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<h2 id="history">History</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 Shubnikov, Alekseĭ Vasilʹevich.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> <a href="/facts/John_Horton_Conway/g3PA1zoK">John Conway</a> calls 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 planigons. 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 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> (or V4.82) means isosceles triangle tiles with one corner with four triangles, and two corners containing eight triangles. </p> <h2 id="construction">Construction</h2> <p>The <a href="/facts/Conway_polyhedron_notation/MVmYMqID">Conway operation</a> of dual interchanges faces and vertices. In <a href="/facts/Archimedean_solids/wYWeHDYT">Archimedean solids</a> and <a href="/facts/Euclidean_tilings_by_convex_regular_polygons/tR5CQACy"><i>k</i>-uniform tilings</a> alike, the new vertex coincides with the center of each <a href="/facts/Regular_polygon/ws89Nhpt">regular face</a>, or the <a href="/facts/Centroid/H6dDbg0h">centroid</a>. In the Euclidean (plane) case; in order to make new faces around each original vertex, the centroids must be connected by new edges, each of which must intersect exactly one of the original edges. Since regular polygons have <a href="/facts/Dihedral_group/1PUlRh4M">dihedral symmetry</a>, we see that these new centroid-centroid edges must be <a href="/facts/Bisection/JTsya7Ca">perpendicular bisectors</a> of the common original edges (e.g. the centroid lies on all edge perpendicular bisectors of a regular polygon). Thus, the edges of <i>k</i>-dual uniform tilings coincide with centroid-to-edge-midpoint line segments of all regular polygons in the <i>k</i>-uniform tilings. </p> Planigon Constructions<table><tbody><tr><th>Centroid-to-Centroid</th><th>12-5 Dodecagram</th></tr><tr><td></td><td></td></tr></tbody></table> <h3>Using the 12-5 Dodecagram (Above)</h3> <p>All 14 uniform usable regular vertex planigons also hail<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> from the <a href="/facts/Dodecagram/cJ5qjQWj">6-5 dodecagram</a> (where each segment subtends 5 π / 6 {\displaystyle 5\pi /6} radians, or 150 degrees). </p><p>The <a href="/facts/Incircle_and_excircles_of_a_triangle/x9QCn9ML">incircle</a> of this dodecagram demonstrates that all the 14 VRPs are <a href="/facts/Tangential_quadrilateral/JBdfotIx">cocyclic</a>, as alternatively shown by <a href="/facts/Circle_packing/nKFWD2ug">circle packings</a>. The ratio of the incircle to the circumcircle is: </p><p> sin π 12 = sin 15 ∘ = 6 − 2 4 ≈ 0.258819 {\displaystyle \sin {\frac {\pi }{12}}=\sin 15^{\circ }={\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}\approx 0.258819} </p><p>and the convex hull is precisely the <a href="/facts/Regular_dodecagon/lKDvfUsm">regular dodecagons</a> in the <a href="/facts/List_of_k-uniform_tilings/Uuxoua41">k-uniform tiling</a>. The equilateral triangle, square, regular hexagon, and regular dodecagon; are shown above with the VRPs. </p><p>In fact, any group of planigons can be constructed from the edges of a 2 k - ( k − 1 ) {\displaystyle 2k{\text{-}}(k-1)} <a href="/facts/Polygram_(geometry)/I4naM8nS">polygram</a>, where k = gcd ( n 1 , … , n m ) {\displaystyle k=\gcd(n_{1},\dots ,n_{m})} and n i {\displaystyle n_{i}} is the number of sides of sides in the RP adjacent to each involved vertex figure. This is because the <a href="/facts/Regular_polygon/ws89Nhpt">circumradius</a> 1 2 csc π n i {\displaystyle {\frac {1}{2}}\csc {\frac {\pi }{n_{i}}}} of any regular n i {\displaystyle n_{i}} -gon (from the vertex to the centroid) is the same as the distance from the center of the polygram to its line segments which intersect at the angle 