tag:blogger.com,1999:blog-71567716175076204922024-03-24T16:33:01.837-07:00Weave AnythingWeaving baskets by computerJames Malloshttp://www.blogger.com/profile/00763341541407040741noreply@blogger.comBlogger295125tag:blogger.com,1999:blog-7156771617507620492.post-161406523667328912024-03-18T12:41:00.000-07:002024-03-18T12:41:56.327-07:00Coding corrugated baskets<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjqhxKLHt5aWXaUTrNRSGuvOpJ8-7efFj4S5KsvQBxLO7Tw8veTXxTcOO0mjZO6_pzVzjpMUoF86F3WklifppqGJeXZ4ib_ClWWEobptulF8ck4YQDSOvykh_WAl4EXycIiVgcBvmaDmshBV5z4RakitxKwLY-CS5QtAK_LIbMrLNMmauI1yjUSLte363c/s1762/C065DC7F-5826-4210-9819-CC47EFE4B986_4_5005_c.jpeg" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="600" data-original-height="214" data-original-width="1762" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjqhxKLHt5aWXaUTrNRSGuvOpJ8-7efFj4S5KsvQBxLO7Tw8veTXxTcOO0mjZO6_pzVzjpMUoF86F3WklifppqGJeXZ4ib_ClWWEobptulF8ck4YQDSOvykh_WAl4EXycIiVgcBvmaDmshBV5z4RakitxKwLY-CS5QtAK_LIbMrLNMmauI1yjUSLte363c/s600/C065DC7F-5826-4210-9819-CC47EFE4B986_4_5005_c.jpeg"/></a></div>
<p>Corrugated baskets present two different coding problems: how one might mark assembly information directly on a custom-punched tape, and how one might write down (on a piece of paper) assembly information for a tape pre-marked with a standard consecutive numbering. The latter is the topic here.</p>
<p>In the standard consecutive numbering, orienting the tape from the smaller to the larger numbers, each segment of the tape (i.e., a segment between crossings) should be marked near its terminal end as shown above, so that the number naturally becomes a sub-address for the nearest crossing.<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhmntGC47a1QcuUoh38GvvY4L8zYECmGI-D5A0FKiPPpvZf403AhRqx8BD6Ttnh00DXYuAsgr3QMTiwkLjGFvJnRlCOrda7hQux1tmA6bzWQ_LzwkwvnRwI-KaYyHapOmrjyDqz6HSOrNVPHN0k3nMJqgLQOF-ndh6ywkDYFStkctq9kGYsiLfrJVj97Dw/s974/369A9E87-39FA-4E17-92AD-49411B346FBC_1_201_a.jpeg" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="929" data-original-width="974" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhmntGC47a1QcuUoh38GvvY4L8zYECmGI-D5A0FKiPPpvZf403AhRqx8BD6Ttnh00DXYuAsgr3QMTiwkLjGFvJnRlCOrda7hQux1tmA6bzWQ_LzwkwvnRwI-KaYyHapOmrjyDqz6HSOrNVPHN0k3nMJqgLQOF-ndh6ywkDYFStkctq9kGYsiLfrJVj97Dw/s400/369A9E87-39FA-4E17-92AD-49411B346FBC_1_201_a.jpeg"/></a></div></p>
<p>All the crossings in a corrugated basket look the same. Rotated so that the converging quadrant is at the bottom, as shown above, they all look like the diagram. The crossing has a binomial address (represented here generically by 0,1): an even sub-address on the left and an odd sub-address on the right. (Those positions might just as well be reversed, but, to make things easier for the basket maker, we will always choose a basepoint for the counting that makes the above diagram correct.)</p>
<p>The oriented geodesic path is always a boundary between white and a tinted region. At each crossing, white switches sides; “black” also switches sides but its tint changes as well. For example, in the diagram, the 0 strand, before this crossing, has white on the right, and the dark tint on the left. After this crossing, white is on the left and the light tint is on the right. Also, the sub-address will have incremented by one and so changed parity.</p>
<p>If we are going to be walking along the path a lot in the oriented direction, it might be good to have the right leg shorter than the left because there is always going to be either a dale on the left or a hill on the right.</p>
<p>0 and 1 in the diagram merely symbolize even and odd sub-addresses, their relative magnitude is unknown: either could be the bigger sub-address and thus the later strand. If, say, the even sub-address is bigger, then we are colliding with the left side of the earlier strand and are potentially closing a dale region, conversely, if the odd address is bigger, then we are colliding with the right side of the earlier strand, and are potentially closing a hill region.</p>
<p>If we always write the full address in "even-odd" order, it is easy to remember on which side of the smaller sub-address the bigger sub-address is approaching.</p>
James Malloshttp://www.blogger.com/profile/00763341541407040741noreply@blogger.com0tag:blogger.com,1999:blog-7156771617507620492.post-70998875905529625762024-03-07T11:22:00.000-08:002024-03-07T11:22:16.148-08:00The ouroboros splice<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhYoIUi_671Y4rj22UPw9-DWeSLLSDMSafyxe9FDVst-2UNIf5DrqkQpWNwfTnqqvC8IeKCGukEB8GwjllrMRZkW4U16atwqXJjFPQ6We9ueT_HtQU5b-1K7m_1dASkeIxR2L5GZH0L3gmhIYOiBi-c77BnmNfxvth8-slj__-FLmgOik88LX0-_0jae9M/s2122/130AB19B-7E55-45DB-9158-3FE712E217FD_1_201_a.jpeg" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="1043" data-original-width="2122" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhYoIUi_671Y4rj22UPw9-DWeSLLSDMSafyxe9FDVst-2UNIf5DrqkQpWNwfTnqqvC8IeKCGukEB8GwjllrMRZkW4U16atwqXJjFPQ6We9ueT_HtQU5b-1K7m_1dASkeIxR2L5GZH0L3gmhIYOiBi-c77BnmNfxvth8-slj__-FLmgOik88LX0-_0jae9M/s400/130AB19B-7E55-45DB-9158-3FE712E217FD_1_201_a.jpeg"/></a></div>
<p>When finishing a unicursal basket a splice is needed to connect the ribbon's head to its tail. The method advocated here is simple, and is sturdy enough for cardstock baskets. The ribbon is cut just long enough at both ends to doubly-cover the first crossing and the last crossing as shown above. The cutting is not done at 90°, but at the angle determined by the two outer notch positions, as seen in the photo below. The first crossing is unlocked, allowing the ribbon’s tail to be threaded under both the last and first crossing. Those crossings are then locked with a doubled thickness of ribbon underneath. Made this way, the splice is scarcely noticeable in the finished basket.</p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgD9sah57YnbRHloiqkfgvOuv7Tlgj61c3hURXY1swfeJ-pfC-GNfOCg17Fbn-pH-I1drvjLnNrrBO0CwoKjBzNRewYdhT_tL9Q0DlhdBV4c-iGC542h2T77EbA_U_drn7xetkfkkIPcn4Cdc2B8YzfsGVKbXVpz_K__geCOG0I_6t8u38tIgK6jgTFiUc/s1000/ourobouros.jpg" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="1000" data-original-width="1000" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgD9sah57YnbRHloiqkfgvOuv7Tlgj61c3hURXY1swfeJ-pfC-GNfOCg17Fbn-pH-I1drvjLnNrrBO0CwoKjBzNRewYdhT_tL9Q0DlhdBV4c-iGC542h2T77EbA_U_drn7xetkfkkIPcn4Cdc2B8YzfsGVKbXVpz_K__geCOG0I_6t8u38tIgK6jgTFiUc/s400/ourobouros.jpg"/></a></div>James Malloshttp://www.blogger.com/profile/00763341541407040741noreply@blogger.com0tag:blogger.com,1999:blog-7156771617507620492.post-23649580483751383032024-03-07T08:43:00.000-08:002024-03-07T11:27:39.832-08:00Locked crossing profiles for cardstock<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh7-mivKbK2HSJyHSoKgpl3GaqUgfzReP0m5dtP8t71Uvu49OIIB5ZYawekeP0g1p_PKlPyK4wnnfWxvCoYgn2GWxr8YoxGsENPKW-q_Z20sU92TtFB7bkdYpDZj0g_G2sTZ-bAoZVfwEUCZ-3J262nP54uLUwHyqf07I1ekJeVWxkx77V_zG3NHT6HFUo/s1477/1708F86E-09C8-461A-8B20-1A69D618F3AA_1_201_a.jpeg" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="1362" data-original-width="1477" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh7-mivKbK2HSJyHSoKgpl3GaqUgfzReP0m5dtP8t71Uvu49OIIB5ZYawekeP0g1p_PKlPyK4wnnfWxvCoYgn2GWxr8YoxGsENPKW-q_Z20sU92TtFB7bkdYpDZj0g_G2sTZ-bAoZVfwEUCZ-3J262nP54uLUwHyqf07I1ekJeVWxkx77V_zG3NHT6HFUo/s400/1708F86E-09C8-461A-8B20-1A69D618F3AA_1_201_a.jpeg"/></a></div>
<p>I have been using the above notch pattern for 2cm-wide 65-lb cardstock weaving elements cut on a Cricut Maker 3 (courtesy of the DC Public Libray's FabLab.) The 7.5° alternately plus or minus rotation of the crossed diagonals makes locked crossings that are 75°/105°. These can be used to corrugate crossings that are nominally 90°/90° (tabby weave) as deficit/excess, as well as crossings that are nominally 60°/120° (kagome weave) as excess/deficit.</p>James Malloshttp://www.blogger.com/profile/00763341541407040741noreply@blogger.com0tag:blogger.com,1999:blog-7156771617507620492.post-14476215749656390352023-08-11T13:09:00.002-07:002023-08-11T13:09:31.122-07:00Notch detail for locked crossings in aluminum<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEimMx7AhBhnTsEPziRvutcGeFB8HdtJsCJL9qERUP1MlFU824WsaKLsZ-hFe3N-fS0zTn8abUYhldFSZjntsV7y6YIsp3Sd8uI-PoMY2ftKgFUz6x49B9dx69vFc1-6WBpu6dG6dB7OqS_QrFQaNvkCJy4i6umEYv0EZUFwZme-J7TCaLhrnRhzvoV4igg/s2044/73DDAFEA-818E-41C7-9072-107B4106BF05_1_201_a.jpeg" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="1835" data-original-width="2044" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEimMx7AhBhnTsEPziRvutcGeFB8HdtJsCJL9qERUP1MlFU824WsaKLsZ-hFe3N-fS0zTn8abUYhldFSZjntsV7y6YIsp3Sd8uI-PoMY2ftKgFUz6x49B9dx69vFc1-6WBpu6dG6dB7OqS_QrFQaNvkCJy4i6umEYv0EZUFwZme-J7TCaLhrnRhzvoV4igg/s400/73DDAFEA-818E-41C7-9072-107B4106BF05_1_201_a.jpeg"/></a></div>
<p>The notch profile above works for aluminum sheet metal having a springy temper, such as is widely sold to the building trades as flashing (9 mil thickness,) trim stock (18 mil thickness,) and gutter stock (27 mil thickness.)</p>
<p>For example, with 9 mil (0.009") aluminum flashing a width of 1.5", tangent circles of 0.125" diameter, and outer rounds of 0.5" diameter are suitable for weaving with locked crossings.</p>
<p>If the weavers are to cross at an angle other than 90-degrees, the diagonals of the crossing rhombus remain perpendicular to each other, but rotate together to align with the crossing's planes of symmetry. The notches translate to maintain tangency.</p>James Malloshttp://www.blogger.com/profile/00763341541407040741noreply@blogger.com0tag:blogger.com,1999:blog-7156771617507620492.post-77536025724741786162023-08-02T13:38:00.001-07:002023-08-02T13:38:33.403-07:00Hill-and-valley weaving of voxel object surfaces<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhBtS3I5MX8vYzMeoMTYUztZAaYLxZpQFyeaHu7_GChBPyKSwHhM0bldICIIhQq2t65uKtjipDQdEhfLMqVZzRjgnb2mfBpVTJ9mNoRzumIsxCR3Ba13HmJ_BUn7EXNLDPiaxStfOzir4VGJl3FzzwLVF0b7CDMIUzAAtyLo3MF5qzBewrLQ0TRnGIYpow/s1066/92BA3FC5-DEC9-4F92-ABA6-6CBA2948E93E_1_201_a.jpeg" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="959" data-original-width="1066" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhBtS3I5MX8vYzMeoMTYUztZAaYLxZpQFyeaHu7_GChBPyKSwHhM0bldICIIhQq2t65uKtjipDQdEhfLMqVZzRjgnb2mfBpVTJ9mNoRzumIsxCR3Ba13HmJ_BUn7EXNLDPiaxStfOzir4VGJl3FzzwLVF0b7CDMIUzAAtyLo3MF5qzBewrLQ0TRnGIYpow/s400/92BA3FC5-DEC9-4F92-ABA6-6CBA2948E93E_1_201_a.jpeg"/></a></div>
<p>Hill-and-valley weaving must follow the medial of a bipartite map. The skeletal surface graphs of voxelized objects (example above) are bipartite. Pasting the truchet tile (below) onto each square face (with corner colors matching) gives a face 3-coloring of the medial graph (the new edges trace the boundaries between colors).</p>
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<p>For example, for a single voxel, i.e., a cube, decorating its six faces with the truchet tile, shows that its 3-colored medial is a cuboctahedron (below) with its square faces colored 'saddle' and its triangular faces colored alternately 'hill' and 'valley'.</p>
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<p>A spherical cuboctahedron (see below; art by Watchduck) is four great circles in an arrangement of maximum symmetry, so the smallest angles between these planes is equal to the dihedral angles of the tetrahedron, or approximately 70.5288 degrees.</p>
