tag:blogger.com,1999:blog-40261574084605424892024-03-18T21:31:07.180-07:00MazePuzzlesThis blog was originally intended to publicize my maze books. In designing mazes for those books, I used many tessellations. Eventually I decided the tessellations were more interesting than the mazes and now the blog is mostly about tessellations.Dessert Survivorhttp://www.blogger.com/profile/04616064444288249273noreply@blogger.comBlogger67125tag:blogger.com,1999:blog-4026157408460542489.post-42377593518294042502022-02-09T18:15:00.001-08:002022-02-09T18:15:53.481-08:00Typefaces based on tessellating patterns<p> It has been more than nine months since I posted here. During that time I have been designing typefaces, investigating <a href="http://ingrimayne.com/fonts2/alternating.htm" target="_blank">what is possible</a> using the OpenType feature Contextual Alternatives (calt). There are few typefaces that use this feature to alternate two character sets, perhaps because the results appeal to a niche market and serious type designers have better uses of their time than to design for a minuscule market. I like the feature because it gives interesting results when there is a geometrical connection between the two character sets that are alternated. My first use of the feature was to alternate letter sets that were based on trapezoids.</p><p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjvmAGUm5-vaWGxnex8ce6WJJO4yTeksZFwT0_OuPXu7mwz31H0lvwPLzagG1Y550DFUJyIuU8C6QQozBb13JCazrdRxNlZZmdwH20JMq4yLnNF0UIlE2spY_dKnz0wZTyVJZlPHAoJ2Me0njyf3jmb_996Hoppdn61duNh5a1i5oV-bvbRbkGSCm4w" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="251" data-original-width="425" src="https://blogger.googleusercontent.com/img/a/AVvXsEjvmAGUm5-vaWGxnex8ce6WJJO4yTeksZFwT0_OuPXu7mwz31H0lvwPLzagG1Y550DFUJyIuU8C6QQozBb13JCazrdRxNlZZmdwH20JMq4yLnNF0UIlE2spY_dKnz0wZTyVJZlPHAoJ2Me0njyf3jmb_996Hoppdn61duNh5a1i5oV-bvbRbkGSCm4w=s16000" /></a></div><br />Notice that the trapezoids that serve as templates on which the letters are formed will tessellate. Tessellations were not on my mind when I designed this face. I was focused only on how the letters fit together in a word. However. I realized later that the template shape was part of a tessellation pattern, in the IH58 type. It can be constructed in several ways. The tops and bottoms can rotate 180º or flip to form neighbors. The sides can be C edges that rotate around their centers to form neighbors, or they can be considered glide edges. In terms of symbols, it could be CCCC, CGCG, CFCF, or GFGF. <p></p><p>The typeface above did not exhaust the possibilities of trapezoidal letters, so I did a second family with a similar template but with letters that look very different.</p><p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgJV43jWyyWNECar6AAOByD9ewluvanlHZ6EcaS8-w0m6T4LMGEtZ3897baXoYg2t6AtbERU-arM9XTjq1LCCgibGO3PXblB-SlEhXK4PQSrfnTE5NqpjuUR0UFkOsyWZvD7SAoAaaTZq1PQrWE9fFxOgk7KKDCb0jgWCAcPin_51Ik78qjkQiTo_dD" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="232" data-original-width="343" src="https://blogger.googleusercontent.com/img/a/AVvXsEgJV43jWyyWNECar6AAOByD9ewluvanlHZ6EcaS8-w0m6T4LMGEtZ3897baXoYg2t6AtbERU-arM9XTjq1LCCgibGO3PXblB-SlEhXK4PQSrfnTE5NqpjuUR0UFkOsyWZvD7SAoAaaTZq1PQrWE9fFxOgk7KKDCb0jgWCAcPin_51Ik78qjkQiTo_dD=s16000" /></a></div><br />Another trapezoidal shape that tessellates is formed by bisecting a regular hexagon. <a href="https://www.fontspring.com/fonts/ingrimayne-type/bihext" target="_blank">In this case</a> I put the straight sides between letters and the interlocking part of the pattern on the top and bottom.<p></p><p></p><div class="separator" style="clear: both; text-align: center;"><div class="separator" style="clear: both; text-align: center;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhg_qR1LMTxRjPFsKfS3-MHnoni4anw4DBKiQIZIWSuRY8Az5vvBlqLDZXP2uGQzlRHLMboo5wpkmLtRodYjfnkRI3qpHbGGjIT6qUoJAWYnPqf_OunPtBIMYeZq4ncf3Vd6E4KdNw1opn_hq8Me_DGQrC1KVqV-4UiRRYlDBLqGE9X0GranJicEsQF" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="330" data-original-width="430" src="https://blogger.googleusercontent.com/img/a/AVvXsEhg_qR1LMTxRjPFsKfS3-MHnoni4anw4DBKiQIZIWSuRY8Az5vvBlqLDZXP2uGQzlRHLMboo5wpkmLtRodYjfnkRI3qpHbGGjIT6qUoJAWYnPqf_OunPtBIMYeZq4ncf3Vd6E4KdNw1opn_hq8Me_DGQrC1KVqV-4UiRRYlDBLqGE9X0GranJicEsQF=s16000" /></a></div></div></div>The trapezoid can be asymmetrical, as <a href="https://www.fontspring.com/fonts/ingrimayne-type/zoidicfun" target="_blank">in the example</a> below where the trapezoid was formed by cutting a rectangle into two equal parts with a slanted line. The pattern here can be formed by flipping the shape over the top and bottom and rotating it around the center of its sides. It fits the Grünbaum-Shepard IH49 classification. <p></p><p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEi2SnDGNkIzmvRQS80ezktxsbjaIylLmN1D11i56A-3j97ZCVH3-IB2hN1DOvDgDQDTIGJJntjLv0h6re0OwCMCARuS7X0nqPAI7Hw7wZh8ZNtE46AUNHL-Y6CwDf3ILT7mMeINYELG3kujM34U16mE_i0li7br-ZZ3C6N_I1sAQbtKOyQV1629lg07" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="242" data-original-width="306" src="https://blogger.googleusercontent.com/img/a/AVvXsEi2SnDGNkIzmvRQS80ezktxsbjaIylLmN1D11i56A-3j97ZCVH3-IB2hN1DOvDgDQDTIGJJntjLv0h6re0OwCMCARuS7X0nqPAI7Hw7wZh8ZNtE46AUNHL-Y6CwDf3ILT7mMeINYELG3kujM34U16mE_i0li7br-ZZ3C6N_I1sAQbtKOyQV1629lg07=s16000" /></a></div><br />I followed up with letters formed with templates of rectangles and parallelograms that give a wave to a line of text. I did not think of tessellations when designing these, but <a href="https://www.fontspring.com/fonts/ingrimayne-type/undulate" target="_blank">the first</a> template shape tessellates as a TCTC pattern (and also TGTG, CGCG, and G1G2G1G2) and belongs to the IH-66 class. In <a href="https://www.fontspring.com/fonts/ingrimayne-type/bannetters" target="_blank">the second pattern</a>, the tops and bottoms can be either center-point-rotation or translated edges and the sides can be either glided or flipped edges.<p></p><p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgxqqgXnVidWlrI3Ah-2zQg01WvPK3AMYzUEz_KkiIKrYdUUF_Tyo3EkoUSKAqEt6LLoGLLxmQxTWI_obPveI5y8dUfwIoydJRiohPbZOjXffgTNFUhm6yY9xZ-ysgDYZYz7F5VUambH-H8wIfpISoOvHgrnPyc0yQc-iBM4NoJ9OrSaNYK7fW5QUl6" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="246" data-original-width="395" src="https://blogger.googleusercontent.com/img/a/AVvXsEgxqqgXnVidWlrI3Ah-2zQg01WvPK3AMYzUEz_KkiIKrYdUUF_Tyo3EkoUSKAqEt6LLoGLLxmQxTWI_obPveI5y8dUfwIoydJRiohPbZOjXffgTNFUhm6yY9xZ-ysgDYZYz7F5VUambH-H8wIfpISoOvHgrnPyc0yQc-iBM4NoJ9OrSaNYK7fW5QUl6=s16000" /></a></div><p></p><p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEiMuQ6BTugjIP_GO-EQ_tt-BP8SVSPPXWbqtIAJj9orJjeX_8co6-L5ikKfDBTj-oUPTa6RQ-yFkDIXu6ml0hgkl-M4C-zychwLhpiXZ1883H5dSLAY7KR7Rc6Amo-YOX0eNy4Sv_VY9B6rnJ-yvmnLHtd9hnr--XqihrWYhw9Km3U8XJRpKrDnbkYT" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="240" data-original-width="344" src="https://blogger.googleusercontent.com/img/a/AVvXsEiMuQ6BTugjIP_GO-EQ_tt-BP8SVSPPXWbqtIAJj9orJjeX_8co6-L5ikKfDBTj-oUPTa6RQ-yFkDIXu6ml0hgkl-M4C-zychwLhpiXZ1883H5dSLAY7KR7Rc6Amo-YOX0eNy4Sv_VY9B6rnJ-yvmnLHtd9hnr--XqihrWYhw9Km3U8XJRpKrDnbkYT=s16000" /></a></div><br />Using alternating characters one can design text that has a wave both horizontally and vertically. The template of the <a href="https://www.fontspring.com/fonts/ingrimayne-type/woven" target="_blank">first of these</a> is a IH-73 pattern (C4C4C4C4 at all corners, though it can also be formed as G1G1G2G2) and the second as an IH-71 pattern (fits both C4C4C4C4 and G1G2G1G2). The template shape in the <a href="https://www.fontspring.com/fonts/ingrimayne-type/billowed" target="_blank">second sample</a> will tessellate in one orientation, two orientations, or four orientations. Only the four-orientation pattern is shown below.