2 π / n i {\displaystyle 2\pi /n_{i}} , since all 2 k - ( k − 1 ) {\displaystyle 2k{\text{-}}(k-1)} polygrams admit incircles of inradii 1 / 2 {\displaystyle 1/2} tangent to all its sides. </p> <h3>Regular Vertices</h3> <p>In <i>Tilings and Patterns</i>, <a href="/facts/Branko_Gr%25C3%25BCnbaum/taVzX1MG">Grünbaum</a> also constructed the Laves tilings using <i>monohedral tiles with</i> <i>regular vertices</i>. A vertex is regular if all angles emanating from it are equal. In other words:<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> </p> <ol><li>All vertices are regular,</li> <li>All Laves planigons are congruent.</li></ol> <p>In this way, all Laves tilings are unique except for the <a href="/facts/Square_tiling/3ym86mvv">square tiling</a> (1 degree of freedom), <a href="/facts/Prismatic_pentagonal_tiling/mH6cUVHj">barn pentagonal tiling</a> (1 degree of freedom), and <a href="/facts/Hexagonal_tiling/HwG5F0YH">hexagonal tiling</a> (2 degrees of freedom): </p> Tiling Variants<table><tbody><tr><th>Square</th><th>Barn Pentagon</th><th>Hexagon</th></tr><tr><td></td><td></td><td></td></tr></tbody></table> <p>When applied to higher dual co-uniform tilings, all dual coregular planigons can be distorted except for the triangles (<a href="/facts/Similarity_(geometry)/Sbd6hEAN">AAA similarity</a>), with examples below: </p> Tiling Variants<table><tbody><tr><td>S2TCH</td><td>I2RFH</td><td>IrDC</td><td>FH (p6)</td><td>sBH (short)</td><td>CB (pgg)</td></tr></tbody></table> <h2 id="derivation-of-all-possible-planigons">Derivation of all possible planigons</h2> <p>For edge-to-edge Euclidean tilings, the <a href="/facts/Internal_angle/jd78Qkbt">interior angles</a> of the convex polygons meeting at a vertex must add to 360 degrees. A regular <i>n</i>-gon has internal angle ( 1 − 2 n ) 180 ∘ {\displaystyle \left(1-{\frac {2}{n}}\right)180^{\circ }} degrees. There are seventeen combinations of regular polygons whose internal angles add up to 360 degrees, each being referred to as a <i>species</i> of vertex; in four cases there are two distinct cyclic orders of the polygons, yielding twenty-one <i>types</i> of vertex. </p><p>In fact, with the vertex (interior) angles 60 ∘ , 90 ∘ , 108 ∘ , 120 ∘ , 128 4 7 ∘ , 135 ∘ , 140 ∘ , 144 ∘ , 147 3 11 ∘ , 150 ∘ , … {\displaystyle 60^{\circ },90^{\circ },108^{\circ },120^{\circ },128{\frac {4}{7}}^{\circ },135^{\circ },140^{\circ },144^{\circ },147{\frac {3}{11}}^{\circ },150^{\circ },\dots } , we can find all combinations of admissible corner angles according to the following rules: </p> <ol><li>Every vertex has at least degree 3 (a degree-2 vertex must have two straight angles or one reflex angle);</li> <li>If the vertex has degree d {\displaystyle d} , the smallest d − 1 {\displaystyle d-1} polygon vertex angles sum to over 180 ∘ {\displaystyle 180^{\circ }} ;</li> <li>The vertex angles add to 360 ∘ {\displaystyle 360^{\circ }} , and must be angles of regular polygons of positive integer sides (of the sequence 60 ∘ , 90 ∘ , 108 ∘ , 120 ∘ , 128 4 7 ∘ , 135 ∘ , 140 ∘ , 144 ∘ , 147 3 11 ∘ , 150 ∘ , … {\displaystyle 60^{\circ },90^{\circ },108^{\circ },120^{\circ },128{\frac {4}{7}}^{\circ },135^{\circ },140^{\circ },144^{\circ },147{\frac {3}{11}}^{\circ },150^{\circ },\dots } ).