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<p>To weave the cuboctahedron weave 'flat' (i.e., without hills and valleys) weavers must cross each other such that the internal angles of the triangular faces are about 70.5288 degrees, a bit wider than the 60 degrees these angles would measure on the plane. To make the triangular faces into hills and valleys, and, correspondingly, the square faces into saddles, we need even wider internal angles in the triangular faces. Below is a hill-and-valley weaving of the cuboctahedron with 100 degree internal angles in the triangular faces. Because of the hills on alternate triangles of the cuboctahedron, the basket appears strongly tetrahedral.</p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhNYQ-8W-tSgm9SanTBi25Zs7SWnL4kRKXwWUvFp-yCtHYdXzidy5yg8cdyK8CTKs7XSNP3xKqkheyI3aUtTx-CQ64L_Kjz3dGWFZPKH34a-3MTdnjwLrmCs_RiNKbjY7kmnd5CKkU8M1qxmdtY69olSBMGGYaSK_0bBrPpre-DBpG_Cll5UX2sFSzd0HY/s1280/hill-and-valley%20cuboctahedron.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="1280" data-original-width="1280" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhNYQ-8W-tSgm9SanTBi25Zs7SWnL4kRKXwWUvFp-yCtHYdXzidy5yg8cdyK8CTKs7XSNP3xKqkheyI3aUtTx-CQ64L_Kjz3dGWFZPKH34a-3MTdnjwLrmCs_RiNKbjY7kmnd5CKkU8M1qxmdtY69olSBMGGYaSK_0bBrPpre-DBpG_Cll5UX2sFSzd0HY/s400/hill-and-valley%20cuboctahedron.png"/></a></div>James Malloshttp://www.blogger.com/profile/00763341541407040741noreply@blogger.com0tag:blogger.com,1999:blog-7156771617507620492.post-23074148479814593962023-07-28T09:46:00.001-07:002023-07-28T09:52:32.345-07:00More on the math of hill-and-valley weaving<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj78KnMqXj5O4-Zkfg83Evjw2LDAsFjYh9WQ0tJ9yOydhgN7RSNukedX6SRd9exvn66hObFM1rVOND_Ae-WIfrULVbTOA-Cl1iIsq_uGTUGuv1M-sbHm0Kc4SmZC891EotgXn47M6Odm37tGuGoAwNWd0Xl58erhTJ1gVElo7QSULAQRZKg6We5rssiKIQ/s1183/Primal%20edge.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="1183" data-original-width="1183" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj78KnMqXj5O4-Zkfg83Evjw2LDAsFjYh9WQ0tJ9yOydhgN7RSNukedX6SRd9exvn66hObFM1rVOND_Ae-WIfrULVbTOA-Cl1iIsq_uGTUGuv1M-sbHm0Kc4SmZC891EotgXn47M6Odm37tGuGoAwNWd0Xl58erhTJ1gVElo7QSULAQRZKg6We5rssiKIQ/s400/Primal%20edge.png"/></a></div>
<p>Above, the quadrilateral domain of a map edge.</p>
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<p>Above, the hill-and-valley coloring of the quadrilateral domain of a map edge via Mrs. Stott's expansion, Me(Me(m)). Black = Up; White = Down; Pink = Saddle.</p>
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<p>Above, the hill-and-valley coloring of the quadrilateral domain of a map edge via Ring, Me(Su(m)). Black = Up; White = Down; Pink = Saddle.</p>
<p>From the <a href="http://weaveanything.blogspot.com/2023/07/the-math-of-hill-and-valley-weaving.html">previous post</a> it is clear that we must start from a bipartite map (a bipartite graph embedded in a surface) in order to design a basket that can wear a hill-and-valley corrugation of its surface. There are many such maps. For example, any quadrangulation of the sphere is a bipartite map. Also, since the 3D cartesian grid of points with integer coordinates can be bicolored (e.g., simply color each vertex according to the parity of x+y+z), the surfaces of polycubes and voxelized objects (image below) are bipartite maps regardless of their (necessarily orientable) topology.</p>
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<p>If we must start from a non-bipartite map, the map operation Radial (a.k.a., Quadrangulation), which doubles the number of edges in the map, always yields a bipartite map--the partition of vertices being respectively, the original vertices and the new vertices that are added in the center of each original face. Taking the Medial of this map (at the cost of another doubling of edges) yields a weavable pattern. So the entire chained operation on a non-bipartite map m is Me(Ra(m)). Since Medial gives the same result whether applied to a map or its dual; and Ra() and Me() of any given map are in fact duals, Me(Ra(m)) = Me(Me(m)) = Me^2(m). This compound operation, "the medial of the medial", or "medial squared", is also known as Mrs. Stott's expansion.</p>
<p>Another way to make a given map bipartite is to subdivide the edges, in other words, insert degree-2 vertices in the middle of each original edge. So the compound map operation needed in this case is Me(Su(M)), a map operation which I have elsewhere called Ring ("Extra ways to see: An artist's guide to map operations." Hyperseeing: Proceedings of ISAMA 2011, pages 111–121, Summer 2011. See chart below.) because it generates weave patterns associated with chainmail.</p>
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James Malloshttp://www.blogger.com/profile/00763341541407040741noreply@blogger.com0tag:blogger.com,1999:blog-7156771617507620492.post-17834994856518874872023-07-27T13:21:00.001-07:002023-07-27T13:21:39.998-07:00The math of hill-and-valley weaving<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjxbfrUw5qoU4EuxA18iCJNMKGmyG1CZZi7X4jLCmxUgtxzwMLuDZWhEK-bJf7b3aWkVFPRJQ-_V-b-m6Ltd2Y3nocP6iV3nE81-II0kU6QDTyop8G9Y3ob-Bg-rDA-wZQGJkmP6cupKQ80wC5AXIY4aK8VY5DeUMfHaGetwH9v4r4tlDcApYAfukf7hZI/s1280/IMG_2297.jpeg" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="896" data-original-width="1280" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjxbfrUw5qoU4EuxA18iCJNMKGmyG1CZZi7X4jLCmxUgtxzwMLuDZWhEK-bJf7b3aWkVFPRJQ-_V-b-m6Ltd2Y3nocP6iV3nE81-II0kU6QDTyop8G9Y3ob-Bg-rDA-wZQGJkmP6cupKQ80wC5AXIY4aK8VY5DeUMfHaGetwH9v4r4tlDcApYAfukf7hZI/s400/IMG_2297.jpeg"/></a></div>
<p>A weaving pattern for any graph drawn on a surface can be derived by taking the medial of the graph, a construction that fills the rhombic domain of each edge in the graph with black and white regions as below, (original edge in red, original vertices in green):</p>
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<p>That construction suffices for ordinary weaving, but not for hill-and-valley texturing because the black regions, which correspond to the locations of the original vertices, must be partitioned into alternating hills and valleys. So we need to start with a bipartite graph (for example, see below: each edge connects a green vertex to a blue vertex), then the medial will inherit the needed bipartition of the black regions into hills and valleys from the bipartition of the original vertices.</p>
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<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi4LixcCt6XNi3ihpKLTaRE-SIAvvPHS9l_aI0PIYjTkULvYH9814fghuTyn2iauIo2euepV0N1C1Ww_GESM1CFfvQdY0ytlmNuP72FIjXORKbuJNmfSRWDdXlmkYDGr-ju1tj1TJkL_GDkKbPIzNWuSkofH-fEE70NrzazxOlpldrhEsjLw_k-NhuoBT8/s1766/C2CF81D7-248F-459E-A28E-E1BE70A84C02_1_201_a.jpeg" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" height="400" data-original-height="1766" data-original-width="1417" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi4LixcCt6XNi3ihpKLTaRE-SIAvvPHS9l_aI0PIYjTkULvYH9814fghuTyn2iauIo2euepV0N1C1Ww_GESM1CFfvQdY0ytlmNuP72FIjXORKbuJNmfSRWDdXlmkYDGr-ju1tj1TJkL_GDkKbPIzNWuSkofH-fEE70NrzazxOlpldrhEsjLw_k-NhuoBT8/s400/C2CF81D7-248F-459E-A28E-E1BE70A84C02_1_201_a.jpeg"/></a></div>James Malloshttp://www.blogger.com/profile/00763341541407040741noreply@blogger.com0tag:blogger.com,1999:blog-7156771617507620492.post-14592028623765843012023-07-27T10:34:00.001-07:002023-07-27T10:34:57.237-07:00Hill-and-valley weaving with locked crossings<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh72VwedCwF2CUTVh6m3c0_hpLbta1Ap6M_jrm7MhfkOWt7gMB-BJFTMzctayO5iPbfvcBa4bEFjmi9nwqAs_QoxX2ghSJyTFDLlD1W3lBJuXF0DDoqfjUlR4X3LH5K-XqS_b5cDbTJ1wxOJQz26nVK2zxf_rPCc0sFeL9KHMv3zC8rwlapve9kDDR4BZo/s3024/C88E6748-669B-47E7-B93B-A364AB42D805.heic" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="3024" data-original-width="3024" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh72VwedCwF2CUTVh6m3c0_hpLbta1Ap6M_jrm7MhfkOWt7gMB-BJFTMzctayO5iPbfvcBa4bEFjmi9nwqAs_QoxX2ghSJyTFDLlD1W3lBJuXF0DDoqfjUlR4X3LH5K-XqS_b5cDbTJ1wxOJQz26nVK2zxf_rPCc0sFeL9KHMv3zC8rwlapve9kDDR4BZo/s400/C88E6748-669B-47E7-B93B-A364AB42D805.heic"/></a></div>
<p>Corrugated surfaces can be woven using <a href="https://weaveanything.blogspot.com/2021/03/weaving-with-locked-crossings.html?m=0">locked crossings</a> if the crossings are locked at some angle other than 90 degrees. The sample above was woven using crossings locked at 100 degrees (80 degrees on the acute side), with the distance between crossings fixed at 3x the width. The resulting hill-and-valley corrugation adds a lot of stiffness to the woven panel. The 1-inch wide cardstock weavers were made by hand using a paper cutter and 0.25-inch diameter hand punch. Below is the pattern used to cut the weavers.</p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjS4CW7L4NfX2gkbZ-o8CFfVjwxpJLZ84s35NocUivYaAGyE966hwtPgQL7DHV1NLfu8AP6NUOsCyeO9DG4bFk1mVnxuawaZ0aiV1NZjJ1oSdSGcOVPwTklJ1I9d4_jEWCg6e5rI6tq7tUMHasP5y0hkcRznUWmqTIZlKgHMhJ4ekZkeBZWcPKOPrGWFYc/s1535/CFA8CA63-866A-4755-BB17-ECCE3490EDCC_1_201_a.jpeg" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" height="400" data-original-height="1535" data-original-width="1140" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjS4CW7L4NfX2gkbZ-o8CFfVjwxpJLZ84s35NocUivYaAGyE966hwtPgQL7DHV1NLfu8AP6NUOsCyeO9DG4bFk1mVnxuawaZ0aiV1NZjJ1oSdSGcOVPwTklJ1I9d4_jEWCg6e5rI6tq7tUMHasP5y0hkcRznUWmqTIZlKgHMhJ4ekZkeBZWcPKOPrGWFYc/s400/CFA8CA63-866A-4755-BB17-ECCE3490EDCC_1_201_a.jpeg"/></a></div>James Malloshttp://www.blogger.com/profile/00763341541407040741noreply@blogger.com0tag:blogger.com,1999:blog-7156771617507620492.post-73811765723531273292023-06-21T13:47:00.000-07:002023-06-21T13:47:02.374-07:00"Bespoke tet" and "Bespoke cube"<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUhv47ab5fs95NWAN4QaZ7wwyu6pPF_SC85X7eNn-6hCFVkVx_jKSoJwMce4Ii47nnbIlJDQZSyx4uFKZpqj4QkkXhM_5WHKXAaIEauzAyjAeym2t2DZ47expVjzlcugXd-LqAbGH1gtSyYoYnR1QSdAcn4Sg-oOPtV6Lr0ABC7N5kk1pCx9Jp6K7F000/s4032/tet-arrangement.HEIC" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" height="400" data-original-height="4032" data-original-width="3024" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUhv47ab5fs95NWAN4QaZ7wwyu6pPF_SC85X7eNn-6hCFVkVx_jKSoJwMce4Ii47nnbIlJDQZSyx4uFKZpqj4QkkXhM_5WHKXAaIEauzAyjAeym2t2DZ47expVjzlcugXd-LqAbGH1gtSyYoYnR1QSdAcn4Sg-oOPtV6Lr0ABC7N5kk1pCx9Jp6K7F000/s400/tet-arrangement.HEIC"/></a></div><div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgE0EPeKnXfFMwEFIe0mWsTO2JXfCBUZ_uuiyS98HdvPDWrkoUNHaSTZYo7fVM9TNc2pPCxoWcGYbZi2VdZjvJ1-rVXN8ln-RL9uycy0u3Dga_gBcbvGkHKt1c1ddYG2OhhIeXUWU4KpmuGn1xKu8aydI1JSD2uIFWJh2kPple9n1gpdonGiO1JvofYMNE/s4032/tet-woven.HEIC" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" height="400" data-original-height="4032" data-original-width="3024" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgE0EPeKnXfFMwEFIe0mWsTO2JXfCBUZ_uuiyS98HdvPDWrkoUNHaSTZYo7fVM9TNc2pPCxoWcGYbZi2VdZjvJ1-rVXN8ln-RL9uycy0u3Dga_gBcbvGkHKt1c1ddYG2OhhIeXUWU4KpmuGn1xKu8aydI1JSD2uIFWJh2kPple9n1gpdonGiO1JvofYMNE/s400/tet-woven.HEIC"/></a></div>
<p>As a demonstration of coding unit weaving in a Cricut layout, the tetrahedron model above, which has a variety of edge lengths and is that sense 'bespoke', is woven together as indicated by its layout.</p>
<p>"Bespoke cube," below, is slightly more complicated model, but still scaled to fit vertically on a 12" long Cricut mat.