<p></p><p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjw7QYSlUT2UYp9N-bUmZnYWhJuvJyjmzY7gVkD1NdhktIk023rBjsVnoYQypLVxCJci-i1CEFd5AxhWlEWo5ktn4A2eDxg85bZkPKHJmkGRCLYFBvzBAGYZL9-vKvDI9cBsSpnmdLDYKkBAkKp_JN-8WVnc-3PR9-sN9Z1uzvpzICVWHKeobRmD0cG" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="259" data-original-width="501" src="https://blogger.googleusercontent.com/img/a/AVvXsEjw7QYSlUT2UYp9N-bUmZnYWhJuvJyjmzY7gVkD1NdhktIk023rBjsVnoYQypLVxCJci-i1CEFd5AxhWlEWo5ktn4A2eDxg85bZkPKHJmkGRCLYFBvzBAGYZL9-vKvDI9cBsSpnmdLDYKkBAkKp_JN-8WVnc-3PR9-sN9Z1uzvpzICVWHKeobRmD0cG=s16000" /></a></div><p></p><p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhSHfiHt7ynKF6xseEoCmWm_LsFyf3K8CAE-aeYbuxL1tSXdDbi7Cs-d5HBYtrKUJ5LcB2ZSP1Y36ju-fBCKU0qYnweGeXpYDQu-u8kmHtGHLJ70KaSBMxg2lOwmGyThzBU8cQZTHaX6WH4UP7-vCxsJuDJlfa8Nl37c5BY2Ezmqf0ZkiFBPM8YZxCN" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="260" data-original-width="507" src="https://blogger.googleusercontent.com/img/a/AVvXsEhSHfiHt7ynKF6xseEoCmWm_LsFyf3K8CAE-aeYbuxL1tSXdDbi7Cs-d5HBYtrKUJ5LcB2ZSP1Y36ju-fBCKU0qYnweGeXpYDQu-u8kmHtGHLJ70KaSBMxg2lOwmGyThzBU8cQZTHaX6WH4UP7-vCxsJuDJlfa8Nl37c5BY2Ezmqf0ZkiFBPM8YZxCN=s16000" /></a></div><br />Both of these typefaces are based on a template of distorted squares. <p></p><p>I suspect a reason that I am willing to pursue this line of inquiry is because I like tessellations more than other type designers. No one else that I know of has designed multiple fonts from shapes that can form tessellation patterns.</p><p>For more information about these and other typefaces with alternating characters, see my type blog at <a href="https://nohypetype.blogspot.com/" target="_blank">nohyptype.blogspot.com</a></p>Dessert Survivorhttp://www.blogger.com/profile/04616064444288249273noreply@blogger.com0tag:blogger.com,1999:blog-4026157408460542489.post-83328489819060432642021-04-20T13:30:00.002-07:002021-04-26T13:05:03.627-07:00Grandpa's Amazing Tale: A Book of Mazes<p>It has been almost three years since I released <i><a href="https://www.amazon.com/Mazes-Escher-Would-Robert-Schenk/dp/1719016712" target="_blank">Mazes Escher Would Like</a></i> and after that book I did not expect to do another. There are so many maze books available that it is difficult to be seen. Also, the software I use is old and no longer runs on currently-available computers. And yet for some reason that I do not fully understand, in April of 2021 I decided to do one more.</p><p>As I was sorthing through old documents on my computer, I stumbled on an old story I had written. I thought was funny and decided to illustrate it with mazes. Most of the mazes I used were old designs, but I did develop over twenty new maze designs for this book. </p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiz0ogRwdUVc3sBAF8anHw96d4szjd7blq5h_wK3zDsDOKrmqOQN1S8SqnQZX6M_1yyGPO17ceav2b6vJUfCyvmXtv0UJypOq2B_wI6QBN3IDsZRKoQxofIzXQ4db_oGHPGYJx2v7xpVQKc/s300/Screen+Shot+2021-04-13+at+1.32.54+PM.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="288" data-original-width="300" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiz0ogRwdUVc3sBAF8anHw96d4szjd7blq5h_wK3zDsDOKrmqOQN1S8SqnQZX6M_1yyGPO17ceav2b6vJUfCyvmXtv0UJypOq2B_wI6QBN3IDsZRKoQxofIzXQ4db_oGHPGYJx2v7xpVQKc/s16000/Screen+Shot+2021-04-13+at+1.32.54+PM.png" /></a></div><p>Yes, I tessellated toilets and made a maze from the pattern. Below is another tessellation pattern I used.</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYDG58rFaXkvhZWka-5UJEr3N7GMVmKD3sh-tnA9BrZmwIHEBkqYLAwsifBcCksr-X_ebGLGmdCbNGhXtfAoO32adQZASjxeGGiOftRp2Y3NNu8DXZCf0kSoHkvo-MuenmeL9v-Y472gFN/s368/Screen+Shot+2021-04-13+at+1.33.18+PM.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="320" data-original-width="368" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYDG58rFaXkvhZWka-5UJEr3N7GMVmKD3sh-tnA9BrZmwIHEBkqYLAwsifBcCksr-X_ebGLGmdCbNGhXtfAoO32adQZASjxeGGiOftRp2Y3NNu8DXZCf0kSoHkvo-MuenmeL9v-Y472gFN/s320/Screen+Shot+2021-04-13+at+1.33.18+PM.png" width="320" /></a></div>The above maze would be trivial if displayed in a friendly typeface but it is one of the most difficult mazes in the book because it is so hard to follow the little passages that connect. It also posed a challenge for the way I construct mazes because the basic pattern is a grid of diamonds, so everything connects diagonally. <div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgrzyI9GN_WwYWm09-Xocr3ttZduEAdu-NanF7YH1Y15BlJRUzApg9iY6YlHm_-YWSfn2l9vR7PGoP-4G6umJmRldMVIltm-PRL_m0I_zG5J97NZ5EctJ-K-1CD7IlIIpW3PP0tes6SJkXo/s329/Screen+Shot+2021-04-13+at+1.33.31+PM.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="323" data-original-width="329" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgrzyI9GN_WwYWm09-Xocr3ttZduEAdu-NanF7YH1Y15BlJRUzApg9iY6YlHm_-YWSfn2l9vR7PGoP-4G6umJmRldMVIltm-PRL_m0I_zG5J97NZ5EctJ-K-1CD7IlIIpW3PP0tes6SJkXo/s16000/Screen+Shot+2021-04-13+at+1.33.31+PM.png" /></a></div>The shape used above was also a challenge for my maze construction set. The basic shape is triangular, with one orientation adjacent to two above and the other with two below. I treated it as a maze of triangles and used two typefaces to render it.<div><br /></div><div>Not everything is tessellation based. I used foot prints to form the walls of a maze that allows passages up and down, right and left, and diagonally.</div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhi4u6UFNgo8Iy0gJOimStU21A4HHE5gUL7WMfJ7Z-J2F_Zt11mQosUsslL0pgVkOg-juZGOQaLty6MpxOAu9EfDfxZRCCae-yagPW_Rj86L-53HuFSLoXUdwNWQTEbNEUvR1Xm-0R7A703/s296/Screen+Shot+2021-04-13+at+1.33.44+PM.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="278" data-original-width="296" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhi4u6UFNgo8Iy0gJOimStU21A4HHE5gUL7WMfJ7Z-J2F_Zt11mQosUsslL0pgVkOg-juZGOQaLty6MpxOAu9EfDfxZRCCae-yagPW_Rj86L-53HuFSLoXUdwNWQTEbNEUvR1Xm-0R7A703/s0/Screen+Shot+2021-04-13+at+1.33.44+PM.png" /></a></div><br /><div>The book is <i>Grandpa's Amazing Tale: A Book of Mazes.</i> It uses mazes to illustrate a silly story told by a grandparent to a granddaughter and involves time travel and dinosaurs, so it is both a story book and a puzzle book. It is intended for children in early elementary school.</div><div><br /></div><div>It is published through Amazons's Kindle Direct Publishing and is available in <a href="https://www.amazon.com/dp/B092P778QS" target="_blank">paperback only at Amazon</a>.</div>Dessert Survivorhttp://www.blogger.com/profile/04616064444288249273noreply@blogger.com0tag:blogger.com,1999:blog-4026157408460542489.post-43252662593545133842020-09-11T04:47:00.002-07:002020-09-11T04:47:58.170-07:00TessieSomeMoreIn August I took a break for updating and creating alphabetic fonts to play with <a href="http://www.tesselmaniac.com/blog/">Tesselmanic! </a>in an effort to create another tessellation font. With a lot of trial and error (mostly error), I was able to come up with over 40 designs that would fit into a typeface. I could not think of a clever name so settled for TessieSomeMore, indicating that it was a continuation of the <a href="http://ingrimayne.com/fonts2/Tessies.html">Tessie series</a> of fonts.<br />
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As I played with the various templates in Tesselmaniac!, I found that sometimes I would create something quite similar to what I had done in the past. One example is shape that resembles a butterfly. Below is the most recent version.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcHEaUUN3cEGgGIYzUtJQdtHyS06L1pAoZHNZnBaB26cSPApGGvGTb-8RtYs5TAqSeO2fqSx5fIoKVdYqb_aG8WpsEIftqbotLACy-LMF5W7aacHi2LKEd_QKHXAW1sYBAxfQmF3aXM3A/s1600/Screen+Shot+2020-09-05+at+10.11.58+AM.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="302" data-original-width="462" height="261" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcHEaUUN3cEGgGIYzUtJQdtHyS06L1pAoZHNZnBaB26cSPApGGvGTb-8RtYs5TAqSeO2fqSx5fIoKVdYqb_aG8WpsEIftqbotLACy-LMF5W7aacHi2LKEd_QKHXAW1sYBAxfQmF3aXM3A/s400/Screen+Shot+2020-09-05+at+10.11.58+AM.png" width="400" /></a></div>
Looking through past efforts, I found something quite similar in <a href="https://www.myfonts.com/fonts/ingrimayne/tessiebugs/">TessieBugs</a>. I kept the new version because it is so easily recognizable as an insect.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhs_TcH93vxB5WPYrXXVj2iNYfcVGMRY7htrNnER6FUWu7pp7ZAfNMSsLpkhd_PpNH0yglqdbcFUphgLQiFGl3Kd626ODFM4bqFzLmqqSkUPzX-NkEQJ0Ygp1ElfShRXOv-nb9T3kbyHaU/s1600/Screen+Shot+2020-09-05+at+10.12.44+AM.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="257" data-original-width="349" height="235" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhs_TcH93vxB5WPYrXXVj2iNYfcVGMRY7htrNnER6FUWu7pp7ZAfNMSsLpkhd_PpNH0yglqdbcFUphgLQiFGl3Kd626ODFM4bqFzLmqqSkUPzX-NkEQJ0Ygp1ElfShRXOv-nb9T3kbyHaU/s320/Screen+Shot+2020-09-05+at+10.12.44+AM.png" width="320" /></a></div>