</li></ol> <p>Using the rules generates the list below: </p> Arrangements of regular polygons around a vertex<table><tbody><tr><th>Degree-6 vertex</th><th>Degree-5 vertex</th><th>Degree-4 vertex</th><th>Degree-3 vertex</th></tr><tr><td> 60 ∘ - 60 ∘ - 60 ∘ - 60 ∘ - 60 ∘ - 60 ∘ ( × 1 ) {\displaystyle 60^{\circ }{\text{-}}60^{\circ }{\text{-}}60^{\circ }{\text{-}}60^{\circ }{\text{-}}60^{\circ }{\text{-}}60^{\circ }~(\times 1)} </td><td> 60 ∘ - 60 ∘ - 60 ∘ - 90 ∘ - 90 ∘ ( × 2 ) {\displaystyle 60^{\circ }{\text{-}}60^{\circ }{\text{-}}60^{\circ }{\text{-}}90^{\circ }{\text{-}}90^{\circ }~(\times 2)} </td><td> 60 ∘ - 60 ∘ - 90 ∘ - 150 ∘ ( × 2 ) {\displaystyle 60^{\circ }{\text{-}}60^{\circ }{\text{-}}90^{\circ }{\text{-}}150^{\circ }~(\times 2)} </td><th> 60 ∘ - 128 4 7 ∘ - 171 3 7 ∘ ( × 1 ) {\displaystyle 60^{\circ }{\text{-}}128{\frac {4}{7}}^{\circ }{\text{-}}171{\frac {3}{7}}^{\circ }~(\times 1)} </th></tr><tr><td></td><td> 60 ∘ - 60 ∘ - 60 ∘ - 60 ∘ - 120 ∘ ( × 1 ) {\displaystyle 60^{\circ }{\text{-}}60^{\circ }{\text{-}}60^{\circ }{\text{-}}60^{\circ }{\text{-}}120^{\circ }~(\times 1)} </td><td> 60 ∘ - 60 ∘ - 120 ∘ - 120 ∘ ( × 2 ) {\displaystyle 60^{\circ }{\text{-}}60^{\circ }{\text{-}}120^{\circ }{\text{-}}120^{\circ }~(\times 2)} </td><th> 60 ∘ - 135 ∘ - 165 ∘ ( × 1 ) {\displaystyle 60^{\circ }{\text{-}}135^{\circ }{\text{-}}165^{\circ }~(\times 1)} </th></tr><tr><td></td><td></td><td> 60 ∘ - 90 ∘ - 90 ∘ - 120 ∘ ( × 2 ) {\displaystyle 60^{\circ }{\text{-}}90^{\circ }{\text{-}}90^{\circ }{\text{-}}120^{\circ }~(\times 2)} </td><th> 60 ∘ - 140 ∘ - 160 ∘ ( × 1 ) {\displaystyle 60^{\circ }{\text{-}}140^{\circ }{\text{-}}160^{\circ }~(\times 1)} </th></tr><tr><td></td><td></td><td> 90 ∘ - 90 ∘ - 90 ∘ - 90 ∘ ( × 1 ) {\displaystyle 90^{\circ }{\text{-}}90^{\circ }{\text{-}}90^{\circ }{\text{-}}90^{\circ }~(\times 1)} </td><th> 60 ∘ - 144 ∘ - 156 ( × 1 ) {\displaystyle 60^{\circ }{\text{-}}144^{\circ }{\text{-}}156~(\times 1)} </th></tr><tr><td></td><td></td><td></td><td> 60 ∘ - 150 ∘ - 150 ∘ ( × 1 ) {\displaystyle 60^{\circ }{\text{-}}150^{\circ }{\text{-}}150^{\circ }~(\times 1)} </td></tr><tr><td></td><td></td><td></td><th> 90 ∘ - 108 ∘ - 162 ∘ ( × 1 ) {\displaystyle 90^{\circ }{\text{-}}108^{\circ }{\text{-}}162^{\circ }~(\times 1)} </th></tr><tr><td></td><td></td><td></td><td> 90 ∘ - 120 ∘ - 150 ∘ ( × 1 ) {\displaystyle 90^{\circ }{\text{-}}120^{\circ }{\text{-}}150^{\circ }~(\times 1)} </td></tr><tr><td></td><td></td><td></td><td> 90 ∘ - 135 ∘ - 135 ∘ ( × 1 ) {\displaystyle 90^{\circ }{\text{-}}135^{\circ }{\text{-}}135^{\circ }~(\times 1)} *</td></tr><tr><td></td><td></td><td></td><th> 108 ∘ - 108 ∘ - 144 ∘ ( × 1 ) {\displaystyle 108^{\circ }{\text{-}}108^{\circ }{\text{-}}144^{\circ }~(\times 1)} </th></tr><tr><td></td><td></td><td></td><td> 120 ∘ - 120 ∘ - 120 ∘ ( × 1 ) {\displaystyle 120^{\circ }{\text{-}}120^{\circ }{\text{-}}120^{\circ }~(\times 1)} </td></tr></tbody></table> <p>*The 90 ∘ - 135 ∘ - 135 ∘ ( × 1 ) {\displaystyle 90^{\circ }{\text{-}}135^{\circ }{\text{-}}135^{\circ }~(\times 1)} cannot coexist with any other vertex types. </p><p>The solution to Challenge Problem 9.46, <i>Geometry</i> (Rusczyk),<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> is in the Degree 3 Vertex column above. A triangle with a <a href="/facts/Hendecagon/FFdXrqzl">hendecagon</a> (11-gon) yields a 13.2-gon, a square with a <a href="/facts/Heptagon/b0qIVNhY">heptagon</a> (7-gon) yields a 9.3333-gon, and a pentagon with a hexagon yields a 7.5-gon). Hence there are 1 ( 1 ) + ( 1 ( 2 ) + 1 ) + ( 3 ( 2 ) + 1 ) + 10 = 21 {\displaystyle 1(1)+(1(2)+1)+(3(2)+1)+10=21} combinations of regular polygons which meet at a vertex. </p> <h3>Planigons in the plane</h3> <p>Only eleven of these angle combinations can occur in a <a href="/facts/List_of_Euclidean_uniform_tilings/c1ALUDMm">Laves Tiling</a> of planigons. </p><p>In particular, if three polygons meet at a vertex and one has an odd number of sides, the other two polygons must be the same. If they are not, they would have to alternate around the first polygon, which is impossible if its number of sides is odd. By that restriction these six cannot appear in any k-dual-uniform tiling: </p> <p>On the other hand, these four can be used in <i>k</i>-dual-uniform