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgrnYTTth20Jv6CEuphtN63oouBYJ2XfAibw6GdoKnl2I-MNOPCUjTj4qJyDNLukQZmKvZSYpyRdnxw6nrVH3vVAzZaoVnhNyJN2_9J6-o_jilrqFudQi1EJXqKl8tC0cr6_zI5aUUwzloXaZc9RqyqgW7hsEKuW0_W2zoJh61C6djTW8_4qyjQ_zZW-Eo/s3981/cube-arrangement.heic" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="1249" data-original-width="3981" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgrnYTTth20Jv6CEuphtN63oouBYJ2XfAibw6GdoKnl2I-MNOPCUjTj4qJyDNLukQZmKvZSYpyRdnxw6nrVH3vVAzZaoVnhNyJN2_9J6-o_jilrqFudQi1EJXqKl8tC0cr6_zI5aUUwzloXaZc9RqyqgW7hsEKuW0_W2zoJh61C6djTW8_4qyjQ_zZW-Eo/s400/cube-arrangement.heic"/></a></div><div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEia76dJEfHtaPi7anrMFedLzWtwD_tADsNkxK6CCGjQqZdDoep2Ry1FXxgnVIbAgmgsGJR_vZma7G-2z612Ew48UQnCDnwwG39VIjVLzg2reu5RK75Da6LNWHISM6bmVhd9_2fTwnVrWn5yMfrWBAX_ZeasJ2cyinRXoiAy_3MzUACF9KC45ujXeaPFg0A/s3024/cube-woven.heic" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="3024" data-original-width="3024" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEia76dJEfHtaPi7anrMFedLzWtwD_tADsNkxK6CCGjQqZdDoep2Ry1FXxgnVIbAgmgsGJR_vZma7G-2z612Ew48UQnCDnwwG39VIjVLzg2reu5RK75Da6LNWHISM6bmVhd9_2fTwnVrWn5yMfrWBAX_ZeasJ2cyinRXoiAy_3MzUACF9KC45ujXeaPFg0A/s400/cube-woven.heic"/></a></div>James Malloshttp://www.blogger.com/profile/00763341541407040741noreply@blogger.com0tag:blogger.com,1999:blog-7156771617507620492.post-43733832617412947002023-06-20T14:25:00.000-07:002023-06-20T14:25:03.718-07:00Coding unit weaving in paper cutting<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgyZ17Aqo_QkFTK8hy-DDVISJOUL5dBhO_WLXOnw1bXW9qbOkaTqG4kT-IjPkUxWf6i3qezjbo0vf2AaVuAF4fQgzt61zk-llam9j2EhxS0a_CBIwDCsnGXnUEX2j0W6TRZPJw25yXvHyP7xdCvxWhl7e4KO5Xu6x_O6HFabisXqa5xMi0-7ePsJvMY9jA/s1280/coding_unit-weaving.jpeg" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" height="400" data-original-height="1280" data-original-width="884" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgyZ17Aqo_QkFTK8hy-DDVISJOUL5dBhO_WLXOnw1bXW9qbOkaTqG4kT-IjPkUxWf6i3qezjbo0vf2AaVuAF4fQgzt61zk-llam9j2EhxS0a_CBIwDCsnGXnUEX2j0W6TRZPJw25yXvHyP7xdCvxWhl7e4KO5Xu6x_O6HFabisXqa5xMi0-7ePsJvMY9jA/s400/coding_unit-weaving.jpeg"/></a></div>
<p>A digital cutting machine such as Cricut is a practical way to make unit-weavers (such as twogs) in all different lengths, allowing more customized shapes to be woven. There is then the problem of how to keep the different lengths sorted and properly sequenced in the weaving process. I find that narrow paper bridges (the J-shapes) in the model above, can keep twogs sequenced in a way that also codes the weaving moves. For example, reading the model above from top to bottom, the pattern is natural seen as open-left, open-right, close-left, close-right, (knowing that the last move closes up the weaving, 'close-right' is deduced as the only possibility) a weaving pattern that builds a tetrahedron. In fact, the short paper bridges would hardly allow any other pattern to be woven.</p> James Malloshttp://www.blogger.com/profile/00763341541407040741noreply@blogger.com0tag:blogger.com,1999:blog-7156771617507620492.post-50099557441694487492023-06-15T09:27:00.002-07:002023-06-15T09:27:40.569-07:00Adams "World in a Square II" on the tetrakis hexahedron<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiaQaSS_BTSazJKNIVOhvSBAx35jmVRhNltKMi6GGPPpoGt8Res4bfm19xxXxfQqptMcMQBhq4GTjkwVVQ_qBOO2TPTNwn4C-JpZP_gQ0kWtQcLAFNQnKZUP3Vl4eZakK6_LWXERNj21vI1XvAHUZ--X6OBBPIkAaGEXDFjqW85lzQGjrw50-DFwzPB/s2142/Screen%20Shot%202023-06-15%20at%2010.29.27%20AM.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="546" data-original-width="2142" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiaQaSS_BTSazJKNIVOhvSBAx35jmVRhNltKMi6GGPPpoGt8Res4bfm19xxXxfQqptMcMQBhq4GTjkwVVQ_qBOO2TPTNwn4C-JpZP_gQ0kWtQcLAFNQnKZUP3Vl4eZakK6_LWXERNj21vI1XvAHUZ--X6OBBPIkAaGEXDFjqW85lzQGjrw50-DFwzPB/s400/Screen%20Shot%202023-06-15%20at%2010.29.27%20AM.png"/></a></div>
<p>The Adams "World in a Square II" map projection tiles with itself to form a seamlessly periodic Earth. We use identically printed and cut pieces that are triangular after folding, but place rubber bands on only half of the triangles. Each triangle has a complete copy of the Earth's surface, the Western Hemisphere is seen on the face with interwoven flaps, the Eastern Hemisphere is seen on the smooth face.</p>
<p>(The Prime Meridian of the Adams tile has been shrunk to 89% of its true length, resulting in folded triangles that are isoceles and slightly acute to approximate the faces of a tetrakis hexahedron. This is, of course, a fudge to avoid calculating a proper conformal projection, what might be called "World in a (slightly rhombic) Square.") </p>
<p>Since there is never a need to reopen (unweave) the variety of triangles that wear the rubber bands, these can show their smooth side to the weaver, and thus show their Eastern Hemisphere to the weaver.</p>
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<p>A quadrangulation can be reduced to an incrementally smaller quadrangulation by a face contraction: shrinking either diagonal dimension of one face to zero. The inverse operation, face expansion, incrementally grows the quadrangulation by opening a path of length 2 into a quadrilateral. Since there is only one sort of face on the cube, and the two diagonals of each face are essentially the same, there is only one way to decrement a cube by face contraction, so there is a particular shape that we can call Cube-1. Likewise, there is only one sort of path of length 2 on the edges of a cube, so there is only one way to increment a cube by face expansion, so there is a particular shape that we can call Cube+1. (The fact that the "squares" in this model are actually composed of four hinged triangles, lets us play this freely with solid geometry.)</p>
<p>In the strip of photos above, we start with Cube+1, then contract 2 faces to reach Cube-1. So what happened to Cube? To reach Cube from Cube-1, we must go back to Cube+1 and contract a single face. This indirection is necessitated by the limits of simulating topological transformations via physical folding. If our ambition was only to reach Cube from Cube-1, then we stuck with one quad's worth of unnecessary surface riding piggy-back on Cube.</p>
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<p>Reducing Cube-1 down to a one-faced quadrangulation (photo strip above) proceeds directly because in each step we we are simply folding down a "flap" formed by a vertex of degree-2.</p>James Malloshttp://www.blogger.com/profile/00763341541407040741noreply@blogger.com0tag:blogger.com,1999:blog-7156771617507620492.post-67270245115937072612023-05-24T11:50:00.000-07:002023-05-24T11:50:00.469-07:00The smallest simple planar quadrangulations<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj2D3VRQea4aQgAxEwY7wSf1c1O_i0aOGYHEHMV_RZxLBSRWFhmZyInm5adxyWGAofszEA131ld7zqDw92dqudPW1zFcGAEjJb-am1ruePkFTQQiJrnp2sqZGjwRZj68h-cZHoZhBvfcgyFakgdfPDdXn0uu6oJ4Gnxpo6BTaqRUzkUQl8xS04W7O-X/s870/E1C1E685-FB97-4ED0-BBF6-9443F7FA814C_1_201_a.jpeg" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="870" data-original-width="870" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj2D3VRQea4aQgAxEwY7wSf1c1O_i0aOGYHEHMV_RZxLBSRWFhmZyInm5adxyWGAofszEA131ld7zqDw92dqudPW1zFcGAEjJb-am1ruePkFTQQiJrnp2sqZGjwRZj68h-cZHoZhBvfcgyFakgdfPDdXn0uu6oJ4Gnxpo6BTaqRUzkUQl8xS04W7O-X/s400/E1C1E685-FB97-4ED0-BBF6-9443F7FA814C_1_201_a.jpeg"/></a></div>
<p>The smallest simple planar (aka, spherical) quadrangulation has 3 vertices, so let's call it Q3. Q3 has one face (exemplifying the rule that all planar quadrangulations have 2 more vertices than faces.) Many authorities, the authors of the <a href="http://combos.org/plantri">plantri program</a> included, do not classify Q3 as a quadrangulation because the boundary of its face is not a cycle, but we include it here. There is also a unique simple planar quadrangulation for 4 vertices, and likewise for 5 vertices, so we will call them Q4 and Q5:</p>
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<p>Now we come to a great divide in the realm of simple planar quadrangulations. There are two simple planar quadrangulations with 6 vertices, so we will have to distinguish them as Q6.1 and Q6.2:
</p>
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<p>But that is not the great divide. Q6.1 and Q6.2 are the smallest simple planar quadrangulations with a separating 4-cycle (aka, a non-facial 4-cycle.) A non-facial 4-cycle outlines a locus where we can imagine two separate quadrangulations have been glued face-to-face at a single quad face. Notice that if we glue two planar quadrangulations (SPQ's) together at more than one face, the result takes us outside the SPQ class: the compound is no longer a quadrangulation of a sphere, either by virtue of achieving a higher topology, or having at least one edge that is not embedded in the surface.</p>
<p>In this sense an SPQ with a non-facial 4-cycle is composite. For example, it is easy to see how both Q6.1 and Q6.2 could have been formed by gluing together two copies of Q5 together at a single face.</p>
<p>Of course the business of gluing quadrangulations together can be iterated, but all subsequent attachments must be arranged in a tree-like pattern. If we ever form a closed circuit of SPQ's, we will have made a torus rather than a sphere.</p>
<p>It is very handy that compositions of SPQ's are always tree-like. A tree graph always has at least two leaf-edges that are attached to the rest of the tree at only one point. So a composite SPQ always has at least two leaf components that are attached to the rest of the composition by a single non-facial 4-cycle. If we possess reduction rules that reduce such a component down to its one non-facial 4-cycle, we will discover that the 4-cycle in question is no longer non-facial, so we can move on to another of its (once again, at least two) leaf components. In this way, as <a href="https://arxiv.org/abs/1606.07662">Fuchs and Gellert</a> point out, we never need to operate on a vertex that lies in a non-facial 4-cycle.</p>