I noticed that if I added a straight segment where the wings touch, I could convert a tessellation that is IH68 to one that has six adjacent tiles and is type IH14.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiovqLD5ayG4MmX7oAhXq6BvlvgO6YByZMsvAklaScBEqaS1S5WjPfDmvhLFFtbp3wWKh7OyPbpLZ-W3l6lE5OXBq-X2PaCHV-5TKwLHcqUEYpeL3nanCsPTXQzmjkHJ4ZAdbP2JGxeWXs/s1600/Screen+Shot+2020-09-07+at+8.45.04+PM.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="221" data-original-width="291" height="303" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiovqLD5ayG4MmX7oAhXq6BvlvgO6YByZMsvAklaScBEqaS1S5WjPfDmvhLFFtbp3wWKh7OyPbpLZ-W3l6lE5OXBq-X2PaCHV-5TKwLHcqUEYpeL3nanCsPTXQzmjkHJ4ZAdbP2JGxeWXs/s400/Screen+Shot+2020-09-07+at+8.45.04+PM.png" width="400" /></a></div>
Another insect design that resembles a past effort is this mite.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjAPfYEVykE-I8f-8j_a9IGqFld8jMN5g9nNZAqu1Sbi5QOsFo8XiqciN6-K_Zg1sczvmIODtjeiPkEH9QomI9jnG4kYG7cWVaP8DBwrPiR3MOozhAJfIV7eQmLKwMklCsMs191TlkvS3o/s1600/Screen+Shot+2020-09-05+at+10.11.50+AM.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="298" data-original-width="521" height="228" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjAPfYEVykE-I8f-8j_a9IGqFld8jMN5g9nNZAqu1Sbi5QOsFo8XiqciN6-K_Zg1sczvmIODtjeiPkEH9QomI9jnG4kYG7cWVaP8DBwrPiR3MOozhAJfIV7eQmLKwMklCsMs191TlkvS3o/s400/Screen+Shot+2020-09-05+at+10.11.50+AM.png" width="400" /></a></div>
However, the old mite had six legs, so the new one is substantially different.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEilpooqigcRVy7WdArLmxvsvXCLSbz3G8JD1khRyKmS-VHvif3eHpplC5GvC1vuVOj6-i_F92dxkNUxdsj7Ua4fzUm68nkg9gfKU6E_eLbyOI8ZD4KpIK6f6lflTkXYmGmHnuapRSVBmv8/s1600/Screen+Shot+2020-09-05+at+10.13.00+AM.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="296" data-original-width="379" height="311" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEilpooqigcRVy7WdArLmxvsvXCLSbz3G8JD1khRyKmS-VHvif3eHpplC5GvC1vuVOj6-i_F92dxkNUxdsj7Ua4fzUm68nkg9gfKU6E_eLbyOI8ZD4KpIK6f6lflTkXYmGmHnuapRSVBmv8/s400/Screen+Shot+2020-09-05+at+10.13.00+AM.png" width="400" /></a></div>
There are certain shapes that many people will stumble on if they play with tessellations enough.<br />
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Creating Escher-like pattern with the isohedral classification IH74 is challenging because there are not a lot of real-world objects that mirror both horizontally and vertically. Below is a stylized moth that fits the classification.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiaGGeiLCvwqaXpPuRfP0Fhx_YULdpOiDPtmrzVdA8fuZ_-wH9YStOVCX1utHnxK6tjJycwYXfp5raWEpsc9dw3htPIEEY5YlVBoWGs5pFiy49R2tcZVtTCiwQgJVku1NL2oxCA6isMcZA/s1600/Screen+Shot+2020-09-07+at+8.44.19+PM.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="210" data-original-width="319" height="263" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiaGGeiLCvwqaXpPuRfP0Fhx_YULdpOiDPtmrzVdA8fuZ_-wH9YStOVCX1utHnxK6tjJycwYXfp5raWEpsc9dw3htPIEEY5YlVBoWGs5pFiy49R2tcZVtTCiwQgJVku1NL2oxCA6isMcZA/s400/Screen+Shot+2020-09-07+at+8.44.19+PM.png" width="400" /></a></div>
Again, the addition of a straight line where the wings touch turns it into a hexagonal figure. It retains all of its symmetry and is now IH17, another classification which is very difficult to use for Escher-like tessellations.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj61q8TO1-q5i8xgrEdDN5IfeqR3aQGpva43kzBZA5mmhwKUsJNq096xhD0HZ9uSzML2hnjzy_wyZLq8MRLFggw5o_Y8457-bDX5A9HEGCEu35iAwsXFTt9AJ1Sv59yquV-wVZbD7CARHQ/s1600/Screen+Shot+2020-09-07+at+8.44.29+PM.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="326" data-original-width="358" height="363" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj61q8TO1-q5i8xgrEdDN5IfeqR3aQGpva43kzBZA5mmhwKUsJNq096xhD0HZ9uSzML2hnjzy_wyZLq8MRLFggw5o_Y8457-bDX5A9HEGCEu35iAwsXFTt9AJ1Sv59yquV-wVZbD7CARHQ/s400/Screen+Shot+2020-09-07+at+8.44.29+PM.png" width="400" /></a></div>
It is always exciting to find a new tessellating shape that is recognizable as an object. TessieSomeMore includes a shape recognizable as a toilet.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg2glSN9Y59AbxNhlcpypvUdo9OnQi0Rm9vQyhGgy7pHGbdudsl1PUruUSeWCeTNbtTtbKx8sCtoMVkt6aIWpvqG4bIJb3x-Zzp5GxaJInyefuYim2Uhl1McYUKFXtr9R4oiWfl1RaJtGE/s1600/Screen+Shot+2020-09-03+at+8.39.37+PM.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="230" data-original-width="366" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg2glSN9Y59AbxNhlcpypvUdo9OnQi0Rm9vQyhGgy7pHGbdudsl1PUruUSeWCeTNbtTtbKx8sCtoMVkt6aIWpvqG4bIJb3x-Zzp5GxaJInyefuYim2Uhl1McYUKFXtr9R4oiWfl1RaJtGE/s1600/Screen+Shot+2020-09-03+at+8.39.37+PM.png" /></a></div>
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Another shape unlike any that I had previous found was one that resembles a creepy insect.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6KkZ_RvaaFfRLKiFZOfq_pGjz1uO31SOM-bRXzSUVkpttOCPqfCfCTQiHsDzKUjMGQSguI4zTjodiCpSZjwVgS9ALTn95f_xikHkRbnRA117B0nYBkWpp9m6nOcFjE9efHwl2L6jZOnU/s1600/Screen+Shot+2020-09-05+at+10.11.35+AM.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="484" data-original-width="531" height="291" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6KkZ_RvaaFfRLKiFZOfq_pGjz1uO31SOM-bRXzSUVkpttOCPqfCfCTQiHsDzKUjMGQSguI4zTjodiCpSZjwVgS9ALTn95f_xikHkRbnRA117B0nYBkWpp9m6nOcFjE9efHwl2L6jZOnU/s320/Screen+Shot+2020-09-05+at+10.11.35+AM.png" width="320" /></a></div>
I tried to find interesting Escher-like tessellations in which the tile had two-fold rotational symmetry without mirror symmetry and came up empty. There are many real-world objects with reflective symmetry but very few with only rotational symmetry.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhGXKH7dDbfRvfortFZ6T5NxFmH0xHpnQ0CWM5nIyA-uH66k5ucLhw8OZfItW6gPBKKNaWnKPEJtpdSvYQXXrlMnJ3Fd4o14j1gI1i1D3uZaoYAWTLjLs9_LKs7RdvpEnaauB1hLV49Kcw/s1600/Screen+Shot+2020-09-05+at+10.23.47+AM.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="320" data-original-width="494" height="258" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhGXKH7dDbfRvfortFZ6T5NxFmH0xHpnQ0CWM5nIyA-uH66k5ucLhw8OZfItW6gPBKKNaWnKPEJtpdSvYQXXrlMnJ3Fd4o14j1gI1i1D3uZaoYAWTLjLs9_LKs7RdvpEnaauB1hLV49Kcw/s400/Screen+Shot+2020-09-05+at+10.23.47+AM.png" width="400" /></a></div>
TessisSomeMore is available from <a href="https://www.myfonts.com/fonts/ingrimayne/tessie-some-more/" target="_blank">myfonts</a>. On the page linked, there is a link to a file that includes samples of all the patterns included in the typeface.<div><br /></div><div>(See also this <a href="https://nohypetype.blogspot.com/2020/09/tessiesomemore.html" target="_blank">post at nohypetype.blogspot.com.</a>)</div>Dessert Survivorhttp://www.blogger.com/profile/04616064444288249273noreply@blogger.com0tag:blogger.com,1999:blog-4026157408460542489.post-33837483945262034692019-12-27T07:53:00.000-08:002020-01-18T19:09:02.474-08:00TessieLetters and an unusual alphabetI have finally returned to and finished a series of TessieLetter typefaces that complement the eleven Tessie typefaces (<a href="https://mazepuzzles.blogspot.com/2018/11/tessellation-typefaces-part-1.html">here</a>, <a href="https://mazepuzzles.blogspot.com/2018/11/tessellating-typefaces-part-2.html">here</a>, <a href="https://mazepuzzles.blogspot.com/2018/12/tessellating-typefaces-part-3.html">here</a>, and <a href="https://mazepuzzles.blogspot.com/2019/01/two-more-tessellation-fonts.html">here</a>) that allow one to create tessellation patterns of birds, animals, bugs, and variety of other shapes. TessieLetters is made up of seven different fonts (each in solid and outline styles) that contain all the letters of the alphabet as well as the numbers. You can find them at fontspring.com: <a href="https://www.fontspring.com/fonts/ingrimayne-type/tessielettersace">here</a>, <a href="https://www.fontspring.com/fonts/ingrimayne-type/tessielettersfq">here</a>, <a href="https://www.fontspring.com/fonts/ingrimayne-type/tessielettersgjkmn">here</a>, <a href="https://www.fontspring.com/fonts/ingrimayne-type/tessielettersll">here</a>, <a href="https://www.fontspring.com/fonts/ingrimayne-type/tessielettersosz">here</a>, <a href="https://www.fontspring.com/fonts/ingrimayne-type/tessieletterstt">here</a>, and <a href="https://www.fontspring.com/fonts/ingrimayne-type/tessieletterssingles">here</a> and at myfonts <span id="goog_1563137456"></span><a href="https://www.myfonts.com/fonts/ingrimayne/tessie-letters/">here</a><span id="goog_1563137457"></span>.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEimGPOGYe-gUsfi9s7ifY1GzvPMou23KOXuu2Pts_nH22d03IDj4rIGO5r0OySHuBFCiOgBsISY3WjDvDOxzXTvip8s3TwnYjBrODfDFSflGpymA6v3EFVkuM9Y7EqICnj4LUYIZStAHJ4/s1600/TessieLettersZ.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEimGPOGYe-gUsfi9s7ifY1GzvPMou23KOXuu2Pts_nH22d03IDj4rIGO5r0OySHuBFCiOgBsISY3WjDvDOxzXTvip8s3TwnYjBrODfDFSflGpymA6v3EFVkuM9Y7EqICnj4LUYIZStAHJ4/s640/TessieLettersZ.png" width="640" /></a></div>
