 5 nonwythoffian <ul><li> </li></ul></li> <li><a href="/facts/Square_tiling/3ym86mvv">Square tiling</a> – 9 colorings: 7 wythoffian, 2 nonwythoffian <ul><li> </li></ul></li> <li><a href="/facts/Hexagonal_tiling/HwG5F0YH">Hexagonal tiling</a> – 3 colorings, all wythoffian <ul><li> </li></ul></li> <li><a href="/facts/Trihexagonal_tiling/spAKhDvd">Trihexagonal tiling</a> – 2 colorings, both wythoffian <ul><li> </li></ul></li> <li><a href="/facts/Snub_square_tiling/hNu3KtEA">Snub square tiling</a> – 2 colorings, both alternated wythoffian <ul><li> </li></ul></li> <li><a href="/facts/Truncated_square_tiling/Mbj1fVzm">Truncated square tiling</a> – 2 colorings, both wythoffian <ul><li> </li></ul></li> <li><a href="/facts/Truncated_hexagonal_tiling/c9dRoGFS">Truncated hexagonal tiling</a> – 1 coloring, wythoffian <ul><li> </li></ul></li> <li><a href="/facts/Rhombitrihexagonal_tiling/RJPMacIY">Rhombitrihexagonal tiling</a> – 1 coloring, wythoffian <ul><li> </li></ul></li> <li><a href="/facts/Truncated_trihexagonal_tiling/qMT9oGKC">Truncated trihexagonal tiling</a> – 1 coloring, wythoffian <ul><li> </li></ul></li> <li><a href="/facts/Snub_hexagonal_tiling/0Km9nP9l">Snub hexagonal tiling</a> – 1 coloring, alternated wythoffian <ul><li> </li></ul></li> <li><a href="/facts/Elongated_triangular_tiling/mH6cUVHj">Elongated triangular tiling</a> – 1 coloring, nonwythoffian <ul><li> </li></ul></li></ol> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Convex_uniform_honeycomb/78IkTl7A">Convex uniform honeycomb</a> – The 28 uniform 3-dimensional tessellations, a parallel construction to the convex uniform Euclidean plane tilings.</li> <li><a href="/facts/Euclidean_tilings_by_convex_regular_polygons/tR5CQACy">Euclidean tilings by convex regular polygons</a></li> <li><a href="/facts/List_of_tessellations/px2Ajkna">List of tessellations</a></li> <li><a href="/facts/Percolation_threshold/z67jYyyF">Percolation threshold</a></li> <li><a href="/facts/Uniform_tilings_in_hyperbolic_plane/khwIB6I1">Uniform tilings in hyperbolic plane</a></li></ul> <h2 id="further-reading">Further reading</h2> <ul><li><a href="/facts/John_Horton_Conway/g3PA1zoK">Conway, John H.</a>; Burgiel, Heidi; <a href="/facts/Chaim_Goodman-Strauss/4IKWbx8w">Goodman-Strauss, Chaim</a> (April 18, 2008). "Chapter 19, <i>Archimedean tilings</i>, table 19.1". <a href="https://web.archive.org/web/20100919143320/https://akpeters.com/product.asp?ProdCode=2205"><i>The Symmetries of Things</i></a>. <a href="/facts/A_K_Peters/kIyEuk1n">A K Peters</a> / <a href="/facts/CRC_Press/duTuH4hz">CRC Press</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-56881-220-5. Archived from <a href="https://akpeters.com/product.asp?ProdCode=2205">the original</a> on September 19, 2010.</li> <li><a href="/facts/H.S.M._Coxeter/D2l5jWGG">Coxeter, H.S.M.</a>; <a href="/facts/Michael_S._Longuet-Higgins/abuxImrj">Longuet-Higgins, M.S.</a>; <a href="/facts/J.C.P._Miller/wPxXrRDR">Miller, J.C.P.</a> (1954). "Uniform polyhedra". <i><a href="/facts/Phil._Trans./a0l7M7RR">Phil. Trans.</a></i> 246 A: 401–450.</li> <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 2–3 <i>Circle packings, plane tessellations, and networks</i>, pp. 34–40).</li> <li>Asaro, Laura; Hyde, John; Jensen, Melanie; Mann, Casey; Schroeder, Tyler. <a href="http://faculty.washington.edu/cemann/uniform-edge-coloring.pdf">"Uniform edge-<i>c</i>-colorings of the Archimedean Tilings"</a> (PDF). <a href="/facts/University_of_Washington/36PG6qkG">University of Washington</a>. (<a href="http://faculty.washington.edu/cemann/">Casey Mann at the University of Washington</a>)</li> <li><a href="/facts/Branko_Gr%25C3%25BCnbaum/taVzX1MG">Grünbaum, Branko</a>; <a href="/facts/G.C._Shephard/5AtJ6sMW">Shepard, Geoffrey</a> (November 1977). <a href="http://www.maa.org/sites/default/files/images/upload_library/22/Allendoerfer/1978/0025570x.di021102.02p0230f.pdf">"Tilings by Regular polygons"</a> (PDF).</li> <li>Seymour, Dale; <a href="/facts/Jill_Britton/0kQ9s7tF">Britton, Jill</a> (1989). <a href="https://archive.org/details/introductiontote0000seym/page/50"><i>Introduction to Tessellations</i></a>. Dale Seymour Publications. pp. <a href="https://archive.org/details/introductiontote0000seym/page/50">50–57, 71-74</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0866514613.</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/UniformTessellation.html">"Uniform tessellation"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li> <li><a href="http://www2u.biglobe.ne.jp/~hsaka/mandara/ue2">Uniform Tessellations on the Euclid plane</a></li> <li><a href="http://www.orchidpalms.com/polyhedra/tessellations/tessel.htm">Tessellations of the Plane</a></li> <li><a href="https://web.archive.org/web/20080602030052/http://www.tess-elation.co.uk/index.htm">David Bailey's World of Tessellations</a></li> <li><a href="https://web.archive.org/web/20060909053826/http://www.uwgb.edu/dutchs/SYMMETRY/uniftil.htm"><i>k</i>-uniform tilings</a></li> <li><a href="http://probabilitysports.com/tilings.html"><i>n</i>-uniform tilings</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. W. H. Freeman and Company. pp. 59, 96. ISBN 0-7167-1193-1. <a href="0-7167-1193-1" target="_blank">0-7167-1193-1</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Conway, John H.; Burgiel, Heidi; Goodman-Strauss, Chaim (April 18, 2008). "Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, Euclidean Plane Tessellations". The Symmetries of Things. A K Peters / CRC Press. p. 288. ISBN 978-1-56881-220-5. Archived from the original on September 19, 2010. <a href="978-1-56881-220-5" target="_blank">978-1-56881-220-5</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Encyclopaedia of Mathematics: Orbit - Rayleigh Equation, 1991 <a href="https://books.google.com/books?id=5rPnCAAAQBAJ&dq=Shubnikov%E2%80%93Laves+tilings&pg=PA169" target="_blank">https://books.google.com/books?id=5rPnCAAAQBAJ&dq=Shubnikov%E2%80%93Laves+tilings&pg=PA169</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Ivanov, A. B. (2001) [1994], "Planigon", Encyclopedia of Mathematics, EMS Press <a href="https://www.encyclopediaofmath.org/index.php?title=Planigon" target="_blank">https://www.encyclopediaofmath.org/index.php?title=Planigon</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> </ol>