<p>Degree-2 vertices are not so useful for enclosing volume, and are easily folded down when they occur; mainly we are only interested in degree-2 vertices when they are necessary stepping stones to growing a less baroque shape. Interest therefore turns to certain more restricted classes of simple planar quadrangulations. Several restricted classes can be generated in <a href="http://combos.org/plantri">plantri</a> and then copy-and-pasted as sparse6 codes to be rendered in 3d at <a href="https://sagecell.sagemath.org/">SageMathCell</a>.</p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiSB3PmkhJ-fEuedF9waPK8rqbFjgzvv6I_bSMGAWUlefYikLGt1iH5-I7JK8FHqc7FpJCPpPEBIpLs18kBIieb3DKS2iX7vOSnvKTDG2sNaUCtu_7IK7Y6ybNrXtJaENhoG0xUja2Fyq0EauvRAvwzz-JzaE3TX7PqR9RvirgfftMPVdYr5yRHijsN/s3222/481EFF38-C08C-49EC-BB4D-E8976099C638_1_201_a.jpeg" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="600" data-original-height="1985" data-original-width="3222" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiSB3PmkhJ-fEuedF9waPK8rqbFjgzvv6I_bSMGAWUlefYikLGt1iH5-I7JK8FHqc7FpJCPpPEBIpLs18kBIieb3DKS2iX7vOSnvKTDG2sNaUCtu_7IK7Y6ybNrXtJaENhoG0xUja2Fyq0EauvRAvwzz-JzaE3TX7PqR9RvirgfftMPVdYr5yRHijsN/s600/481EFF38-C08C-49EC-BB4D-E8976099C638_1_201_a.jpeg"/></a></div>
<p>The Online Encyclopedia of Integer Sequences has enumerations of these SPQ classes:
A113201: simple planar quadrangulations;
A078666: simple planar quadrangulations, minimum degree-3;
A007022: simple planar quadrangulations, 3-connected;
A002880: simple planar quadrangulations, 3-connected, without separating 4-cycles;
<p>Per <a href="https://www.researchgate.net/profile/Atsuhiro-Nakamoto/publication/281733364_Generating_3-connected_quadrangulations_on_surfaces/links/5bdf6c5a92851c6b27a78d79/Generating-3-connected-quadrangulations-on-surfaces.pdf">Nagashima et al.</a>, a simple quadrangulation (>Q5) with no separating 4-cycle is 3-connected; and if it is 3-connected it is also minimum degree-3, so A002880 more generally counts all simple planar quadrangulations with no separating 4-cycles having at least 6 vertices. The count is unaffected by adding "3-connected," or "minimum-degree-3" as constraints.</p>
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James Malloshttp://www.blogger.com/profile/00763341541407040741noreply@blogger.com0tag:blogger.com,1999:blog-7156771617507620492.post-44978387812033736112023-05-03T13:15:00.000-07:002023-05-03T13:15:04.774-07:00Growing by unfolding the 3-connected planar quadrangulations<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiqrKZq_s0mEu7ykeP3zMlloiqFWXnSzVMBn5dbd0axx-Xp4IqepJLgLnSuDRbrdusjrNRRl_uQGanVDA3AMMPgZkbqodAyZsRhF_AHSVCl_WWLSoPTOD5Xjooweeu9rRo21ZD9pDA7aOCRExYJvdB90BtCuS9xBVGO18_Ld_5DGcq0UVVTD9dv20Lv/s1096/A47C2735-754B-4DF2-A50A-27987D5BF674_1_201_a.jpeg" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="364" data-original-width="1096" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiqrKZq_s0mEu7ykeP3zMlloiqFWXnSzVMBn5dbd0axx-Xp4IqepJLgLnSuDRbrdusjrNRRl_uQGanVDA3AMMPgZkbqodAyZsRhF_AHSVCl_WWLSoPTOD5Xjooweeu9rRo21ZD9pDA7aOCRExYJvdB90BtCuS9xBVGO18_Ld_5DGcq0UVVTD9dv20Lv/s600/A47C2735-754B-4DF2-A50A-27987D5BF674_1_201_a.jpeg"/></a></div>
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Building on work by <a href="https://www.sciencedirect.com/science/article/pii/0012365X89901593">Batagelj</a>, <a href="http://users.cecs.anu.edu.au/~bdm/papers/quad2b.pdf">Brinkmann et al. </a> proved that the 3-connected planar quadrangulations (3CPQ's) can be derived from the pseudo-double wheels, W<sub>2k</sub> for k >= 3, via two local expansion operations P<sub>1</sub> and P<sub>3</sub>, illustated above. An example of a pseudo-double wheel, W<sub>14</sub>, is illustrated below.
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P<sub>1</sub> is basically a surgery on the surface. By slitting edges and vertices along any path of three vertices, a quad-shaped wound opens that can be healed with new surface. Applied without restriction, this operation can generate any quadrangulation, but Brinkmann et al. enforce restrictions that make it possible to realize P<sub>1</sub> by unfolding triangulated quadrangles. P<sub>3</sub> can also be realized by a twist unfolding of triangulated quads, popping a 2-sided square into a cube in a single move.
Here is an example that starts from a square (2 faces, and not 3-connected) and ends with 16 faces, and 3-connected.<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiga_QKdhj2nghAPJEgmlEKKY-T1Yq4kqfx5ePNzZ5xqAGU60xRL5a3dN8kdpSOyuwKSTgPv5JgJucfPKEIMfUHysVs4eOP46C14xlM7GBSE31aPxKGjWfOh5FUpFAHf8SZ9DOYaLGhVhJdrqFGz46XD5GPVW1WEj-qWVmcFlqRdQ3dFXlq5nCPK3cu/s1000/2-to-16.jpg" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="848" data-original-width="1000" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiga_QKdhj2nghAPJEgmlEKKY-T1Yq4kqfx5ePNzZ5xqAGU60xRL5a3dN8kdpSOyuwKSTgPv5JgJucfPKEIMfUHysVs4eOP46C14xlM7GBSE31aPxKGjWfOh5FUpFAHf8SZ9DOYaLGhVhJdrqFGz46XD5GPVW1WEj-qWVmcFlqRdQ3dFXlq5nCPK3cu/s400/2-to-16.jpg"/></a></div>
James Malloshttp://www.blogger.com/profile/00763341541407040741noreply@blogger.com0tag:blogger.com,1999:blog-7156771617507620492.post-71659088742588045372022-08-11T14:10:00.007-07:002022-08-12T13:25:42.587-07:00Fold-up baskets and 3-connected planar quadrangulations<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjd0Va_tz3k5E7rk86YseCZsEc5xKIlARkV55comf0ahw9UMjvF5PquH7L1d8qSrBhN0sARuRfNvAmiveLSgBh9XoYcFcBjmlhQSKL9gKJG-N_o_chWOskJHxe29cJShC8eDy-IeSlAfUXl-7PhsNiDSsCFVj5xkpPVi2qLP0eBmdcDNdAr9OpE9KkK/s584/FDB4B51B-4442-4C20-9BEE-D02EB17C5C00_1_201_a.jpeg" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="525" data-original-width="584" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjd0Va_tz3k5E7rk86YseCZsEc5xKIlARkV55comf0ahw9UMjvF5PquH7L1d8qSrBhN0sARuRfNvAmiveLSgBh9XoYcFcBjmlhQSKL9gKJG-N_o_chWOskJHxe29cJShC8eDy-IeSlAfUXl-7PhsNiDSsCFVj5xkpPVi2qLP0eBmdcDNdAr9OpE9KkK/s600/FDB4B51B-4442-4C20-9BEE-D02EB17C5C00_1_201_a.jpeg"/></a></div>
<p>Physical surfaces like paper are more restricted in folding than mathematical surfaces because physical surfaces cannot pass through one another or coincide. Perhaps there are maps on the sphere where no physical (a.k.a., self-avoiding) folding of the corresponding fold-up basket is possible? In limited experiments, it seems more common that no <i>un</i>-folding</i> is possible. An example is the skeletal map of a hexagonal prism. This map is bipartite and polyhedral (planar and 3-vertex connected) so it should have an unfolded configuration containing positive volume; however, when assembled from 45-90-45 triangles in a flat configuration, it is seemingly impossible to fully unfold the surface because the two 6-sided faces (each with 6 right angles around the face center) are too hyperbolic to be happy in the unfolded configuration. </p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-0I2HW9208CfsJ7xDYP19PBtgPWFWa0Di5obvxyXEtillq_jt3Jgw0zPc0XHHPt6CpewYZjklyBxTirJquLyN2FA3qLkQNwvVw-XLz2e_eTYjTdEbw9zrfRaDZq1Wl6AiC0qfrI4MDoy-gyU7qOpK6C8KKwvwQAQm5wC3aw8woeXn5_VrrPM0PUhJ/s1330/1C7E3E8F-85F4-4633-A4FB-96B46E73E8F9_1_201_a.jpeg" style="display: block; padding: 1em 0; text-align: center; "><img alt="bicolored hexagonal prism" border="0" width="400" data-original-height="868" data-original-width="1330" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-0I2HW9208CfsJ7xDYP19PBtgPWFWa0Di5obvxyXEtillq_jt3Jgw0zPc0XHHPt6CpewYZjklyBxTirJquLyN2FA3qLkQNwvVw-XLz2e_eTYjTdEbw9zrfRaDZq1Wl6AiC0qfrI4MDoy-gyU7qOpK6C8KKwvwQAQm5wC3aw8woeXn5_VrrPM0PUhJ/s400/1C7E3E8F-85F4-4633-A4FB-96B46E73E8F9_1_201_a.jpeg"/></a></div>
<p>Similarly, maps that are only 2-connected (maps derived by inserting white vertices in edges can be no more than 2-connected) will have at least some portions that contain zero volume--even when fully un-folded--unless triangular faces are deformed to non-planarity (example below).
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEga4thzfKBfSqiDtj_ugbrYFdf9koN2dlKR6YMhZtaOuImfp6pDvXeq2XCY0dKROXl-x5rYWjtKhZN-BDe62dkjGcWYkVYWAXAUVyyx22xI-lZ3A9CvLBY-Zpxk1v5DwGgdC9zGOu3VSTVaCeAA4IModcNv27BzsjqC5NZbJmKbRr7-JIm1m5v85yJM/s1901/FCB39FBD-E86A-4BEA-8C40-BDDDA7F399B0.heic" style="display: block; padding: 1em 0; text-align: center; "><img alt="surface model with triangular faces deformed to non-planarity" border="0" width="400" data-original-height="1901" data-original-width="1901" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEga4thzfKBfSqiDtj_ugbrYFdf9koN2dlKR6YMhZtaOuImfp6pDvXeq2XCY0dKROXl-x5rYWjtKhZN-BDe62dkjGcWYkVYWAXAUVyyx22xI-lZ3A9CvLBY-Zpxk1v5DwGgdC9zGOu3VSTVaCeAA4IModcNv27BzsjqC5NZbJmKbRr7-JIm1m5v85yJM/s400/FCB39FBD-E86A-4BEA-8C40-BDDDA7F399B0.heic"/></a></div>
<p>These observations draw attention to the 3-connected planar quadrangulations (3CPQ's,) which are maps that are polyhedral and quad-faced, as the best candidates for fold-up baskets. <i>Voxel surfaces</i> without topological holes (top image) are a familiar example of this class.</p>
<p>Some basic properties of quadrangulations and 3CPQ's in particular:</p>
<p>Planar quadrangulations are necessarily bipartite.</p>
<p>A <i>simple</i> (i.e., no loops or parallel edges) planar quadrangulation has connectivity either 2 or 3. Note in this regard that the path of three vertices (below) is not considered a simple planar quadrangulation.</p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiCniHJCcRUgM-Ayd-oVwYG8EuMozoseleiuT8JoB7cSBw51soZlcO4CyJ4ug_o7ug11cYyyaWJucAfJYm0oDeyTH5Bk5RLX7bVUO_v1S7JJ6Po_sQGiUWu0DXOHfBC4v-tsNX4ArIi-_ydfb7OqiwAUsyIAZq7BL4rFD-v_J102WkiZERN-T1NFvhY/s1235/28B665B2-7D62-472C-B341-8A13C8FFFE5A_1_201_a.jpeg" style="display: block; padding: 1em 0; text-align: center; "><img alt="bicolored path of threee vertices" border="0" width="320" data-original-height="653" data-original-width="1235" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiCniHJCcRUgM-Ayd-oVwYG8EuMozoseleiuT8JoB7cSBw51soZlcO4CyJ4ug_o7ug11cYyyaWJucAfJYm0oDeyTH5Bk5RLX7bVUO_v1S7JJ6Po_sQGiUWu0DXOHfBC4v-tsNX4ArIi-_ydfb7OqiwAUsyIAZq7BL4rFD-v_J102WkiZERN-T1NFvhY/s320/28B665B2-7D62-472C-B341-8A13C8FFFE5A_1_201_a.jpeg"/></a></div>
<p>3-connected quadrangulations are necessarily simple.</p>
<p>Given 1 of its 2 possible bicolorings, a planar quadrangulation is the <i>radial</i> of the underlying, <i>pre-radial</i> map revealed by drawing the black-vertex to black-vertex diagonal in each quad, and then erasing the original edges. The pre-radial will always be planar; but might not be 3-connected, simple, or bipartite. The pair of pre-radials derived from alternate bicolorings of the same quadrangulation are dual maps. If there is a face in the pre-radial whose boundary walk is not a cycle, then the radial has multiple edges.