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<a href="https://www.fontspring.com/fonts/ingrimayne-type/tessieletterssingles">One of the seven</a> in the series contains only shapes that can be tessellated using a single key. Somewhat surprisingly, I was able to find ways to do the complete alphabet in this way, though some of the letters require a bit of imagination. (U, T, and G are not ideal.) A breakthrough came when I figured out a way to form the letter P. The same shape works for lower-case b, d, q and letters 6 and 9. Below is a picture of the entire one-key tessellating alphabet.<br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhUgXWCGwhxlSiiOgObWMmtcnLFWvf_n4Rk8gW8Y1_AiLHFVqRrHFg3oi1WL3lhysvBjPAJKtTi62HJpM2oc1uTzbN3W3AVOVd_QD0pI1un_h-Lv84nW7u0deS35w7g3CkymvZjQObhQQ8/s1600/Screen+Shot+2019-12-08+at+8.37.44+PM.png" imageanchor="1"><img border="0" height="328" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhUgXWCGwhxlSiiOgObWMmtcnLFWvf_n4Rk8gW8Y1_AiLHFVqRrHFg3oi1WL3lhysvBjPAJKtTi62HJpM2oc1uTzbN3W3AVOVd_QD0pI1un_h-Lv84nW7u0deS35w7g3CkymvZjQObhQQ8/s640/Screen+Shot+2019-12-08+at+8.37.44+PM.png" width="640" /></a><br />
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All these patterns would fit as Heesch types TTTT or TTTTTT. Many have symmetry that would allow them to fit in other Heesch types as well. Also, for several letters (such as f, h, m, n, s, and z) there are multiple shapes that work.<br />
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I am unaware of anyone else who has constructed an alphabet with this property.<br />
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In the spirit of the season:<br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi4Y8rkZqNqpL8oolkMoNPHpHRPYVG8KFKf93BvCGVkMZ9YITEOtw11Yw8m4wL1i1D1fSyPh-qcALj59ghkITaOb7H8Z5jOMRCrn2ujlN490KWp3_igKFKnpmWcQMCjwzNUjhm6IzyD74Q/s1600/Screen+Shot+2019-12-10+at+7.34.43+PM.png" imageanchor="1"><img border="0" height="266" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi4Y8rkZqNqpL8oolkMoNPHpHRPYVG8KFKf93BvCGVkMZ9YITEOtw11Yw8m4wL1i1D1fSyPh-qcALj59ghkITaOb7H8Z5jOMRCrn2ujlN490KWp3_igKFKnpmWcQMCjwzNUjhm6IzyD74Q/s640/Screen+Shot+2019-12-10+at+7.34.43+PM.png" width="640" /></a><br />
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(Cross posted at <a href="https://nohypetype.blogspot.com/2019/12/tessieletters-and-usual-alphabet.html">nohypetype.blogspot.com</a>)Dessert Survivorhttp://www.blogger.com/profile/04616064444288249273noreply@blogger.com0tag:blogger.com,1999:blog-4026157408460542489.post-81539409170331857772019-01-24T09:16:00.003-08:002019-01-24T09:18:01.847-08:00Tessellating Typefaces (part 4)The addition of two more fonts of Escher-like tessellations at myfonts.com brings the total of the new tessie series to eleven. Four are of birds, one of other animals, one of bugs, one of puzzle pieces, and four of everything else.<br />
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One of the recent additions is <a href="https://www.myfonts.com/fonts/ingrimayne/tessiebugs/">TessieBugs</a>. It contains tessellating butterflies, moths, ants, and other creepy, crawly insects.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhVPKZ3GMcCrc7XpCphIEJtAz8lqcOVXaESAbFZ722kcT08XuWtXEaVqpEWrX27Z_g07ncndImpMNoQQZ8n4fJtSVuQJdrXyE75rDIWXmHZrch7ps_J2X2eS01L-pno51NXPtaFJsvJQHA/s1600/TessieBugsFlag.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhVPKZ3GMcCrc7XpCphIEJtAz8lqcOVXaESAbFZ722kcT08XuWtXEaVqpEWrX27Z_g07ncndImpMNoQQZ8n4fJtSVuQJdrXyE75rDIWXmHZrch7ps_J2X2eS01L-pno51NXPtaFJsvJQHA/s1600/TessieBugsFlag.png" /></a></div>
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The other is TessieOddsNends, a hodgepodge of things that did not make it into any of the other ten faces.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgciVx0XXeMzEC3_MI3h7TQ6OIsIfc71yAjwidq5OqkHOC58ht_M2PguWwvv4rKyFIs7H8ubjJEYJFlfj7nD1UygpJOtyn2ClIGc2TlwBBxy6gXZVkq4yEWlNGfN0MmamjZ6OjSUOri8sU/s1600/TessieOddsNendsFlag.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgciVx0XXeMzEC3_MI3h7TQ6OIsIfc71yAjwidq5OqkHOC58ht_M2PguWwvv4rKyFIs7H8ubjJEYJFlfj7nD1UygpJOtyn2ClIGc2TlwBBxy6gXZVkq4yEWlNGfN0MmamjZ6OjSUOri8sU/s1600/TessieOddsNendsFlag.png" /></a></div>
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Each contains two styles, a solid style that must be colored in order to see the shapes (after all, a tessellation fills the plane with no gaps or overlaps), and an outline style that can be used alone or layered over the solid style.</div>
Dessert Survivorhttp://www.blogger.com/profile/04616064444288249273noreply@blogger.com0tag:blogger.com,1999:blog-4026157408460542489.post-67684932192179229442018-12-06T12:13:00.002-08:002018-12-06T12:13:34.888-08:00Tessellating Typefaces (part 3)Three more tessellating typefaces have been added to Myfonts.com. Included is a fourth typeface of birds, <a href="https://www.myfonts.com/fonts/ingrimayne/tessiextrabirds/">TessieXtraBirds</a>. When playing with <a href="http://www.tesselmaniac.com/tess/Home.html">TesselManiac</a> and before it TesselMania, I have had the tendency to see bird possibilities much more readily than other shapes.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjVzxwerX13NdeS6xliR1dDaRlkxTRaYHqeywjtBQkmXajDM6wgi69ZXJqm2_CaAxcZ9l2hiGtrptxr1OtHP5jRSBegPgCBNsZqrrt8dnGpQzgjG_qrb-sVbyv3VJD4ZqIxEIwT-bNraf0/s1600/TessieExtraBordsPoster.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjVzxwerX13NdeS6xliR1dDaRlkxTRaYHqeywjtBQkmXajDM6wgi69ZXJqm2_CaAxcZ9l2hiGtrptxr1OtHP5jRSBegPgCBNsZqrrt8dnGpQzgjG_qrb-sVbyv3VJD4ZqIxEIwT-bNraf0/s640/TessieExtraBordsPoster.png" width="640" /></a></div>
The other two typefaces have a variety of shapes and no clear theme as their names indicate. Below is a poster for <a href="https://www.myfonts.com/fonts/ingrimayne/tessiemorestuff/">TessieMoreStuff</a>. The plane shape has four edges that are all shaped with identical center-point rotation. The poster has them in three different tiling arrangements.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjBApNU5F1sBIW6XsYlXFm5bDL1LKeoE0C3Nz5pTodWmouNTeXF1f4N4xJYXDx39bs3Q-2Qv0kLHoISFXgygNvJVcVT3rRisyzIjYy-5iWaJXc6WakD63VbJdWZlbZPMho8764Xzrnipmg/s1600/TessieMoreStuffPanes.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjBApNU5F1sBIW6XsYlXFm5bDL1LKeoE0C3Nz5pTodWmouNTeXF1f4N4xJYXDx39bs3Q-2Qv0kLHoISFXgygNvJVcVT3rRisyzIjYy-5iWaJXc6WakD63VbJdWZlbZPMho8764Xzrnipmg/s640/TessieMoreStuffPanes.png" width="640" /></a></div>
The musical notes from <a href="http://www.myfonts.com/fonts/ingrimayne/tessiemiscellaneous/">TessieMiscellaneous</a> are a based on a triangle. It is a pattern unlike any that I have seen elsewhere.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjoyko1SBqDByYxIEbYW-hmhdxywp715zlYWCM07y7Qf_5k-KP5szVZFxTSNiPaYUorfCXyXbRAmQ3aiPRQYaUb79j7l-2vhj_gxfBy1Oq2-LErZAp1UiZTDi1SP4Ik7Bwd-fZGSHy1FWI/s1600/TessieMiscMusicPoster.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjoyko1SBqDByYxIEbYW-hmhdxywp715zlYWCM07y7Qf_5k-KP5szVZFxTSNiPaYUorfCXyXbRAmQ3aiPRQYaUb79j7l-2vhj_gxfBy1Oq2-LErZAp1UiZTDi1SP4Ik7Bwd-fZGSHy1FWI/s640/TessieMiscMusicPoster.png" width="640" /></a></div>
Each family contains two styles, a solid style that must be colored and an outline style that can be used alone or layered over the solid style (as shown in the two top illustrations). Each has a sample file that shows what key combination are needed to produce the various patterns that are possible.<br />
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(Cross posted at <a href="https://nohypetype.blogspot.com/2018/12/tessellating-typefaces-part-3.html">nohypetype.blogspot.com</a>)Dessert Survivorhttp://www.blogger.com/profile/04616064444288249273noreply@blogger.com0tag:blogger.com,1999:blog-4026157408460542489.post-8413722181482930182018-11-20T09:40:00.002-08:002019-01-24T09:18:17.755-08:00Tessellating Typefaces (part 2)Three more tessellating typefaces have been added to Myfonts.com. Included is another typeface of birds, <a href="https://www.myfonts.com/fonts/ingrimayne/tessiemorebirds/">TessieMoreBirds</a>. Many of the bird shapes in this typeface are of swimming birds. I noticed that combining two of the swimming birds in the tiling below would result in both swimming and flying birds (shown on the right). (The flying birds will not tessellate alone.)<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjuqkOCaBFXoueligOTBaMRtfB7icdOcACBO3N8yn1mmIUJokZinNdxy_WGSd18TEjilwyhyOmhdup_x-u5STQA6V8ov9S_jonZta09_QD6FGnfa34WhNAoYet1fpH_ScnKDMxZOqNV67k/s1600/TessieMoreBirdsPoster2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="720" data-original-width="1440" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjuqkOCaBFXoueligOTBaMRtfB7icdOcACBO3N8yn1mmIUJokZinNdxy_WGSd18TEjilwyhyOmhdup_x-u5STQA6V8ov9S_jonZta09_QD6FGnfa34WhNAoYet1fpH_ScnKDMxZOqNV67k/s640/TessieMoreBirdsPoster2.png" width="640" /></a></div>