tilings: </p> <p>Finally, assuming unit side length, all regular polygons and usable planigons have side-lengths and areas as shown below in the table: </p> Regular Polygons and Planigons<table><tbody><tr><th colspan="8"></th></tr><tr><th colspan="2">Regular Polygons</th><th colspan="6">Planigons</th></tr><tr><th>Triangle</th><td>Area: 3 4 {\displaystyle {\frac {\sqrt {3}}{4}}} <p>Side Lengths: 1</p></td><th>V3.122(O)</th><td>Area: 1 + 7 4 3 {\displaystyle 1+{\frac {7}{4{\sqrt {3}}}}} <p>Side Lengths: 2 + 3 , 1 + 2 3 {\displaystyle 2+{\sqrt {3}},1+{\frac {2}{\sqrt {3}}}} </p></td><th>V32.62(I)</th><td>Area: 2 3 {\displaystyle {\frac {2}{\sqrt {3}}}} <p>Side Lengths: 3 , 2 3 , 1 3 {\displaystyle {\sqrt {3}},{\frac {2}{\sqrt {3}}},{\frac {1}{\sqrt {3}}}} </p></td><th>V44(s)</th><td>Area: 1<p>Side Lengths: 1</p></td></tr><tr><th>Square</th><td>Area: 1<p>Side Lengths: 1</p></td><th>V4.6.12(3)</th><td>Area: 3 4 + 3 3 2 {\displaystyle {\frac {3}{4}}+{\frac {3{\sqrt {3}}}{2}}} <p>Side Lengths: 1 + 3 , 3 + 3 2 , 1 + 3 2 {\displaystyle 1+{\sqrt {3}},{\frac {3+{\sqrt {3}}}{2}},{\frac {1+{\sqrt {3}}}{2}}} </p></td><th>V(3.6)2(R)</th><td>Area: 2 3 {\displaystyle {\frac {2}{\sqrt {3}}}} <p>Side Lengths: 2 3 {\displaystyle {\frac {2}{\sqrt {3}}}} </p></td><th>V32.4.3.4(C)</th><td>Area: 1 2 + 3 4 {\displaystyle {\frac {1}{2}}+{\frac {\sqrt {3}}{4}}} <p>Side Lengths: 1 2 + 1 2 3 , 1 3 {\displaystyle {\frac {1}{2}}+{\frac {1}{2{\sqrt {3}}}},{\frac {1}{\sqrt {3}}}} </p></td></tr><tr><th>Hexagon</th><td>Area: 3 3 2 {\displaystyle {\frac {3{\sqrt {3}}}{2}}} <p>Side Lengths: 1</p></td><th>V32.4.12(S)</th><td>Area: 3 4 + 5 4 3 {\displaystyle {\frac {3}{4}}+{\frac {5}{4{\sqrt {3}}}}} <p>Side Lengths: 3 + 3 2 , 1 + 2 3 , 1 2 + 1 2 3 , 1 3 {\displaystyle {\frac {3+{\sqrt {3}}}{2}},1+{\frac {2}{\sqrt {3}}},{\frac {1}{2}}+{\frac {1}{2{\sqrt {3}}}},{\frac {1}{\sqrt {3}}}} </p></td><th>V3.42.6(r)</th><td>Area: 1 2 + 1 3 {\displaystyle {\frac {1}{2}}+{\frac {1}{\sqrt {3}}}} <p>Side Lengths: 1 + 3 2 , 2 3 , 1 , 1 2 + 1 2 3 {\displaystyle {\frac {1+{\sqrt {3}}}{2}},{\frac {2}{\sqrt {3}}},1,{\frac {1}{2}}+{\frac {1}{2{\sqrt {3}}}}} </p></td><th>V33.42(B)</th><td>Area: 1 2 + 3 4 {\displaystyle {\frac {1}{2}}+{\frac {\sqrt {3}}{4}}} <p>Side Lengths: 1 , 1 2 + 1 2 3 , 1 3 {\displaystyle 1,{\frac {1}{2}}+{\frac {1}{2{\sqrt {3}}}},{\frac {1}{\sqrt {3}}}} </p></td></tr><tr><th>Octagon</th><td>Area: 2 + 2 2 {\displaystyle 2+2{\sqrt {2}}} <p>Side Lengths: 1</p></td><th>V3.4.3.12(T)</th><td>Area: 3 4 + 5 4 3 {\displaystyle {\frac {3}{4}}+{\frac {5}{4{\sqrt {3}}}}} <p>Side Lengths: 1 + 2 3 , 1 2 + 1 2 3 {\displaystyle 1+{\frac {2}{\sqrt {3}}},{\frac {1}{2}}+{\frac {1}{2{\sqrt {3}}}}} </p></td><th>V3.4.6.4(D)</th><td>Area: 1 2 + 1 3 {\displaystyle {\frac {1}{2}}+{\frac {1}{\sqrt {3}}}} <p>Side Lengths: 1 + 3 2 , 1 2 + 1 2 3 {\displaystyle {\frac {1+{\sqrt {3}}}{2}},{\frac {1}{2}}+{\frac {1}{2{\sqrt {3}}}}} </p></td><th>V36(H)</th><td>Area: 3 2 {\displaystyle {\frac {\sqrt {3}}{2}}} <p>Side Lengths: 1 3 {\displaystyle {\frac {1}{\sqrt {3}}}} </p></td></tr><tr><th>Dodecagon</th><td>Area: 6 + 3 3 {\displaystyle 6+3{\sqrt {3}}} <p>Side Lengths: 1</p></td><th>V63(E)</th><td>Area: 3 3 4 {\displaystyle {\frac {3{\sqrt {3}}}{4}}} <p>Side Lengths: 3 {\displaystyle {\sqrt {3}}} </p></td><th>V34.6(F)</th><td>Area: 7 4 3 {\displaystyle {\frac {7}{4{\sqrt {3}}}}} <p>Side Lengths: 2 3 , 1 3 {\displaystyle {\frac {2}{\sqrt {3}}},{\frac {1}{\sqrt {3}}}} </p></td><th>V4.82(i)</th><td>Area: 3 4 + 1 2 {\displaystyle {\frac {3}{4}}+{\frac {1}{\sqrt {2}}}} <p>Side Lengths: 1 + 1 2 , 1 2 + 1 2 {\displaystyle 1+{\frac {1}{\sqrt {2}}},{\frac {1}{2}}+{\frac {1}{\sqrt {2}}}} </p></td></tr></tbody></table> <h2 id="number-of-dual-uniform-tilings">Number of Dual Uniform