</p>
<p>A 3CPQ has at least 8 vertices of degree 3.</p>
<p>The smallest 3CPQ is the cube (8 vertices.)</p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgHKpis5T_yu_4kStawWzHL_g21Gib53nPLD5aQCXXAn6DnLFNBpV5-GWb_XZ4AuH1HJZzyMdX-ThOO2Q-nddF-4FNn6VKCbUgw3tWhQ6h8vHYXMJMDhqhVCBxYnAkMZ1bG4V_eZ3scRJUXfM1t3jykD6xBn1II2IhWBIepRGGg7EwWbZwF_vVrL-2J/s1918/8DFE7F51-9B75-454E-BB94-2D54EFA34B66_1_201_a.jpeg" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="1724" data-original-width="1918" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgHKpis5T_yu_4kStawWzHL_g21Gib53nPLD5aQCXXAn6DnLFNBpV5-GWb_XZ4AuH1HJZzyMdX-ThOO2Q-nddF-4FNn6VKCbUgw3tWhQ6h8vHYXMJMDhqhVCBxYnAkMZ1bG4V_eZ3scRJUXfM1t3jykD6xBn1II2IhWBIepRGGg7EwWbZwF_vVrL-2J/s600/8DFE7F51-9B75-454E-BB94-2D54EFA34B66_1_201_a.jpeg"/></a></div>
<p>There is no 3CPQ with 9 vertices. The only 3CPQ with 10 vertices has sparse6 code, “:I`ACPpaIOyFJSxn”. (Sparse6 codes can be viewed at the <a href="https://sagecell.sagemath.org/">Sage Cell Server</a>.) This object is known as the pseudo-double wheel of 8 faces, or W<sub>8</sub>. Notice that the cube can be seen as W<sub>6</sub>, and it is the smallest pseudo-double wheel that is 3-connected.
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj5yh3TDu1f_wQFTRLQXG2zveRp4670PuA2cMnJj3WyR22JG-fH6cshhGyPyxC-J3xUVQcOFkTyLo3DVVAHXHkdCus4fYCq-RZFX5eD3ictF7vvO5tIrt5bpsmjmZwSNToyhMCcqujMlsuDagsWmzdxbh8CIpWWBcpPnERReQuSy7oIroZftF-prx7M/s1907/1D93486E-45E1-4104-9B61-E832042EF695_1_201_a.jpeg" style="display: block; padding: 1em 0; text-align: center; "><img alt="pseudo double wheel of 10 faces" border="0" width="400" data-original-height="1713" data-original-width="1907" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj5yh3TDu1f_wQFTRLQXG2zveRp4670PuA2cMnJj3WyR22JG-fH6cshhGyPyxC-J3xUVQcOFkTyLo3DVVAHXHkdCus4fYCq-RZFX5eD3ictF7vvO5tIrt5bpsmjmZwSNToyhMCcqujMlsuDagsWmzdxbh8CIpWWBcpPnERReQuSy7oIroZftF-prx7M/s600/1D93486E-45E1-4104-9B61-E832042EF695_1_201_a.jpeg"/></a></div>
<p>Sparse6 codes for 3CPQ's can be generated at <a href="http://combos.org/plantri">The Combinatorial Object Server</a> which uses Brinkmann and McKay's plantri software.</p>
<p>There is just one 3CPQ with 11 vertices: ":J`ACPpaIOxsOUiBe".</p>
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<p>Among the three 3CPQ's with 12 vertices is ":K`ACGcaMGhBeQiRUO^", a.k.a., the double cube.</p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUuEVfUnu3xA0Qopm9_LAeE013fnC-NQwBc0X7xgV3Bo2hyUDCHNX4H1vaRlBuwFtMDdNqTn5zidlN1BnO9Bpei6RHYakKIK2oXVyYQgG81TUfeMKkFoH8GCarFqhMgDNfMOAVa_qXrQKUg2tKD2R3q1ZVB_vC3kbysa5XxUYgMEbA9ID1O2sbELzj/s2290/8DB4AA73-F26F-4549-ADEF-888CD6D22551_1_201_a.jpeg" style="display: block; padding: 1em 0; text-align: center; "><img alt="double cube" border="0" width="400" data-original-height="1743" data-original-width="2290" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUuEVfUnu3xA0Qopm9_LAeE013fnC-NQwBc0X7xgV3Bo2hyUDCHNX4H1vaRlBuwFtMDdNqTn5zidlN1BnO9Bpei6RHYakKIK2oXVyYQgG81TUfeMKkFoH8GCarFqhMgDNfMOAVa_qXrQKUg2tKD2R3q1ZVB_vC3kbysa5XxUYgMEbA9ID1O2sbELzj/s600/8DB4AA73-F26F-4549-ADEF-888CD6D22551_1_201_a.jpeg"/></a></div>
The number of 3-connected planar quadrangulations with n faces forms integer sequence <a href="http://oeis.org/A007022">A007022</a> in the Online Encyclopedia of Integer Sequences. The maps dual to 3CPQ's are simple, 4-regular, 3-connected planar maps: the same integer sequence counts those with n vertices.
James Malloshttp://www.blogger.com/profile/00763341541407040741noreply@blogger.com0tag:blogger.com,1999:blog-7156771617507620492.post-2075263862822465772022-08-04T10:02:00.001-07:002022-08-04T10:02:29.752-07:00Geoweaving: Fold-Up Baskets from Dessins d'Enfants<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEim9D02ihVLmXhIw5otwpR2v9Dbe7TuKW_ja2Cj7KT8eyPGwI61ajJ9_04UtbyH496V2d6mGamOnkmtOaUbp1sNH-qpTd35ezWPrGkzGVBIZySEz1DGSy51PF3-DP9NdzAXbzUhQQS4PFPANlwTtY61XovnucAZFi9AUfb_7NFaJ-TrEws4TkgLboVj/s3024/77ECE720-477D-464A-B67C-DE40273DBB0C.heic" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="3024" data-original-width="3024" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEim9D02ihVLmXhIw5otwpR2v9Dbe7TuKW_ja2Cj7KT8eyPGwI61ajJ9_04UtbyH496V2d6mGamOnkmtOaUbp1sNH-qpTd35ezWPrGkzGVBIZySEz1DGSy51PF3-DP9NdzAXbzUhQQS4PFPANlwTtY61XovnucAZFi9AUfb_7NFaJ-TrEws4TkgLboVj/s400/77ECE720-477D-464A-B67C-DE40273DBB0C.heic"/></a></div>
I just gave a talk on this topic at <a href="https://www.bridgesmathart.org/b2022/">Bridges Aalto 2022</a>. Dessins d'enfants (children's drawings) are famous in number theory (be sure to catch Gareth Jones' <a href="https://www.youtube.com/watch?v=Bkva3x8wgZU">short talk</a> on that topic.) For our purposes, a dessin is a drawing on a closed surface such as the sphere (the only surface we will consider) drawn according to certain rules.
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<p>On the sphere, the prime rule is that the drawing must be connected, in other words, an ant scared to walk on the unmarked surface would still be able to walk from any part of the drawing to any other. That sort of drawing can be done by not lifting the pencil while drawing, or by drawing in any old way and going back later to add "bridges" for Mr. Ant.</p>
<p>Once you have a connected drawing:</p>
<p>1. Add black dots where lines end or cross.</p>
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<p>2. Add white dots in the middle of the edges.</p>
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<p>Now you have a proper dessin. There are a lot of dessins, and a more general way to make them. The next to last column in this table counts all dessins on the sphere having the same number of black-to-white edges.</p>
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<p>The usefulness of dessins, for our purposes, is that they describe all the arrangements by which the Adams World Tile can seamlessly cover a closed surface (for us, the sphere.) What is the Adams World Tile? Oscar Sherman Adams invented it in 1929, calling his map projection "World in a Square II."</p>
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<p>The beauty of the Adams World Tile is that multiple tiles can be tiled together to model a seamless periodic Earth.</p>
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<p>Put a black dot at the South corner of the Adams World Tile, and a white dot at the North corner, and the dessin makes the arrangement of tiles explicit.</p>
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Maker details can be found in my slides for the <a href="https://www.slideshare.net/jmallos/geoweaving-foldup-baskets-from-dessins-denfants">Bridges talk</a> and <a href="https://archive.bridgesmathart.org/2022/bridges2022-159.html#gsc.tab=0">my paper</a> in the Bridges Archive.James Malloshttp://www.blogger.com/profile/00763341541407040741noreply@blogger.com0tag:blogger.com,1999:blog-7156771617507620492.post-23216059666359075802022-02-09T11:26:00.000-08:002022-02-09T11:26:01.765-08:00Math, Symmetry, and Weaving: the 3 dual pairs of hypermap representationsAs described in a <a href="http://weaveanything.blogspot.com/2022/02/the-6-representations-of-hypermaps.html">previous post</a>, the classical hypermap representations come in three dual-pairs. Using the nomenclature of that earlier post, the dual-pairs are: Belyi-James, Walsh-Chess, and Cori-Quad. I am going to refer to these pairs of hypermap representations as: Symmetry, Math, and Weaving. Here is why.
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEiFxcD7Ul54P4g9ABYKa-NsEtPI808e-nOhZ9vdV990Unl8j-cnqlrfzlvWU-0XzZyCY-M3UXS1PVDz7y7l8E4wefO-ORa8u9MTVlmtyNdEpUvV3cn1qMPiHlZOZThXss93JOfuyz8CIGTQe1-coPQ-itU7U_8wcPVujJuq5CatA35QB7nS8K_CPV9B=s2004" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="1020" data-original-width="2004" src="https://blogger.googleusercontent.com/img/a/AVvXsEiFxcD7Ul54P4g9ABYKa-NsEtPI808e-nOhZ9vdV990Unl8j-cnqlrfzlvWU-0XzZyCY-M3UXS1PVDz7y7l8E4wefO-ORa8u9MTVlmtyNdEpUvV3cn1qMPiHlZOZThXss93JOfuyz8CIGTQe1-coPQ-itU7U_8wcPVujJuq5CatA35QB7nS8K_CPV9B=s400"/><div>The 'Symmetry' pair: Belyi and James</div></a></div>
Because they expose the full symmetry of hypermaps--hypervertices, hyperfaces, and hyperedges are interchangeable roles--these two representations are undoubtedly the most fundamental, but perhaps the least useful. In the Belyi (a.k.a., the canonical triangulation) and the James representations the six Lins trialities are just the permutations of three colors. That is too much symmetry be saying anything useful about something with less symmetry.
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgj9eYpnSYtjEhHvAq8nA4wtXV1r9B5if8J3rFTHcNed08rJ0VqVOpfcZj1VQ15idaTiW-W65g4xWVxRQQTmKZ2VYhGaYI5Xfs9xghPd388sPD2uGYjv4ACqjos3ObTD7AV9qTtx_2R4JxQUW1SluMNMvfvt5Q4l72fLSnCpR7ch-9kIbv-Ujxejb9p=s2018" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="1026" data-original-width="2018" src="https://blogger.googleusercontent.com/img/a/AVvXsEgj9eYpnSYtjEhHvAq8nA4wtXV1r9B5if8J3rFTHcNed08rJ0VqVOpfcZj1VQ15idaTiW-W65g4xWVxRQQTmKZ2VYhGaYI5Xfs9xghPd388sPD2uGYjv4ACqjos3ObTD7AV9qTtx_2R4JxQUW1SluMNMvfvt5Q4l72fLSnCpR7ch-9kIbv-Ujxejb9p=s400"/><div>The 'Math' dual-pair: Walsh and Chess</div></a></div>
Math is a monochromatic world. Color and color names are avoided if possible. These two hypermap representations, Walsh and Chess, get by with just two colors (black and white) by associating specific graph elements (faces, vertices, respectively) exclusively with the third color--which therefore never needs naming. Walsh and Chess are in fact the most common hypermap representations in the math literature. See <a href="https://arxiv.org/pdf/1403.5371.pdf">Bernardi and Fusy</a> for a use of Chess.