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<a href="http://www.myfonts.com/fonts/ingrimayne/tessiespinners/">TessieSpinners</a> has a variety of shapes that have suggest spinning or rotating.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEihIHi2X8QXtIgEVs3PggAyhl-2eXP26_0x-U7YLuOhjGlSGM_XW_oo0GQ4JTy0FOOs7S_L_9PEY7C6qGJ6ko7Mu0AmsFvHQuSbxqcb4Mltb8vDJUvBre8TlypgasAaRsZKl2PoFxwW-Ns/s1600/TessieSpinners2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="720" data-original-width="1440" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEihIHi2X8QXtIgEVs3PggAyhl-2eXP26_0x-U7YLuOhjGlSGM_XW_oo0GQ4JTy0FOOs7S_L_9PEY7C6qGJ6ko7Mu0AmsFvHQuSbxqcb4Mltb8vDJUvBre8TlypgasAaRsZKl2PoFxwW-Ns/s640/TessieSpinners2.png" width="640" /></a></div>
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Finally, <a href="https://www.myfonts.com/fonts/ingrimayne/tessieanimals/">TessieAnimals</a> has tessellating shapes that resemble fish and mammals.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgy28mWYHUlmZ9Za9vBSu_3_VAGl8J3KGGVfPfQM-oS4kbZRkMR8lAa2nw6vJSuwHA29XCpzL0tT1cqn6IubPHZbdUCephVPd1cHThRi9iNDOYCTn_3Ch4JuVy6ppYF9pX-EexVakRisHY/s1600/TessieAnimalsPoster.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="720" data-original-width="1440" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgy28mWYHUlmZ9Za9vBSu_3_VAGl8J3KGGVfPfQM-oS4kbZRkMR8lAa2nw6vJSuwHA29XCpzL0tT1cqn6IubPHZbdUCephVPd1cHThRi9iNDOYCTn_3Ch4JuVy6ppYF9pX-EexVakRisHY/s640/TessieAnimalsPoster.png" width="640" /></a></div>
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Part 1, with three other tessellating typefaces, <a href="http://mazepuzzles.blogspot.com/2018/11/tessellation-typefaces-part-1.html">is here</a>.<br />
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(Cross posted to <a href="https://nohypetype.blogspot.com/2018/11/tessellating-typefaces-part-2.html">NoHypeType.blogspot.com</a>)Dessert Survivorhttp://www.blogger.com/profile/04616064444288249273noreply@blogger.com0tag:blogger.com,1999:blog-4026157408460542489.post-17243079551077999462018-11-09T14:35:00.001-08:002018-11-11T10:14:36.315-08:00Tessellation typefaces (part 1)Over the past twenty years I have developed/created/found a lot of tessellations patterns as I have designed <a href="https://www.amazon.com/Robert-Schenk/e/B001KDZPXI/ref=dp_byline_cont_pop_book_1">maze books, coloring books, an activity book, and a book about tessellations</a>. A few years ago I decided to make a couple of typefaces from some of this material, but those typefaces were <a href="https://www.myfonts.com/fonts/ingrimayne/tessie-dingies/">only outlines</a>. This past year I again decided to try to use this material, which has since grown, in typefaces. After a dead end or two, I have mostly finished the effort and am now putting the resulting typefaces on myfonts.com. In this new effort I have both outline and solid versions of each pattern so that the user has much greater flexibility than with the first efforts. To aid the user in using the fonts, for each there is a gallery file in a pdf format that shows what characters are needed to get each tessellation pattern.<br />
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<a href="https://www.myfonts.com/fonts/ingrimayne/tessiepuzzlepieces/">TessiePuzzlePieces</a> contain puzzle pieces. The reason I created these shapes can be found at the link.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhQGj4fPmWze41q9dOGP8X218D31mCanUx5B6i5_oiMEPL8kl1pauB5fzAPcYgeFhv11UQwppbM9EAwill57dxn-LOaQEmAZHMxz1D8Q6K7gmcuA_zYg1VE2VN7v9r3U24tFGbdziMsAMM/s1600/TessiePuzzlePiecesPoster.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="720" data-original-width="1440" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhQGj4fPmWze41q9dOGP8X218D31mCanUx5B6i5_oiMEPL8kl1pauB5fzAPcYgeFhv11UQwppbM9EAwill57dxn-LOaQEmAZHMxz1D8Q6K7gmcuA_zYg1VE2VN7v9r3U24tFGbdziMsAMM/s400/TessiePuzzlePiecesPoster.png" width="400" /></a></div>
<a href="https://www.myfonts.com/fonts/ingrimayne/tessiestandingbirds/">TessieStandingBirds</a> was a result from a quest to see how many different Heesch types I could illustrate with birds standing on the backs of other birds.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhxBJr2Ou3uoeBGX4KNpsg1bNuT-Arc5w6VeMq9nk2KYRDwnKEe9gARk2Mz2Kd3h6unqFHkqd7xqjESdO1aWzOayXrffMgG_sBlNrSv7XNir7USnZaQ3i8KwUMYRyBotLuKkzurLp-JjAU/s1600/TessieStandingBirdsPoster.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="720" data-original-width="1440" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhxBJr2Ou3uoeBGX4KNpsg1bNuT-Arc5w6VeMq9nk2KYRDwnKEe9gARk2Mz2Kd3h6unqFHkqd7xqjESdO1aWzOayXrffMgG_sBlNrSv7XNir7USnZaQ3i8KwUMYRyBotLuKkzurLp-JjAU/s400/TessieStandingBirdsPoster.png" width="400" /></a></div>
<a href="https://www.myfonts.com/fonts/ingrimayne/tessieflyingbirds/">TessieFlyingBirds</a> has a large variety of flying birds. I tried to limit the use of characters that are not readily reached from the standard keyboard, so some flying bird shapes are on other Tessie fonts that are in the process of being put on Myfonts.com.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj3WbGy1B3SEmLDsqiS-nMS3-rfCzUlTWB67iz1N3Iwe0DZBrL-guF-ua6I1ein1QZF61z1R2rN9Z5um7Y2oRQZ0CScuVX75wVfVmXmciX1_3b85du2lUcZPR52qSb4Sn116ohWlGgZbW0/s1600/TessieFlyingBirdsPoster.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="720" data-original-width="1440" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj3WbGy1B3SEmLDsqiS-nMS3-rfCzUlTWB67iz1N3Iwe0DZBrL-guF-ua6I1ein1QZF61z1R2rN9Z5um7Y2oRQZ0CScuVX75wVfVmXmciX1_3b85du2lUcZPR52qSb4Sn116ohWlGgZbW0/s400/TessieFlyingBirdsPoster.png" width="400" /></a></div>
To get a tessellation pattern can require as few as one character but usually two, three, four, or six are required. The pattern above requires two characters because alternate rows are indented.<br />
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This illustration below shows three ways these fonts can be used. On the left is the solid style by itself, in the middle the solid style is overlaid with the outline style, and on the right the outline style is alone This shape is unusual because there are two ways to tile it. In the top six row of the picture, birds are flying both to the left and to the right. In the bottom four rows birds are flying only to the left.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgTUbYdQtFuqyiqj2r2iTjYNPIxC0P17qQlQyHMEZJgOtBW8wR1ywZVOc3el2MCISjquxtr1Uoe4x-sLqmCezRNa61xDhwVYJVR8veJeYIqaSzFH5vgPNdU9M9qnbc1y7UDI5G-0uvRamk/s1600/TessieFlyingBirdsFlag.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="400" data-original-width="400" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgTUbYdQtFuqyiqj2r2iTjYNPIxC0P17qQlQyHMEZJgOtBW8wR1ywZVOc3el2MCISjquxtr1Uoe4x-sLqmCezRNa61xDhwVYJVR8veJeYIqaSzFH5vgPNdU9M9qnbc1y7UDI5G-0uvRamk/s320/TessieFlyingBirdsFlag.png" width="320" /></a></div>
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For those who like the technical details of tessellations, the top birds are tiled as Heesch type TGGTCC while those on the bottom are tiled as Heesch type TG1G1TG2G2.<br />
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(Cross posted to <a href="https://nohypetype.blogspot.com/2018/11/tessellation-typefaces-part-1.html">nohypetype.blogspot.com</a>)<br />
<br />Dessert Survivorhttp://www.blogger.com/profile/04616064444288249273noreply@blogger.com0tag:blogger.com,1999:blog-4026157408460542489.post-78149510965736027032018-08-03T19:35:00.000-07:002018-08-03T19:35:13.983-07:00Butterflies in Mazes Escher Would LikeHere are some further notes on the mazes in <i><a href="http://mazepuzzles.blogspot.com/2018/06/mazes-escher-would-like.html">Mazes Escher Would Like</a></i>.<br />