Tilings</h2> <p>Every dual uniform tiling is in a 1:1 correspondence with the corresponding uniform tiling, by construction of the planigons above and superimposition. </p> <i>k</i>-dual-uniform, <i>m</i>-Catalaves tiling counts<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a><table><tbody><tr><th rowspan="2"></th><th colspan="8"><i>m</i>-Catalaves</th></tr><tr><th></th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>Total</th></tr><tr><th rowspan="7"><i>k</i>-dual-uniform</th><th><a href="/facts/List_of_Euclidean_uniform_tilings/c1ALUDMm">1</a></th><td>11</td><td colspan="5"></td><td>11</td></tr><tr><th><a href="/facts/List_of_k-uniform_tilings/Uuxoua41">2</a></th><td>0</td><td>20</td><td colspan="4"></td><td>20</td></tr><tr><th><a href="/facts/List_of_k-uniform_tilings/Uuxoua41">3</a></th><td>0</td><td>22</td><td>39</td><td colspan="3"></td><td>61</td></tr><tr><th><a href="/facts/List_of_k-uniform_tilings/Uuxoua41">4</a></th><td>0</td><td>33</td><td>85</td><td>33</td><td colspan="2"></td><td>151</td></tr><tr><th><a href="/facts/List_of_k-uniform_tilings/Uuxoua41">5</a></th><td>0</td><td>74</td><td>149</td><td>94</td><td>15</td><td></td><td>332</td></tr><tr><th><a href="/facts/List_of_k-uniform_tilings/Uuxoua41">6</a></th><td>0</td><td>100</td><td>284</td><td>187</td><td>92</td><td>10</td><td>673</td></tr><tr><th>Total</th><td>11</td><td>∞</td><td>∞</td><td>∞</td><td>∞</td><td>∞</td><td>∞</td></tr></tbody></table> <p>Such periodic tilings may be classified by the number of <a href="/facts/Group_action_(mathematics)/Vh9VtUUw">orbits</a> of vertices, edges and tiles. If there are <i>k</i> orbits of planigons, a tiling is known as <i>k</i>-dual-uniform or <i>k</i>-isohedral; if there are <i>t</i> orbits of dual vertices, as <i>t</i>-isogonal; if there are <i>e</i> orbits of edges, as <i>e</i>-isotoxal. </p><p><i>k</i>-dual-uniform tilings with the same vertex faces can be further identified by their <a href="/facts/Wallpaper_group/egxEaC8q">wallpaper group</a> symmetry, which is identical to that of the corresponding <i>k</i>-uniform tiling. </p><p>1-dual-uniform tilings include 3 regular tilings, and 8 Laves tilings, with 2 or more types of regular degree vertices. There are 20 2-dual-uniform tilings, 61 3-dual-uniform tilings, 151 4-dual-uniform tilings, 332 5-dual-uniform tilings and 673 6--dualuniform tilings. Each can be grouped by the number <i>m</i> of distinct vertex figures, which are also called <i>m</i>-Archimedean tilings.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> </p><p>Finally, if the number of types of planigons is the same as the uniformity (<i>m</i> = <i>k</i> below), then the tiling is said to be dual <i>Krotenheerdt</i>. In general, the uniformity is greater than or equal to the number of types of vertices (<i>m</i> ≥ <i>k</i>), as different types of planigons necessarily have different orbits, but not vice versa. Setting <i>m</i> = <i>n</i> = <i>k</i>, there are 11 such dual tilings for <i>n</i> = 1; 20 such dual tilings for <i>n</i> = 2; 39 such dual tilings for <i>n</i> = 3; 33 such dual tilings for <i>n</i> = 4; 15 such dual tilings for <i>n</i> = 5; 10 such dual tilings for <i>n</i> = 6; and 7 such dual tilings for <i>n</i> = 7. </p> <h2 id="regular-and-laves-tilings">Regular and Laves tilings</h2> <p>The 3 regular and 8 semiregular Laves tilings are shown, with planigons colored according to area as in the construction: </p> <h2 id="higher-dual-uniform-tilings">Higher Dual Uniform Tilings</h2> <h3>Insets of Dual Planigons into Higher Degree Vertices</h3> <ul><li>A degree-six vertex can be replaced by a center regular hexagon and six edges emanating thereof;</li> <li>A degree-twelve vertex can be replaced by six deltoids (a center deltoidal hexagon) and twelve edges emanating thereof;</li> <li>A degree-twelve vertex can be replaced by six Cairo pentagons, a center hexagon, and twelve edges emanating thereof (by dissecting the degree-6 vertex in the center of the previous example).