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgOKtRDEiR5TzpffIXA5Cr7mK38etr30FPACtr59HJv-FUHMrqU7dT3bErPn4Q8EqKrsrdQ2x4nruATJEIWIjf2u7ruuB0d0NnKWNTdnfCT5crUj1xHiuLQxjSLgmv3yO0KWXqh63WX-9n4E3J7KZfOPgyJ94A5bxpD75-iEcQVGYY29pEp84urwSdE=s2002" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="1024" data-original-width="2002" src="https://blogger.googleusercontent.com/img/a/AVvXsEgOKtRDEiR5TzpffIXA5Cr7mK38etr30FPACtr59HJv-FUHMrqU7dT3bErPn4Q8EqKrsrdQ2x4nruATJEIWIjf2u7ruuB0d0NnKWNTdnfCT5crUj1xHiuLQxjSLgmv3yO0KWXqh63WX-9n4E3J7KZfOPgyJ94A5bxpD75-iEcQVGYY29pEp84urwSdE=s400"/><div>The 'Weaving' dual-pair: Cori and Quad</div></a></div>
These two hypermap representations, Cori and Quad, are 4-regular (respectively, on vertices, and on faces) and are readable as weaving diagrams.
James Malloshttp://www.blogger.com/profile/00763341541407040741noreply@blogger.com0tag:blogger.com,1999:blog-7156771617507620492.post-66637740721579753152022-02-08T13:44:00.000-08:002022-02-08T13:44:23.771-08:00Visualizing dessins as flow fields
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEiPVVddjfHbPxIW0NKaiGNEgR4PkhZelB06cCxYq1v_fwgVKFq3CoZkegO8o0jNVyNx24VFhQ29DJPwPLxcCkMVrg1xpptm0k5NoXlPaA5JuwfSST5tOhdfYxw6C3LQ_ymptAUvFKfINNT0UoHa4HOLIVwErg8MT5pISa2CPwP3kro1rtj8RWMgFy4P=s800" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="800" data-original-width="800" src="https://blogger.googleusercontent.com/img/a/AVvXsEiPVVddjfHbPxIW0NKaiGNEgR4PkhZelB06cCxYq1v_fwgVKFq3CoZkegO8o0jNVyNx24VFhQ29DJPwPLxcCkMVrg1xpptm0k5NoXlPaA5JuwfSST5tOhdfYxw6C3LQ_ymptAUvFKfINNT0UoHa4HOLIVwErg8MT5pISa2CPwP3kro1rtj8RWMgFy4P=s400"/><div>A dessin with a source/sink flow sketched in.</div></a></div>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEg_uo8Asb3xtbGFf5D1lIgNiLPO5pmM8MkyKxeSrclP3pXJFVKh-pyD3H3h_0hw1parsOulmXSeTcDMaeaJT61yq_PI_7HquHM-eZPsSUOdHDMvCWHqZ8oNbg-cdokj3MShh6jGmKNkFVjz1gsrx_Aiq8AIbD2jCWo58Ezl26fbL4WAZ8-OcwLv1-vz=s800" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="800" data-original-width="800" src="https://blogger.googleusercontent.com/img/a/AVvXsEg_uo8Asb3xtbGFf5D1lIgNiLPO5pmM8MkyKxeSrclP3pXJFVKh-pyD3H3h_0hw1parsOulmXSeTcDMaeaJT61yq_PI_7HquHM-eZPsSUOdHDMvCWHqZ8oNbg-cdokj3MShh6jGmKNkFVjz1gsrx_Aiq8AIbD2jCWo58Ezl26fbL4WAZ8-OcwLv1-vz=s400"/><div>The underlying dessin.</div></a></div>
<p>Maybe the easiest way to visualize how a dessin (a bicolored graph embedded in a surface) classifies Riemann surfaces is to sketch a flow field where, say, the black vertices are sources and the white vertices are sinks. Notice in the sketch above that the flow lines only reach nearest neighbors bordering a given face. Seeing the sketched flow lines as lines of longitude on the Earth (say from North to South) or as lines of constant phase angle on the Riemann sphere (from 0 to infinity) suggests a particular way in which multiple instances of the Riemann sphere can cover the surface.James Malloshttp://www.blogger.com/profile/00763341541407040741noreply@blogger.com0tag:blogger.com,1999:blog-7156771617507620492.post-4455898580264683332022-02-08T11:59:00.002-08:002022-02-08T12:10:45.476-08:00The 6 representations of hypermaps<p>Hypergraphs are generalizations of graphs where a (hyper)edge is now a multiset (of any cardinality) of vertices. By contrast, in a simple graph an edge is a unique, cardinality-2 set of vertices. In a multigraph (parallel edges being allowed) uniqueness is no longer required. In a general graph (self-loops being allowed) an edge is a cardinality-2 multiset of vertices. By allowing the multiset of a hyperedge to have any cardinality, hypergraphs make a vast generalization of ordinary graphs: a hyperedge can connect any number of vertices, each with any degree of multiplicity. For example, there are only two graphs on one edge, but an infinity of hypergraphs on one hyperedge.</p>
<p>Surprisingly, when hypergraphs are embedded in surfaces they are easy to draw. For example, in the Walsh representation of hypermaps, they are simply bipartite, ordinary maps, embedded and drawn the same old way, but endowed with a proper 2-coloring of the vertices: the vertices have been colored black or white such that no edge connects two vertices of the same color. On many ordinary maps such a bicoloring is not possible; a map is said to be bipartite if such a bicoloring is possible.</p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiG9lz4Av2C2_d-gKLOzGLUAXSy81tD8Mh0VryUrkPOctbjoA2Zf9l2Vntviv1yQrJrUKEP90f4Yx5y8zigPihutWZkQWGafob-KQ3oEMvirwwJ-MES96yLtIQevbPFB8ny_fwgM3qoQMi2_h3GPSUILu3aQy-dunFodKpXpGQ9PCevHTcU6ff5e0IO/s827/fullsizeoutput_6db4.jpeg" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" height="320" data-original-height="827" data-original-width="768" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiG9lz4Av2C2_d-gKLOzGLUAXSy81tD8Mh0VryUrkPOctbjoA2Zf9l2Vntviv1yQrJrUKEP90f4Yx5y8zigPihutWZkQWGafob-KQ3oEMvirwwJ-MES96yLtIQevbPFB8ny_fwgM3qoQMi2_h3GPSUILu3aQy-dunFodKpXpGQ9PCevHTcU6ff5e0IO/s320/fullsizeoutput_6db4.jpeg"/></a></div>
<p>Given an ordinary map, we can make it bipartite, and, in fact, bicolored, by inserting a white vertex in the middle of each edge. (In doing so, we have doubled the number of edges in the graph, and moved deeper into the combinatorial explosion of maps--the illusion that hypermaps are a subset of ordinary maps is just that.) This way of drawing a hypergraph lets us stay in the vocabulary of ordinary graphs if we wish. We just need to keep in mind that the white vertices represent generalized edges, octopi that can connect to any number of vertices with any multiplicity. For example, above is a hypermap expressing the idea that a particular friendship involves Peter, Paul, and Mary... and Peter twice.</p>
In addition to the Walsh representation, there are five more classical representations of hypermaps. Only Walsh, Cori, and James are named for their first proponents, T.R.S. Walsh, Robert Cori, and Lynne D. James. For brevity, I have taken the liberty to call the "canonical triangulation" that arises in the study of Belyi functions,'Belyi'; and the checkerboard coloring lately championed by Bernardi and Fusy, 'Chess'; and what may be termed the 'canonical quadrangulation'--the canonical triangulation less the edges of the underlying bipartite graph, 'Quad'.
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-T03dVz7LPpt_ef99x8px0_8GDLxf5Ub1eDYegiGe_x9nS66GGAmRZ64FcsV1pEeK9kxwanC8oNAHjapWKGdoZZ04haV5hp3_TPrV9vrs8C9xaKiXLUlYjIav-uA8tnECFvM8OWlPgMhTqJYPW0MmaHIoAEaeF81bOYpcgMSWL6uDRRAPMnwyJMlt/s1600/Snapshot%202012-02-22%2009-24-39.jpg" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="958" data-original-width="1600" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-T03dVz7LPpt_ef99x8px0_8GDLxf5Ub1eDYegiGe_x9nS66GGAmRZ64FcsV1pEeK9kxwanC8oNAHjapWKGdoZZ04haV5hp3_TPrV9vrs8C9xaKiXLUlYjIav-uA8tnECFvM8OWlPgMhTqJYPW0MmaHIoAEaeF81bOYpcgMSWL6uDRRAPMnwyJMlt/s400/Snapshot%202012-02-22%2009-24-39.jpg"/></a></div>
<p>The chart above is explained in this <a href="https://weaveanything.blogspot.com/2012/02/relations-between-hypermap.html">earlier post.</a></p>
Taking Walsh, which is the most intuitive and easiest to talk about with old vocabulary, as the ancestor, the other five are easily described as descendants by map operations. In fact, besides Dual, the only two map operations needed are Medial ('Ambo' in Conway polyhedron notation) and Kis (central triangulation of faces, also called stellation, or omnicapping.) Note: it is an error to use the same color (white) for faces and vertices in the Walsh representation. Once that is corrected (Walsh faces here are pink) all the other representations can receive their coloring by descent. That is, for example, if a face ultimately derives by map operations from a black vertex in Walsh, it is colored black. Both Cori and its dual, Quad, can be seen as diagrammatic of basket weaving, the one, as sparse weaving, the other as dense weaving. In those interpretations, the colors black and white code the same helical handedness.