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One surprise I found was a hexagonal pattern that looked like it fit into a rectangular grid but when I made a maze with it, I found that it drifted. Below is a moth pattern that does not drift. The edges are labeled clockwise, with 1 being the horizontal edge on the right. Notice that to go down the column, one follows the pattern of 35263526 etc.<br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhn5OcrVqQBB63TA-wonzIzTptmYy0k8PGPJ09oUCSM5R1jRITpzu4z9DfvtPs3cIeruVPZ6PHXBYbSJRpMfNVFj-6YQe-TUycNXvBY1H6NRIK0Q6aLe0OB-rKwjMFigAe2er7Am0sc_0Y/s1600/stablebutterflies.jpg" imageanchor="1"><img border="0" height="551" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhn5OcrVqQBB63TA-wonzIzTptmYy0k8PGPJ09oUCSM5R1jRITpzu4z9DfvtPs3cIeruVPZ6PHXBYbSJRpMfNVFj-6YQe-TUycNXvBY1H6NRIK0Q6aLe0OB-rKwjMFigAe2er7Am0sc_0Y/s640/stablebutterflies.jpg" width="640" /></a><br />
Below is a different butterfly pattern with its six edges labeled in the same way. Notice following the 35263526 route takes down and to the right.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhCbEH7mp6fAwA578wxFSUnt4uwGSpeFBtPla6YDJIugiDZ4LAzIrp56TrBIEpcKKuqKoCd8fHmS8tBlEaFWgJLtDLv4J1ibv886L6cp5SeelnVt9TjI0EICHqZ6M-WbSv-otpbxXc6IxQ/s1600/slantingButter.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhCbEH7mp6fAwA578wxFSUnt4uwGSpeFBtPla6YDJIugiDZ4LAzIrp56TrBIEpcKKuqKoCd8fHmS8tBlEaFWgJLtDLv4J1ibv886L6cp5SeelnVt9TjI0EICHqZ6M-WbSv-otpbxXc6IxQ/s1600/slantingButter.jpg" /></a></div>
One of my maze-making programs allows a correction to be made so that the maze will line up correctly.<br />
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Perhaps the most interesting pattern that I have found in the past few years is of butterflies such as the ones below.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhyR37zYTVaLLNBGIZ9G27rqI-BlTJ2iyoULsZyLrH4aGpQgvgys3SSG81sOQFwlCzQT7kjQGnm5TI80JYLebIhAkGSdV4JP0gVs-UQ9ssofBJc7PKDupPSUfB5dyy2KdHgsdQNC7TOVac/s1600/butterfliesTCCTCC.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhyR37zYTVaLLNBGIZ9G27rqI-BlTJ2iyoULsZyLrH4aGpQgvgys3SSG81sOQFwlCzQT7kjQGnm5TI80JYLebIhAkGSdV4JP0gVs-UQ9ssofBJc7PKDupPSUfB5dyy2KdHgsdQNC7TOVac/s1600/butterfliesTCCTCC.jpg" /></a></div>
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<span style="text-align: start;">The shape of the tile has mirror symmetry but it does not fit into any of the symmetrical types of Grünbaum and Shepard's classification. The tiling above is type CG1CG2G1G2. Because of the symmetry of the the shape, the translation block appears to be two rather than four. </span></div>
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There is another tiling possible, a TCCTCC type, shown below. To see the difference between the two, follow a diagonal from southwest to northeast. Below all the shapes have the same orientation along this diagonal. Above there are two oriented the same, then two different, etc. </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhqqH7JLQ1Lbt3BqSodF_RKvlIrDm9egrRbjgW9an49wWpzt7Bu91FGc5394DxrSG7vEB_TbBZiBinxQk1V77WG7lrXzZQMCQnsKPU0S8T5NJEYmiE7CUZjEGhMeXfC5mh_SPaMFUj52rQ/s1600/ButterfliesCG1CG2G1G2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhqqH7JLQ1Lbt3BqSodF_RKvlIrDm9egrRbjgW9an49wWpzt7Bu91FGc5394DxrSG7vEB_TbBZiBinxQk1V77WG7lrXzZQMCQnsKPU0S8T5NJEYmiE7CUZjEGhMeXfC5mh_SPaMFUj52rQ/s1600/ButterfliesCG1CG2G1G2.jpg" /></a></div>
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Some of the other shapes that are included in the book have appeared in posts on this blog during the past few years and many have appeared in <a href="http://ingrimayne.com/mazes/coloringindex.htm">coloring books</a> or in <i><a href="http://mazepuzzles.blogspot.com/2018/07/the-ongoing-journey.html">Exploring Tessellations</a></i>, but not in other <a href="http://ingrimayne.com/mazes/mazeindex.htm">maze books</a>.</div>
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Seventy-nine of the 83 mazes use tessellations. Ten are geometric or abstract shapes, one is a symbol, 13 are mammals including people, 7 are other vertebrates, 11 are invertebrates, 6 are non-living objects, and 32 are bird shapes.</div>
Dessert Survivorhttp://www.blogger.com/profile/04616064444288249273noreply@blogger.com0tag:blogger.com,1999:blog-4026157408460542489.post-5347281543083089772018-07-18T05:20:00.000-07:002018-07-18T05:20:17.190-07:00Further maze book revisionsAfter updating <a href="http://mazepuzzles.blogspot.com/2018/05/a-couple-of-revisions-to-tessellating.html">tessellating maze books in May</a>, I <a href="http://mazepuzzles.blogspot.com/2018/07/the-ongoing-journey.html">updated</a> some other books. While doing that I discovered patterns that made simple mazes quite difficult to solve. I could not resist the temptation to include some and so did a second round of revisions and updating. Below is a sample, though not an actual maze from either book. All the paths go through the corner swirls.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg79HRpUso3rr9hZkiI5GIZK7rfBxSTaF9Hd3dwiO5QyNhwU9kCe3-aklCenGuqK50uOeivyRu6dVi91cVJr1JlsA4RXWpu1g2SdPdSREbb0K-GjRIskrMzg97aOHqMWCXwLoIK5oCSq8E/s1600/Screen+Shot+2018-07-08+at+7.14.02+AM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg79HRpUso3rr9hZkiI5GIZK7rfBxSTaF9Hd3dwiO5QyNhwU9kCe3-aklCenGuqK50uOeivyRu6dVi91cVJr1JlsA4RXWpu1g2SdPdSREbb0K-GjRIskrMzg97aOHqMWCXwLoIK5oCSq8E/s1600/Screen+Shot+2018-07-08+at+7.14.02+AM.png" /></a></div>
The updating resulted in the elimination of a number of common geometric tessellations that, while they made decent mazes, were not of much interest to those looking for visually-interesting tessellation. <i>Tantalizing Tessellating Mazes</i> had four additional pages changed and <i>More Tessellating Mazes</i> had 12 additional pages changed. As a result of the two 2018 revisions, both books have 39 new mazes and 45 holdover mazes.<br />
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In the maze above there are two shapes that tessellate. The top one is an example of isohedral class IH71. All edges are shaped the same and there is mirroring over a diagonal line. The bottom shape is an example of class IH61. The tile has two-fold rotational symmetry and uses the same edge as the top shape.<br />
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(If the maze above is presented with big openings through the sides rather than the little openings through the curved corners, it is trivially easy to solve. See below.)<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh5erAjcxeQ-J1s6t_14hWub_U2jK_XGaFBNGyjij5gAUIRqUFfvssly_guUFUX9R79UguPx5ZPmYzYi4R7fuH6O5fhzQLWP7eealDI4VhWGrsuCB-IjFos6E5HVTwMpEEUrtPXxljLocg/s1600/Screen+Shot+2018-07-08+at+4.20.17+PM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh5erAjcxeQ-J1s6t_14hWub_U2jK_XGaFBNGyjij5gAUIRqUFfvssly_guUFUX9R79UguPx5ZPmYzYi4R7fuH6O5fhzQLWP7eealDI4VhWGrsuCB-IjFos6E5HVTwMpEEUrtPXxljLocg/s1600/Screen+Shot+2018-07-08+at+4.20.17+PM.png" /></a></div>
<br />Dessert Survivorhttp://www.blogger.com/profile/04616064444288249273noreply@blogger.com0tag:blogger.com,1999:blog-4026157408460542489.post-86521674373113475432018-07-08T05:30:00.000-07:002018-07-08T05:30:35.652-07:00RevisionsThe introduction to <i><a href="http://mazepuzzles.blogspot.com/2016/10/a-final-coloring-book-of-tessellations.html">A Final Coloring Book of Tessellations</a></i> noted that "as I stumble on additional patterns ... I can use the better discoveries to update and revise this and the previous books." In the past few weeks I have revised <i><a href="http://mazepuzzles.blogspot.com/2012/11/tessellation-byproducts.html">A Tessellating Coloring Book</a></i>, <i><a href="http://mazepuzzles.blogspot.com/2015/10/more-tessellations-coloring-book.html">More Tessellations Coloring Book</a></i>, and <i>A Final Coloring Book</i>. New patterns (and thus deleted patterns) were six for the first book, eleven for the second, and seven for the last.<br />
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The three books contain over 300 tessellation designs and each is unique to one book. The first two of the books listed above were designed before the adult-coloring-book fad hit and were meant for children. The last book was intended for adults but it lacks the fussiness that many of the adult colorers seem to desire.<br />