</li></ul> <table><tbody><tr><th colspan="5"></th></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><th>Minor</th><th>Major</th><th>Full</th><th colspan="2">Substitutions</th></tr><tr><th colspan="5">Dual Processes (Insets)</th></tr></tbody></table> <p>This is done above for the <a href="/facts/3-4-6-12_tiling/UuUETn4A">dual</a> of the 3-4-6-12 tiling. The corresponding uniform process is <a href="/facts/Euclidean_tilings_by_convex_regular_polygons/tR5CQACy">dissection</a>, and is shown <a href="/facts/3-4-6-12_tiling/UuUETn4A">here</a>. </p> <h3>2-Dual-Uniform</h3> <p>There are 20 tilings made from 2 types of planigons, the dual of <a href="/facts/Euclidean_tilings_by_convex_regular_polygons/tR5CQACy">2-uniform tilings</a> (Krotenheerdt Duals): </p> <h3>3-Dual-Uniform</h3> <p>There are 39 tilings made from 3 types of planigons (Krotenheerdt Duals): </p> <h3>4-Dual-Uniform</h3> <p>There are 33 tilings made from 4 types of planigons (Krotenheerdt Duals): </p> <h3>5-Dual-Uniform</h3> <p>There are 15 5-uniform dual tilings with 5 unique planigons: </p> <h3>Krotenheerdt duals with six planigons</h3> <p>There are 10 6-uniform dual tilings with 6 unique planigons: </p> <h3>Krotenheerdt duals with seven planigons</h3> <p>There are 7 7-uniform dual tilings with 7 unique planigons: </p><p>The last two dual uniform-7 tilings have the same vertex types, even though they look nothing alike! </p><p>From n ≥ 8 {\displaystyle n\geq 8} onward, there are no uniform <i>n</i> tilings with <i>n</i> vertex types, or no uniform <i>n</i> duals with <i>n</i> distinct (semi)planigons.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> </p> <h2 id="fractalizing-dual-k-uniform-tilings">Fractalizing Dual <i>k</i>-Uniform Tilings</h2> <p>There are many ways of generating new k-dual-uniform tilings from other k-uniform tilings. Three ways is to scale by 1 + 3 , 2 + 3 , 3 + 3 {\displaystyle 1+{\sqrt {3}},2+{\sqrt {3}},3+{\sqrt {3}}} as seen below: </p> Fractalizing Examples<table><tbody><tr><th colspan="5"></th></tr><tr><th></th><th>Original</th><th>Semi-Fractalization</th><th>Truncated Hexagonal Tiling</th><th>Truncated Trihexagonal Tiling</th></tr><tr><th>DualFractalizing</th><td></td><td></td><td></td><td></td></tr></tbody></table> <h3>Large Fractalization</h3> <p>To enlarge the planigons V32.4.12 and V3.4.3.12 using the truncated trihexagonal method, a scale factor of 2 ( 3 + 3 ) {\displaystyle 2(3+{\sqrt {3}})} must be applied: </p> <h3>Big Fractalization</h3> <p>By two 9-uniform tilings in <a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> a big fractalization is achieved by a scale factor of 3 in all planigons. In the case of s,C,B,H its own planigon is in the exact center: </p><p>The two 9-uniform tilings are shown below, fractalizations of the <a href="/facts/Demiregular_tiling/KnYbIzIY">demiregulars</a> DC and DB, and a general example on S2TC: </p> <table><tbody><tr><th>9-Uniform</th><th>S2TC Big Fractalization</th></tr><tr><td> 3Ir3Ds2B (of DB) 3Ir4DsC (of DC)</td><td> S2TCBig Fractalization</td></tr></tbody></table> <h2 id="miscellaneous">Miscellaneous</h2> <h3>Centroid-Centroid Construction</h3> <p>Dual co-uniform tilings (red) along with the originals (blue) of selected tilings.