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiNgucd8QXCy5IHleZvQ9RKchljgO_6yyvTfyX5T0C0vjqdWk8DWbEICNOwj2d2aco7y0sWU5hDQyZCPHwYLkuXNRA5E5oWuyxmE6UmQxn-N6LXEKQL-xdfDPY7JQZBAYssW8Kj2kS10UYMYdKcp1Vwlajb_ibl9XIzhIMOkx1FNvyXHMu9Xzu1ZVly/s800/subdivide_color.jpg" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="800" data-original-width="800" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiNgucd8QXCy5IHleZvQ9RKchljgO_6yyvTfyX5T0C0vjqdWk8DWbEICNOwj2d2aco7y0sWU5hDQyZCPHwYLkuXNRA5E5oWuyxmE6UmQxn-N6LXEKQL-xdfDPY7JQZBAYssW8Kj2kS10UYMYdKcp1Vwlajb_ibl9XIzhIMOkx1FNvyXHMu9Xzu1ZVly/s400/subdivide_color.jpg"/><div>Walsh representation</div></a></div>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjXTl9Zz9SY_QbzztCflbNeOCOF0XPaSicrP3eVswL40UzbfbMHNUuDsYiGkTqXpF-9BMpFs2EmrLgUs-FCaJSI2SpJq5E8vttqTII77wUF136_c9-t1rppE9PzK6PEbDXdDQwUQtFda0JaeCCvg3oWtiCJ18TbcxtSKsDVCN4XBsmUCat2iycl3p6O=s800" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="800" data-original-width="800" src="https://blogger.googleusercontent.com/img/a/AVvXsEjXTl9Zz9SY_QbzztCflbNeOCOF0XPaSicrP3eVswL40UzbfbMHNUuDsYiGkTqXpF-9BMpFs2EmrLgUs-FCaJSI2SpJq5E8vttqTII77wUF136_c9-t1rppE9PzK6PEbDXdDQwUQtFda0JaeCCvg3oWtiCJ18TbcxtSKsDVCN4XBsmUCat2iycl3p6O=s400"/><div>Belyi representation = Kis(Walsh)</div></a></div>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjVs_BGeY52NWTNLOOO2Uxo2OPTfIwVM8eG_S5ZSb-ufKsXDy3Ee1lVB-qBk12Ht7FDaf-hqFRU4twgDvYJbtVGwMeJzQ_5C2am9k447bZhuo7Ip9CkMWtaN2iryxxVzq_f72JOSONFMZU_Ynpvhp7AOIjzMlA9odbfDowhJvWP5Zj9T8BK2pXJDVtZ=s800" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="800" data-original-width="800" src="https://blogger.googleusercontent.com/img/a/AVvXsEjVs_BGeY52NWTNLOOO2Uxo2OPTfIwVM8eG_S5ZSb-ufKsXDy3Ee1lVB-qBk12Ht7FDaf-hqFRU4twgDvYJbtVGwMeJzQ_5C2am9k447bZhuo7Ip9CkMWtaN2iryxxVzq_f72JOSONFMZU_Ynpvhp7AOIjzMlA9odbfDowhJvWP5Zj9T8BK2pXJDVtZ=s400"/><div>Cori representation = Medial(Walsh)</div></a></div>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEiUjQS-v9ZUtcfotEOC2e_60Bv6nVfIYpSZA23gbKAo_g_543VsoWHJHttWH5NrmlWbxRza6IYh7iDejAcR7eVEhFUGOAkInSm-z1_H7J5Wvpef-eJBNKlXrlc585bPiBEyqo0Phw5SwBmNQvbYkaW-CwB0rN5UiteH6vu3SH6o6YoYMbtDQ-sVJaLL=s800" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="800" data-original-width="800" src="https://blogger.googleusercontent.com/img/a/AVvXsEiUjQS-v9ZUtcfotEOC2e_60Bv6nVfIYpSZA23gbKAo_g_543VsoWHJHttWH5NrmlWbxRza6IYh7iDejAcR7eVEhFUGOAkInSm-z1_H7J5Wvpef-eJBNKlXrlc585bPiBEyqo0Phw5SwBmNQvbYkaW-CwB0rN5UiteH6vu3SH6o6YoYMbtDQ-sVJaLL=s400"/><div>Chess representation = Dual(Walsh)</div></a></div>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEiI3VDlRqxo1yCC0lAqI9RvptBQwIKfItQ2xJQqqBj5ToTiqMQBYzHcWWVs_DWV1YTEsaLOE4cuHc4p-l7ZkdC8zeyxwKO08XSSRN5xDLRrF01ExCSClPGjL4OS17vz67HMM7jtxi1ictjz4feBGb7dkdbWtF_xKJWQI3So0o7082aXfsvhJ4wqj5uj=s800" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="800" data-original-width="800" src="https://blogger.googleusercontent.com/img/a/AVvXsEiI3VDlRqxo1yCC0lAqI9RvptBQwIKfItQ2xJQqqBj5ToTiqMQBYzHcWWVs_DWV1YTEsaLOE4cuHc4p-l7ZkdC8zeyxwKO08XSSRN5xDLRrF01ExCSClPGjL4OS17vz67HMM7jtxi1ictjz4feBGb7dkdbWtF_xKJWQI3So0o7082aXfsvhJ4wqj5uj=s400"/><div>James representation = Dual(Belyi)</div></a></div>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEi-Qzq4nF-3WlzL7EkB8FkQDHifCobHB6e0-pyCd8XQkTY1uqRi-9m3rWrIrmkwt5H8Pcf1keRbtw9gsc9xFp3r1FFxjjWzM5uNA6BuJF8n3R-eK81A4TA2se3z6oTWEZXkeEdjUCghBXpuZEBPidFsQUOfIVvrwM38Knay-jFQDh-S0VAgQquI6gjb=s800" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="800" data-original-width="800" src="https://blogger.googleusercontent.com/img/a/AVvXsEi-Qzq4nF-3WlzL7EkB8FkQDHifCobHB6e0-pyCd8XQkTY1uqRi-9m3rWrIrmkwt5H8Pcf1keRbtw9gsc9xFp3r1FFxjjWzM5uNA6BuJF8n3R-eK81A4TA2se3z6oTWEZXkeEdjUCghBXpuZEBPidFsQUOfIVvrwM38Knay-jFQDh-S0VAgQquI6gjb=s400"/><div>Quad representation = Dual(Cori)</div></a></div>
James Malloshttp://www.blogger.com/profile/00763341541407040741noreply@blogger.com0tag:blogger.com,1999:blog-7156771617507620492.post-22648486013187601912021-09-20T14:01:00.001-07:002021-09-20T14:01:18.320-07:00Ramification at {0, ±√3}<div class="separator" style="clear: both;"><a href="https://1.bp.blogspot.com/-bTD-K5QicHE/YUiqOy1kcuI/AAAAAAAAJzw/-nhKvPfg2ZYqY-qL0LzucoxsrPJbbjq0gCLcBGAsYHQ/s1000/PlusMinus.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="573" data-original-width="1000" src="https://1.bp.blogspot.com/-bTD-K5QicHE/YUiqOy1kcuI/AAAAAAAAJzw/-nhKvPfg2ZYqY-qL0LzucoxsrPJbbjq0gCLcBGAsYHQ/s400/PlusMinus.png"/></a></div>
<p>For some purposes it is preferable to have ramification points that are equally spaced in the geographic metric, for example, ramification points located at {0, ±√3} (see diagram above) rather than at {0, 1, ∞}. These will not be Belyi functions, but will have analogous uses. For example, if the three points of ramification are equally spaced around the real-number great circle, their colors can be permuted without the geographic distortion found in four of the <a href="http://weaveanything.blogspot.com/2021/09/belyi-functions-that-permute-vertex.html">vertex-color permuting Belyi functions</a>.</p>
<p>Therefore, it may be desirable to find the unique Mobius transformation, S, that maps {0, 1, ∞}, respectively, to {-√3, 0, √3}; and its inverse, S<sup>-1</sup>, as well. It is easier to start with the inverse since it is a mapping to {0, 1, ∞} like we saw in <a href="http://weaveanything.blogspot.com/2021/09/mobius-maps-to-0-1.html">the previous post</a>, in this case:</p>
<p>z<sub>0</sub> = -√3</p>
<p>z<sub>1</sub> = 0</p>
<p>z<sub>∞</sub> = √3</p>
<p>S<sup>-1</sup>(z) = ((z-z<sub>0</sub>)*(z<sub>1</sub>-z<sub>∞</sub>))/((z-z<sub>∞</sub>)*(z<sub>1</sub>-z<sub>0</sub>))</p>
<p>= ((z+√3)*(-√3))/((z-√3)*(√3))</p>
<p>= -(z+√3)/(z-√3)</p>
<p>= (-z-√3)/(z-√3)</p>
<p>So: a = -1; b = -√3; c = 1; d = -√3</p>
<p>From Michael P. Pitchman's <a href="https://mphitchman.com/geometry/section3-4.html">web chapter on Mobius transformations</a>:</p>
<div class="separator" style="clear: both;"><a href="https://1.bp.blogspot.com/-Wj-PBfEy6cU/YUizF2B7AyI/AAAAAAAAJz4/aWmf55Df5ZQe2vy6yjncHrMK5fM9zO7DQCLcBGAsYHQ/s1000/InverseMobius.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="466" data-original-width="1000" src="https://1.bp.blogspot.com/-Wj-PBfEy6cU/YUizF2B7AyI/AAAAAAAAJz4/aWmf55Df5ZQe2vy6yjncHrMK5fM9zO7DQCLcBGAsYHQ/s400/InverseMobius.png"/></a></div>
<p>So, S(z) = (√3*z-√3)/(z+1) = √3*(z-1)/(z+1</p>
<p>Domain-coloring visualization of S:</p>
<div class="separator" style="clear: both;"><a href="https://1.bp.blogspot.com/-3SknTv5ViXY/YUjsrckGrWI/AAAAAAAAJ0A/8dF1bYj88pkDtx9ZORKKma-idw01gsQkwCLcBGAsYHQ/s1000/ForwardMorph.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="500" data-original-width="1000" src="https://1.bp.blogspot.com/-3SknTv5ViXY/YUjsrckGrWI/AAAAAAAAJ0A/8dF1bYj88pkDtx9ZORKKma-idw01gsQkwCLcBGAsYHQ/s400/ForwardMorph.png"/></a></div>
<p>Domain-coloring visualization of S<sup>-1</sup>:</p>
<div class="separator" style="clear: both;"><a href="https://1.bp.blogspot.com/-B3bRMNcoeQo/YUjtDvzzr4I/AAAAAAAAJ0I/cm2Grh8MbC8VVPsSEwMiFaz9AvDD8DQmACLcBGAsYHQ/s1000/InverseMorph.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="501" data-original-width="1000" src="https://1.bp.blogspot.com/-B3bRMNcoeQo/YUjtDvzzr4I/AAAAAAAAJ0I/cm2Grh8MbC8VVPsSEwMiFaz9AvDD8DQmACLcBGAsYHQ/s400/InverseMorph.png"/></a></div>
<p>Domain-coloring visualization of their composition (Identity):</p>
<div class="separator" style="clear: both;"><a href="https://1.bp.blogspot.com/-GXL1ZPLD2ME/YUjtg5SE5XI/AAAAAAAAJ0Q/wCylcfoZrYsTI0FR62Pnnx1-oHmQN5ZNQCLcBGAsYHQ/s1000/Composition.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="500" data-original-width="1000" src="https://1.bp.blogspot.com/-GXL1ZPLD2ME/YUjtg5SE5XI/AAAAAAAAJ0Q/wCylcfoZrYsTI0FR62Pnnx1-oHmQN5ZNQCLcBGAsYHQ/s400/Composition.png"/></a></div>
<p>Domain-coloring visualization of S∘Tetrahedron∘S<sup>-1</sup>:</p>
<div class="separator" style="clear: both;"><a href="https://1.bp.blogspot.com/-kZt-xn1zj3Y/YUjzzDja2EI/AAAAAAAAJ0Y/lfdH8nOAL9osiSYZAE7Jyquf2i0Jr0Y6QCLcBGAsYHQ/s1000/ForTInv.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="500" data-original-width="1000" src="https://1.bp.blogspot.com/-kZt-xn1zj3Y/YUjzzDja2EI/AAAAAAAAJ0Y/lfdH8nOAL9osiSYZAE7Jyquf2i0Jr0Y6QCLcBGAsYHQ/s400/ForTInv.png"/></a></div>
<p>In the above view of a tetrahedron, the North Pacific now represents vertices, Antarctica now represents mid-edges, and the Sahara represents face centers. It's easy to see that there are four faces (Sahara's) and six edges (Antarctica's), but the four vertices (North Pacifics) are harder to see.</p>
James Malloshttp://www.blogger.com/profile/00763341541407040741noreply@blogger.com1tag:blogger.com,1999:blog-7156771617507620492.post-57669522415733836252021-09-20T08:31:00.002-07:002021-09-20T09:31:48.395-07:00Mobius maps to {0, 1, ∞}<p>Belyi functions are unique up to Mobius transformations, which are conformal maps from the Riemann sphere to the Riemann sphere that preserve, not merely infinitesimal circles, but all circles. The fate of any three distinct points determines a Mobius function. A particularly simple case is when we know which three points will map to 0, 1, and ∞. Namely, we seek a Mobius transformation S, such that:</p>
<p>S(z<sub>0</sub>) = 0</p>
<p>S(z<sub>1</sub>) = 1</p>
<p>S(z<sub>∞</sub>) = ∞</p>
<p>Then S = ((z-z<sub>0</sub>)*(z<sub>1</sub>-z<sub>∞</sub>))/((z-z<sub>∞</sub>)*(z<sub>1</sub>-z<sub>0</sub>)).</p>
<p>For example, the <a href="http://weaveanything.blogspot.com/2021/09/belyi-functions-that-permute-vertex.html">vertex-color permuting Belyi functions</a> can be derived this way.</p>
James Malloshttp://www.blogger.com/profile/00763341541407040741noreply@blogger.com0tag:blogger.com,1999:blog-7156771617507620492.post-56107476292497215542021-09-16T13:56:00.003-07:002021-09-16T15:18:54.688-07:00Belyi functions that permute vertex colorsThe canonical triangulation of a dessin has vertices of three colors: black for original vertices, white for edge centers, 'star' for face centers. The simplest Belyi functions that permute these colors in the 6 possible ways are the 6 transformations Coxeter, in Regular Polytopes, gave as an example of the operation of the symmetric group on three elements:
<div class="separator" style="clear: both;"><a href="https://1.bp.blogspot.com/-nnePRVWFMQk/YUOxLqFe4cI/AAAAAAAAJzI/EfF7Fu38TmAwpmTjSwpjfDgr95QlUV5hQCLcBGAsYHQ/s1000/Coxeter-regular-polytopes.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="600" data-original-height="521" data-original-width="1000" src="https://1.bp.blogspot.com/-nnePRVWFMQk/YUOxLqFe4cI/AAAAAAAAJzI/EfF7Fu38TmAwpmTjSwpjfDgr95QlUV5hQCLcBGAsYHQ/s600/Coxeter-regular-polytopes.png"/></a></div>