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They all still have some weak designs and some geometric tilings that are in the public domain so if in the future I find additional interesting tessellating shapes, there may be more revisions.Dessert Survivorhttp://www.blogger.com/profile/04616064444288249273noreply@blogger.com0tag:blogger.com,1999:blog-4026157408460542489.post-2493700774174975302018-07-05T06:52:00.000-07:002018-07-05T06:52:00.766-07:00The ongoing journey<i><a href="http://mazepuzzles.blogspot.com/2015/10/a-journey-through-heesch-types-and.html">Exploring Tessellations: A Journey through Heesch Types And Beyond</a></i> was published in October, 2015 and then for the next year was <a href="http://mazepuzzles.blogspot.com/2016/01/an-unfinished-journey.html">repeatedly</a> <a href="http://mazepuzzles.blogspot.com/2016/08/take-outs.html">updated</a> as mistakes were corrected, new material added, and sections reorganized. The last of those updates was at the end of 2016. Since that last update I have designed an <a href="http://mazepuzzles.blogspot.com/2017/01/new-for-2017.html">activity book</a> and <a href="http://mazepuzzles.blogspot.com/2018/06/mazes-escher-would-like.html">another maze book</a> as well as <a href="http://mazepuzzles.blogspot.com/2018/05/a-couple-of-revisions-to-tessellating.html">updating two earlier</a> maze books. The work on these books provided enough new material to justify going back and reviewing the content and organization of <i>Exploring Tessellations</i>.<br />
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This revision of June 2018 adds many new tilings as examples and deletes some old tilings that were substandard. It also reorganizes the section on Heesch types. The original organization of that section was based on efforts to find standing birds that tessellated. That whimsical organization was replaced with a more standard organization that stresses Heesch families. Heesch and his co-author showed that their list of 28 types consisted of nine main types from which the other 19 types could be derived by shrinking edges to zero. For example, type TTTT results when one of the TT pairs in type TTTTTT shrinks away.<br />
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These chess pieces were added as an example of isohedral class IH69. The tile has symmetry over a diagonal. It simultaneously fits Heesch types CCCC, G1G1G2G2, and CCGG. In this example two edges are straight, but that is not a requirement of class IH69.</div>
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This next image has been in the book from the start as an illustration of isohedral class IH91. It also has reflective symmetry. It is based on an isosceles triangle and one edge must be straight for it to have that symmetry. It fits both Heesch type CCC and CGG.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgA3tZYERiReYs5dd0zbyE5Fip3BRQonesATXbE1cAIbG_udAFBvlXQrlZ7MDybWIYSbfX9BO-4p96a1Be9cknTPycuZMSny0ODux5zBcLcx7gMTtRZ2hYW8CSZsb0hAdpccZOqpjzhHkQ/s1600/ChessThrdee.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgA3tZYERiReYs5dd0zbyE5Fip3BRQonesATXbE1cAIbG_udAFBvlXQrlZ7MDybWIYSbfX9BO-4p96a1Be9cknTPycuZMSny0ODux5zBcLcx7gMTtRZ2hYW8CSZsb0hAdpccZOqpjzhHkQ/s1600/ChessThrdee.jpg" /></a></div>
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Finally, below is a new example of Heesch type CC3C3 (IH39), which is the hardest of the Heesch type to form into Escher-like tessellations. It could be a leaf on a twig or a rosebud.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhYj2TbLpzMOhe-gO_btnz3S7ChfxrFI1wYOMlfHIO_mfU47nODZnw_IIfxD86PWC9JtNSaBKS9zoDY3CDgLpH-Tl7Vyq_hz2zv0UGyp6oDi1jLj41bhqSvdRESvf5iE-eLRy4pfDYWKLg/s1600/Ivy.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhYj2TbLpzMOhe-gO_btnz3S7ChfxrFI1wYOMlfHIO_mfU47nODZnw_IIfxD86PWC9JtNSaBKS9zoDY3CDgLpH-Tl7Vyq_hz2zv0UGyp6oDi1jLj41bhqSvdRESvf5iE-eLRy4pfDYWKLg/s1600/Ivy.jpg" /></a></div>
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The book has hundreds of other illustrations and is one of the few sources that explain both the Heesch classification and the isohedral classification of <a href="https://www.amazon.com/Tilings-Patterns-Second-Dover-Mathematics/dp/0486469816/">Grünbaum and Shepard</a>.</div>
Dessert Survivorhttp://www.blogger.com/profile/04616064444288249273noreply@blogger.com0tag:blogger.com,1999:blog-4026157408460542489.post-44680706652020466582018-06-05T19:32:00.000-07:002018-06-05T19:33:11.908-07:00Mazes Escher would like Since <i><a href="http://mazepuzzles.blogspot.com/2015/03/holiday-mazes-new-maze-book.html">Holiday Mazes</a></i> was published in early 2015, I have continued to develop tessellations for <a href="http://ingrimayne.com/mazes/coloringindex.htm">several coloring books</a>, an <a href="http://mazepuzzles.blogspot.com/2017/01/new-for-2017.html">activities book</a>, and <i><a href="http://mazepuzzles.blogspot.com/2016/08/take-outs.html">Exploring Tessellations: A Journey through Heesch Types And Beyond</a></i>. Recently I realized that I had more than enough new tessellation designs for another maze book focused on tessellations.<br />
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The result is <i>Mazes Escher Would Like</i>. The name is descriptive because the most of the mazes use Escher-like tessellations, that is, shapes that both tessellate and resemble real world objects. It is also an obvious attempt to have a title that might turn up in search results on Amazon, which is the only place that will be selling the book.<br />
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The book contains 83 mazes moderately difficult mazes suitable for older children and adults. Something new for my maze books is a table showing the Heesch types and isohedral classifications of each pattern used.<br />
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There are 36 IH groups represented and 24 of the 27 Heesch types. The Heesch types missing are C3C3C3C3C3C3, C3C3C3C3, and C3C3C6C6.<br />
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The book has a lot of bird patterns (32 of the 83) because for some reason birds pop out when I am looking for tessellations.<br />
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One of the odder bird designs in the book is the one below, shown in the sample maze that I use to keep track of the many maze typefaces I have. It makes an interesting maze.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhIH4jykyLyFWpiHfd4NG39wshH-R2_wmihlhc7l3sOE2GC-E6OnsLYd8v5Z6Svs0IHgdBhv1E-URNkPLKOeV0aFyFrlq2nIKRvGnWPWqGM6mfT_96BB-f0szFjBc-2Axma57yZp7WXiLU/s1600/strangebirds..jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhIH4jykyLyFWpiHfd4NG39wshH-R2_wmihlhc7l3sOE2GC-E6OnsLYd8v5Z6Svs0IHgdBhv1E-URNkPLKOeV0aFyFrlq2nIKRvGnWPWqGM6mfT_96BB-f0szFjBc-2Axma57yZp7WXiLU/s1600/strangebirds..jpg" /></a></div>
It presented a challenge because my maze generation program only allows triangle-based mazes to have two orientations but this pattern has four. The solution was to print it using two typefaces. My maze generating programs were written in the over twenty years ago in a defunct computer language for an obsolete operating system that no longer runs on modern hardware. Sometimes it takes some effort and creativity to get the output I want.<br />
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In late 2015 I was playing with a group of tilings that I dubbed the "<a href="http://mazepuzzles.blogspot.com/2015/11/the-fabfours-family-of-typefaces.html">Fab Fours</a>" and which found a home in the <i><a href="http://mazepuzzles.blogspot.com/2015/11/delightful-designs-coloring-book-of.html">Delightful Designs: A Coloring Book of Magical Patterns</a></i>. I tried to use a few shapes that I had found then to make mazes. An example is a three-edged shape that makes for a visually attractive maze.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhH-lYSzXk5Iim9iFQgFQ4vAYiCwZiGIsbEgu5O0gGY3qfzTA6nhyZKPiV9czP3LmRXZSZJev9cmFvueIk-NPSOj3wlmqeAxVf_Z85FMlNYy7km8ap9QcelBxXZSNkFHpS-kL2t_KknE5Q/s1600/Confusing.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhH-lYSzXk5Iim9iFQgFQ4vAYiCwZiGIsbEgu5O0gGY3qfzTA6nhyZKPiV9czP3LmRXZSZJev9cmFvueIk-NPSOj3wlmqeAxVf_Z85FMlNYy7km8ap9QcelBxXZSNkFHpS-kL2t_KknE5Q/s1600/Confusing.jpg" /></a></div>