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a><a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> Generated by centroid-edge midpoint construction by polygon-centroid-vertex detection, rounding the angle of each co-edge to the nearest 15 degrees. Since the unit size of tilings varies from 15 to 18 pixels and every regular polygon slightly differs,<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> there is some overlap or breaks of dual edges (an 18-pixel size generator incorrectly generates co-edges from five 15-pixel size tilings, classifying some squares as triangles). </p> <h3>Other Edge-Edge Construction Comparisons</h3> <p>Other edge-edge construction comparisons. Rotates every 3 seconds. </p> Comparisons<table><tbody><tr><th>SDB</th><th>3IrB</th><th>TDDC</th><th>IIRF</th><th>rFBH</th><th>OOOOT</th><th>3SrFCBH</th><th>O33STIr2C2B</th></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table> <h3>Affine Linear Expansions</h3> <p>Below are affine linear expansions of other uniform tilings, from the original to the dual and back: </p> Affine Linear Expansions<table><tbody><tr><th>8-Uniform 3STDC</th><th>12-Uniform 3STRrD</th><th>12-Uniform O3STIrCB</th><th>13-Uniform All Slab</th><th>16-Uniform OSTEIrCB</th><th>24-Uniform All Planigons</th></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table> <p>The first 12-uniform tiling contains all planigons with three types of vertices, and the second 12-uniform tiling contains all types of edges. </p> <h3>Optimized Tilings</h3> <p>If a {\displaystyle a} - b {\displaystyle b} tiling means a {\displaystyle a} dual uniform, b {\displaystyle b} Catalaves tiling, then there exists a 11-9 tiling,<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> a 13-10 tiling, 15-11 tiling, a 19-12 tiling, two 22-13 tilings, and a 24-14 tiling. Also exists a 13-8 slab tiling and a 14-10 non-clock tiling. Finally, there are 7-5 tilings using all clock planigons:<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a> </p> <table><tbody><tr><th>11-9</th><th>13-10</th><th>15-11</th><th>19-12</th><th>22-13</th></tr><tr><td> OSTRrD2sC2B</td><td> 3S2IRr3DFCBH</td><td> 3STEIRrFCB5H</td><td> O3ST3Rr2D3FsCB3H</td><td> O32ST2EIRr4D2FCB4H O32ST2EIRr3DFC2B5H</td></tr><tr><th>24-14</th><th>13-8 Slab</th><th>14-10 Non Clock</th><th colspan="2">7-5 All Clock</th></tr><tr><td> O32S3TEIRr2DFsC2B6H</td><td> EI2Rr2F2s2B2H</td><td> EIRr3DFsCB2H</td><td> O33STB</td><td> O32ST2D</td></tr></tbody></table> <h3>Circle Packing</h3> <p class="note">See also: <a href="/facts/Circle_packing/nKFWD2ug">Circle packing § Other_packings</a></p> <p>Each uniform tiling corresponds to a circle packing, in which circles of diameter 1 are placed at all vertex points, corresponding to the planigons.<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a> Below are the circle packings of the Optimized Tilings and all-edge tiling: </p> <h3>5-dual-uniform 4-Catalaves tilings</h3> <p>A slideshow of all 94 5-dual-uniform tilings with 4 distinct planigons. Changes every 6 seconds, cycles every 60 seconds. </p> <h3>Clock Tilings</h3><p> All tilings with regular dodecagons in <a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a> are shown below, alternating between uniform and dual co-uniform every 5 seconds:</p> All tilings with <a href="/facts/Dodecagon/lKDvfUsm">regular dodecagons</a> are shown below, alternating between uniform and dual co-uniform every 5 seconds. <h3>65 <i>k</i>-Uniform Tilings</h3><p> A comparison of 65 <i>k</i> uniform tilings in <a href="/facts/Euclidean_tilings_by_convex_regular_polygons/tR5CQACy">uniform planar tilings</a> and their dual uniform tilings. The two lower rows coincide and are to scale:</p> A comparison of 65 <i>k</i> uniform tilings in <a href="/facts/Euclidean_tilings_by_convex_regular_polygons/tR5CQACy">uniform planar tilings</a> and their dual uniform tilings. The two lower rows coincide and are to scale. <ul><li><a href="https://onlinelibrary.wiley.com/doi/abs/10.1002/(SICI)1099-0526(199907/08)4:6%3C31::AID-CPLX7%3E3.0.CO;2-D">Planigon tessellation