<p>Here's what they look like in geographic domain coloring (geographic conventions same as previous post.) All these functions describe a half-edge or <i>brin</i> in different positions/orientations: its black vertex is in each case coincident with Antarctica, its white vertex coincident with Null Island.</p>
<p>Identity, z:</p>
<div class="separator" style="clear: both;"><a href="https://1.bp.blogspot.com/-wsjO65NaveM/YUOsvBi-8WI/AAAAAAAAJyY/ifm8SuGLN9Eq5S8kTrM38aGrJ6HFT2k1QCLcBGAsYHQ/s1000/Identity%2Bpair.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="600" data-original-height="501" data-original-width="1000" src="https://1.bp.blogspot.com/-wsjO65NaveM/YUOsvBi-8WI/AAAAAAAAJyY/ifm8SuGLN9Eq5S8kTrM38aGrJ6HFT2k1QCLcBGAsYHQ/s400/Identity%2Bpair.png"/></a></div>
<p>Dual, 1/z:</p>
<div class="separator" style="clear: both;"><a href="https://1.bp.blogspot.com/-HHcsMCrn8lo/YUOtAiD-MdI/AAAAAAAAJyg/z0e348OiF9Izb5VznJqnY04dK5-uW1kMACLcBGAsYHQ/s1000/1overzPair.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="600" data-original-height="502" data-original-width="1000" src="https://1.bp.blogspot.com/-HHcsMCrn8lo/YUOtAiD-MdI/AAAAAAAAJyg/z0e348OiF9Izb5VznJqnY04dK5-uW1kMACLcBGAsYHQ/s400/1overzPair.png"/></a></div>
<p>1-z:</p></p></p>
<div class="separator" style="clear: both;"><a href="https://1.bp.blogspot.com/-GAoFU1mp-ps/YUOtRSligTI/AAAAAAAAJyo/3WuTxnWRMgAhyLCFy1qTRlyerCgfDpUMwCLcBGAsYHQ/s1000/1minuszPair.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="600" data-original-height="502" data-original-width="1000" src="https://1.bp.blogspot.com/-GAoFU1mp-ps/YUOtRSligTI/AAAAAAAAJyo/3WuTxnWRMgAhyLCFy1qTRlyerCgfDpUMwCLcBGAsYHQ/s400/1minuszPair.png"/></a></div>
<p>z/(z-1):</p></p>
<div class="separator" style="clear: both;"><a href="https://1.bp.blogspot.com/-mDCBUwFdhbk/YUOt0MPV8WI/AAAAAAAAJyw/dlb-RNWZL48_iyVCEFpI2hTRTrN2k1fNwCLcBGAsYHQ/s1000/zoverzminus1Pair.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="600" data-original-height="501" data-original-width="1000" src="https://1.bp.blogspot.com/-mDCBUwFdhbk/YUOt0MPV8WI/AAAAAAAAJyw/dlb-RNWZL48_iyVCEFpI2hTRTrN2k1fNwCLcBGAsYHQ/s400/zoverzminus1Pair.png"/></a></div>
<p>1/(1-z):</p>
<div class="separator" style="clear: both;"><a href="https://1.bp.blogspot.com/-zFGQyBPGmoY/YUOudMsxkuI/AAAAAAAAJy4/4w26MsZjh0EBwDgdUsJbKnaMgwEC5APFQCLcBGAsYHQ/s1000/1over1minuszPair.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="600" data-original-height="501" data-original-width="1000" src="https://1.bp.blogspot.com/-zFGQyBPGmoY/YUOudMsxkuI/AAAAAAAAJy4/4w26MsZjh0EBwDgdUsJbKnaMgwEC5APFQCLcBGAsYHQ/s400/1over1minuszPair.png"/></a></div>
<p>(z-1)/z:</p>
<div class="separator" style="clear: both;"><a href="https://1.bp.blogspot.com/-NP3tK0QUMxE/YUOu6S1XS_I/AAAAAAAAJzA/wHFNAmqMNxoyHs0HjUeQ4zuiVVNWCXEtwCLcBGAsYHQ/s1000/zminus1overzPair.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="600" data-original-height="501" data-original-width="1000" src="https://1.bp.blogspot.com/-NP3tK0QUMxE/YUOu6S1XS_I/AAAAAAAAJzA/wHFNAmqMNxoyHs0HjUeQ4zuiVVNWCXEtwCLcBGAsYHQ/s400/zminus1overzPair.png"/></a></div>
James Malloshttp://www.blogger.com/profile/00763341541407040741noreply@blogger.com0tag:blogger.com,1999:blog-7156771617507620492.post-81564188209329887962021-09-16T11:36:00.002-07:002021-09-16T11:36:30.781-07:00Geographic Domain Coloring & Belyi functions<div class="separator" style="clear: both;"><a href="https://1.bp.blogspot.com/-r_bXviN65s4/YUOA60i3XJI/AAAAAAAAJx4/I7eGwsI3OKQLxToZlJHOlBgvDRcLgVjZgCLcBGAsYHQ/s1000/Tetrapair.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="499" data-original-width="1000" src="https://1.bp.blogspot.com/-r_bXviN65s4/YUOA60i3XJI/AAAAAAAAJx4/I7eGwsI3OKQLxToZlJHOlBgvDRcLgVjZgCLcBGAsYHQ/s400/Tetrapair.png"/></a></div>
<p>Domain coloring is a popular way to visualize functions that map the complex plane to the complex plane (equivalently, the Riemann sphere to the Riemann sphere). The range space (the plane or sphere the function maps <i>to</i>) is given a patterned coloring and those colors are mapped back to the domain space (the plane or sphere the function maps <i>from</i>) by evaluating the function at every pixel in the domain. This of course involves evaluating the function a million times or so, but computers make it easy.</p>
<p>For the image above, the function (-64*((1/z)^3+1)^3/(((1/z)^3-8)^3*(1/z)^3)), a Belyi function for the tetrahedron, is interpreted as a Riemann-sphere to Riemann-sphere mapping. The range Riemann sphere was decorated with a map of the globe oriented in a particular way: South Pole at 0, North Pole at ∞ (on the Riemann sphere, ∞ is the point antipodal to 0,) and Null Island (shorthand for the point off the coast of Africa at 0° latitude, 0° longitude) is at 1 (on the Riemann sphere, 1 is halfway around from 0 to ∞.) The picture below shows the range sphere in the same projection used above.</p>
<div class="separator" style="clear: both;"><a href="https://1.bp.blogspot.com/-JYqeRXAJX4Q/YUOIk4lzdHI/AAAAAAAAJyE/IZizXl0SAxINN5KTYFAhk5JNGfAc4CM4wCLcBGAsYHQ/s1000/Identity%2Bpair.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="501" data-original-width="1000" src="https://1.bp.blogspot.com/-JYqeRXAJX4Q/YUOIk4lzdHI/AAAAAAAAJyE/IZizXl0SAxINN5KTYFAhk5JNGfAc4CM4wCLcBGAsYHQ/s400/Identity%2Bpair.png"/></a></div>
<p>Imagine the two polar projections to be hinged at 1, then imagine closing them together like a face-down book to form a two-disk map of the world.</p>
<p>Always keep in mind that <i>domain</i> coloring produces a view of the <i>domain</i>. In other words, the Belyi function begins with the hollow, non-physical, decorated sphere in the top picture, and stretches and folds it, in perfect registration, to cover the globe in the bottom picture. Pretty remarkable!<p>
The <i>dessin</i>, in this case a tetrahedron, is found by tracing in the domain (top picture) all the pre-images of the line segment [0,1] in the range (bottom picture). This geographic domain coloring makes: vertices correspond to Antarctica's, faces correspond to Arctic Oceans, and edges to Antarctica-to-Antarctica sea voyages that pass between two copies of Africa. James Malloshttp://www.blogger.com/profile/00763341541407040741noreply@blogger.com0tag:blogger.com,1999:blog-7156771617507620492.post-8346635524518395362021-06-17T13:49:00.000-07:002021-06-17T13:49:09.102-07:00Weave-like meshing<div class="separator" style="clear: both;"><a href="https://1.bp.blogspot.com/-zaMJqSIa_0Q/YMuzJcFx8_I/AAAAAAAAJtM/hhNxg5KjkMQ0Wimrhl7ypPmNntb3GjULACLcBGAsYHQ/s1875/2A054838-86CF-4113-8DAC-6FC944767744_1_201_a.jpeg" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="480" data-original-height="924" data-original-width="1875" src="https://1.bp.blogspot.com/-zaMJqSIa_0Q/YMuzJcFx8_I/AAAAAAAAJtM/hhNxg5KjkMQ0Wimrhl7ypPmNntb3GjULACLcBGAsYHQ/s400/2A054838-86CF-4113-8DAC-6FC944767744_1_201_a.jpeg"/></a></div>
Computer graphics has come a long way toward generating surface meshes reminiscent of basket weaving. The image above illustrates (on the right) the technique of authors Wenzel Jakob, Marco Tarini, Daniele Panozzo, and Olga Sorkine-Hornung, called <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.723.7864&rep=rep1&type=pdf">Instant Field-Aligned Meshing.</a> To me the meshing has an almost musical quality. James Malloshttp://www.blogger.com/profile/00763341541407040741noreply@blogger.com0tag:blogger.com,1999:blog-7156771617507620492.post-37522509598532457092021-06-17T13:29:00.003-07:002021-06-17T13:29:40.079-07:00Using isometries in design and fabrication<div class="separator" style="clear: both;"><a href="https://1.bp.blogspot.com/-lYzhN2S_fLo/YMutrW7owZI/AAAAAAAAJtE/knnMuB6gxFwnMNnMj1AJ-BZ4d10QkgIvACLcBGAsYHQ/s1726/20BABE8E-1A56-4A77-AB50-E00758CF7991_1_201_a.jpeg" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="1144" data-original-width="1726" src="https://1.bp.blogspot.com/-lYzhN2S_fLo/YMutrW7owZI/AAAAAAAAJtE/knnMuB6gxFwnMNnMj1AJ-BZ4d10QkgIvACLcBGAsYHQ/s400/20BABE8E-1A56-4A77-AB50-E00758CF7991_1_201_a.jpeg"/></a></div>
There's <a href="https://repository.kaust.edu.sa/handle/10754/669048">a cool new paper</a> by Caigui Jiang, Hui Wang, Victor Ceballos Inza, Felix Dellinger, Florian Rist, Johannes Wallner, and Helmut Pottmann. Would you have believed an independently-designed, sleek, freeform shape like in the picture above could be assembled from a small alphabet of curved surface patches? The secret is to allow the patches to bend isometrically, that is with constant intrinsic (Gaussian) curvature--as thin shells naturally tend to do--vastly extending the design space of a small set of tools. Bravo!James Malloshttp://www.blogger.com/profile/00763341541407040741noreply@blogger.com0tag:blogger.com,1999:blog-7156771617507620492.post-26323750739346554032021-05-27T07:32:00.000-07:002021-05-27T07:32:38.849-07:00Customizing TPMS to fit<div class="separator" style="clear: both;"><a href="https://1.bp.blogspot.com/-s16oMgXkmb0/YK-op9eRfZI/AAAAAAAAJqg/BH77pH0IQBMVAofIrQCcPLD6MnRPe_dlwCPcBGAsYHg/s1463/Screen%2BShot%2B2021-05-27%2Bat%2B9.46.58%2BAM.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="320" data-original-height="1136" data-original-width="1463" src="https://1.bp.blogspot.com/-s16oMgXkmb0/YK-op9eRfZI/AAAAAAAAJqg/BH77pH0IQBMVAofIrQCcPLD6MnRPe_dlwCPcBGAsYHg/s320/Screen%2BShot%2B2021-05-27%2Bat%2B9.46.58%2BAM.png"/></a></div>
<p>A recent paper on 3D printing is likely also relevant to 3D weaving: "<a href="http://irc.cs.sdu.edu.cn/TPMSInjection/index.html">Strong 3D Printing by TPMS Injection</a>" by Xin Yan, Cong Rao, Lin Lu, Andrei Sharf, Haisen Zhao, and Baoquan Chen. The authors show how to modify a level-set approximation of a triply-periodic minimal surface (TPMS) to custom fit the shape and interior stress levels of a sculptural object. <a href="https://twitter.com/alisonmartin57?lang=en">Alison Grace Martin</a> has already demonstrated that many TPMS can be woven.</p>
<div class="separator" style="clear: both;"><a href="https://1.bp.blogspot.com/-hjA-oTbUjk8/YK-pi3TVLcI/AAAAAAAAJq0/YhbqDnpdezUsO1JzFJqZFoiyJ28qsloZgCPcBGAsYHg/s1010/Screen%2BShot%2B2021-05-27%2Bat%2B9.49.01%2BAM.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" height="320" data-original-height="1010" data-original-width="995" src="https://1.bp.blogspot.com/-hjA-oTbUjk8/YK-pi3TVLcI/AAAAAAAAJq0/YhbqDnpdezUsO1JzFJqZFoiyJ28qsloZgCPcBGAsYHg/s320/Screen%2BShot%2B2021-05-27%2Bat%2B9.49.01%2BAM.png"/></a></div>
James Malloshttp://www.blogger.com/profile/00763341541407040741noreply@blogger.com0