cellular automata</a> Alexander Korobov, 30 September 1999</li> <li><a href="https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=im&paperid=3749&option_lang=eng"> B. N. Delone, “Theory of planigons”</a>, Izv. Akad. Nauk SSSR Ser. Mat., 23:3 (1959), 365–386</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 2010-09-19. <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> <li id="fn:5"><p>"THE BIG LIST SYSTEM OF TILINGS OF REGULAR POLYGONS". THE BIG LIST SYSTEM OF TILINGS OF REGULAR POLYGONS. Retrieved 2019-08-30. <a href="https://www.biglist-tilings.com/" target="_blank">https://www.biglist-tilings.com/</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><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:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Rusczyk, Richard. (2006). Introduction to geometry. Alpine, CA: AoPS Inc. ISBN 0977304523. OCLC 68040014. <a href="0977304523" target="_blank">0977304523</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>"n-Uniform Tilings". probabilitysports.com. Retrieved 2019-06-21. <a href="http://probabilitysports.com/tilings.html" target="_blank">http://probabilitysports.com/tilings.html</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>k-uniform tilings by regular polygons Archived 2015-06-30 at the Wayback Machine Nils Lenngren, 2009 [verification needed] <a href="http://www2.math.uu.se/research/pub/Lenngren1.pdf" target="_blank">http://www2.math.uu.se/research/pub/Lenngren1.pdf</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>"11,20,39,33,15,10,7 - OEIS". oeis.org. Retrieved 2019-06-26. <a href="https://oeis.org/search?q=11,20,39,33,15,10,7&sort=&language=&go=Search" target="_blank">https://oeis.org/search?q=11,20,39,33,15,10,7&sort=&language=&go=Search</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>"Tessellation catalog". zenorogue.github.io. Retrieved 2022-03-21. <a href="https://zenorogue.github.io/tes-catalog/?c=k-uniform/" target="_blank">https://zenorogue.github.io/tes-catalog/?c=k-uniform/</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>"n-Uniform Tilings". probabilitysports.com. Retrieved 2019-06-21. <a href="http://probabilitysports.com/tilings.html" target="_blank">http://probabilitysports.com/tilings.html</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>J. E. Soto Sánchez, On Periodic Tilings with Regular Polygons, PhD Thesis, IMPA, Aug 2020. <a href="http://chequesoto.info/thesis.html" target="_blank">http://chequesoto.info/thesis.html</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>"n-Uniform Tilings". probabilitysports.com. Retrieved 2019-06-21. <a href="http://probabilitysports.com/tilings.html" target="_blank">http://probabilitysports.com/tilings.html</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>"n-Uniform Tilings". probabilitysports.com. Retrieved 2019-06-21. <a href="http://probabilitysports.com/tilings.html" target="_blank">http://probabilitysports.com/tilings.html</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>"Tessellation catalog". zenorogue.github.io. Retrieved 2022-03-21. <a href="https://zenorogue.github.io/tes-catalog/?c=k-uniform/" target="_blank">https://zenorogue.github.io/tes-catalog/?c=k-uniform/</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>J. E. Soto Sánchez, On Periodic Tilings with Regular Polygons, PhD Thesis, IMPA, Aug 2020. <a href="http://chequesoto.info/thesis.html" target="_blank">http://chequesoto.info/thesis.html</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> <li id="fn:18"><p>"n-Uniform Tilings". probabilitysports.com. Retrieved 2019-06-21. <a href="http://probabilitysports.com/tilings.html" target="_blank">http://probabilitysports.com/tilings.html</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li> </ol>