It is not an Escher-like tessellation, but Escher also dabbled in various geometric shapes. (There are four mazes at the end that are not framed with tessellations. Three of them have over/under paths and I wanted to include a few just in case this is the last maze book I ever design.)Dessert Survivorhttp://www.blogger.com/profile/04616064444288249273noreply@blogger.com0tag:blogger.com,1999:blog-4026157408460542489.post-29786465340370463442018-05-29T06:11:00.000-07:002018-05-29T06:11:38.347-07:00A couple of revisions to tessellating maze booksWhen I designed <i><a href="http://mazepuzzles.blogspot.com/2012/02/they-are-out.html">Tantalizing Tessellating Mazes</a></i> in 2011 and 2012 I not only had enough tessellation patterns to produce 70 mazes for this book, but had enough left over to do another 70 mazes in <i><a href="http://mazepuzzles.blogspot.com/2012/06/more-tessellating-mazes.html">More Tessellating Mazes</a></i>. Many of the tessellations were common geometric shapes. At the time I did not see this as a shortcoming because I was more focused on mazes than on tessellations.<br />
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In the six years since the publication of these two books I have devoted considerable time to finding <a href="http://mazepuzzles.blogspot.com/2015/04/exploring-tessellating-birds.html">new tessellation</a> patterns. Most of what I have found has been used in other maze books, but I early in 2018 I realized that I still had enough unused material to revise and upgrade both <i>Tantalizing Tessellating Mazes</i> and <i>More Tessellating Mazes</i>.<br />
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Both books had 70 pages of mazes and 35 pages of solutions, with two solutions per page. The revisions shrink the solutions so that four fit on a page. This change frees up space for an additional fourteen mazes with no change in page count. In addition I dropped patterns that I thought were the least interesting and replaced them with new patterns and mazes. As a result, both books now have a greater percentage of Escher-like mazes, that is, mazes based on shapes that resemble real world objects rather than geometric, abstract shapes. In <i>Tantalizing Tessellating Mazes</i> 35 of the mazes are new and 49 are carried over from the previous edition. The numbers for <i>More Tessellating Mazes</i> are 27 new and 57 holdovers.<br />
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The order of mazes has been changed. The new mazes are slightly more difficult than the mazes they replace, but they still seem best described as fairly easy and appropriate for ages 8 and older. Both books have many bird tessellations because I seem prone to find bird shapes as I toy with <i><a href="http://tesselmaniac.com/tess/Home.html">Tesselmaniac!</a></i>. (<i>Tantalizing Tessellating Mazes</i> has 29 bird tessellations and <i>More Tessellating Mazes</i> has 21.)<br />
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Some of the shapes used in the books have previously <a href="https://mazepuzzles.blogspot.com/2012/11/tessellation-byproducts.html">appeared</a> on this blog. Here is a bird tessellation illustrated with a small sample maze that I used to keep track of maze typefaces.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbp2Tvchhnn6SyOTt3Oydcu7DPbRK6eY55eh_wcQ0VOfTj-cDllJMN9RGIAVeSRG7_cdg4CmgsDfLprfLqnbcV9K8MtxTPXtAn7fVIkXnvhxkmmkMUGo-6PJ9ATiIj1e7acXd2odbOrSE/s1600/Screen+Shot+2018-05-27+at+12.19.09+PM.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="571" data-original-width="478" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbp2Tvchhnn6SyOTt3Oydcu7DPbRK6eY55eh_wcQ0VOfTj-cDllJMN9RGIAVeSRG7_cdg4CmgsDfLprfLqnbcV9K8MtxTPXtAn7fVIkXnvhxkmmkMUGo-6PJ9ATiIj1e7acXd2odbOrSE/s400/Screen+Shot+2018-05-27+at+12.19.09+PM.jpg" width="333" /></a></div>
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The pattern below comes from the quilting world. This maze allows not only horizontal and vertical passages but also passages through the corners. The shape is nothing special but the corner passages make even small mazes a challenge. (This is not a maze from the book but a sample maze to illustrate the pattern.)<br />
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Both books are available on Amazon and links to them are in the side margin.</div>
Dessert Survivorhttp://www.blogger.com/profile/04616064444288249273noreply@blogger.com0tag:blogger.com,1999:blog-4026157408460542489.post-20238650186387158662018-05-05T15:23:00.000-07:002018-05-05T15:23:01.639-07:00Tessellating activitiesIt has been over a year since I last published a book via CreateSpace, and since then CreateSpace has ceased to sell books; all sales are now through Amazon. If you click on any CreateSpace link in a previous post, it will take you to an Amazon page.<br />
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I have not spent much time or effort with tessellations (or mazes) since <i><a href="https://www.amazon.com/Tessellating-Animals-Activity-Book-Coloring/dp/1541173449/">Tessellating Animals Activity Book</a></i> was published in January 2017. A <a href="http://mazepuzzles.blogspot.com/2017/01/new-for-2017.html">post at the time</a> tried to indicate what kind of activities were in the book, but the illustrations were of partial puzzles that could not be solved. The sample pages that Amazon shows are even less revealing of what is in the book.<br />
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Below are examples of some of the types of puzzles in <i>Tessellating Animals Activity Book</i>. First is a mini-Sudoku. Normal Sudoku puzzles are 9-by-9 grids, but simpler versions can be constructed with 4-by-4 or 6-by-6 grids. The puzzle below uses numbers 1 through 6. All rows and all columns must have each number only once. In addition, there are six boxes indicated by the shading and each of those boxes must also have each of the six numbers only once.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWFrqdyq9VGo04M-4eyBuqHx5nHX2o24tiGhFYtxauXddwRr0XGnQUbuJBHBSS4V4G7vWtH7PvkxGbDr80zOvaG-4t9rfcbTjdPtLCIPyLQmAwnabfTKiNplYND6PZ2sNCL6n4CLA9ihE/s1600/SudokuPuzzle.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="586" data-original-width="542" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWFrqdyq9VGo04M-4eyBuqHx5nHX2o24tiGhFYtxauXddwRr0XGnQUbuJBHBSS4V4G7vWtH7PvkxGbDr80zOvaG-4t9rfcbTjdPtLCIPyLQmAwnabfTKiNplYND6PZ2sNCL6n4CLA9ihE/s1600/SudokuPuzzle.jpg" /></a></div>
The book contains 14 mini-Sudoku puzzles. (The bird-head shape was in preliminary versions of the book but was dropped before it was published because it is a low-quality design.)<br />
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A second type of puzzle is a word search, which also use grids. The grid here is provided by a weird shape that was never in any version of the book but which works well for this type of puzzle. Words can be vertical, horizontal, or diagonal and they may be reversed. Find the following words: reflect, valence, hexagon, rotate, escher, glide, flips, edge. The letters not used in any of these words will spell another word that is related to tessellations.<br />
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There are ten word-search puzzles in the book and all are larger than this one.</div>
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There are eleven pages of decoder puzzles in the book and most of those pages have more than one puzzle. In the puzzle below the letters have been replaced by their position in the alphabet: A=1, B=2, C=3, etc. </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEidwaQreSjFH9w3fonQOJWJ23BCY_5yj3WoEZCF97ZhaaZ0EmPBZz0QE081vubOJACqxfjcNxmEnc5U_UUUZqLtP0gPKL7bOjLIf30F17GwH338LMItcZuSFPHIW2DGebUVK7dDhqSpox8/s1600/codepuzzlewitholdbirds.jpg" imageanchor="1"><img border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEidwaQreSjFH9w3fonQOJWJ23BCY_5yj3WoEZCF97ZhaaZ0EmPBZz0QE081vubOJACqxfjcNxmEnc5U_UUUZqLtP0gPKL7bOjLIf30F17GwH338LMItcZuSFPHIW2DGebUVK7dDhqSpox8/s640/codepuzzlewitholdbirds.jpg" width="552" /></a></div>
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This shape was a first attempt to find a <a href="http://mazepuzzles.blogspot.com/2015/04/exploring-tessellating-birds.html">standing bird that fit Heesch type TGTG</a>. It is not very good and I have found better ways to illustrate the type.</div>
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The book contains a number of mazes, some of the traditional variety but others that are coded. They are simpler than the coded mazes in the book <i><a href="http://mazepuzzles.blogspot.com/2013/02/really-this-is-maze.html">Hidden Path Mazes: Decode to Solve.</a></i></div>
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The various puzzles and activities are supplemented with explanations of basic characteristics of tessellations.</div>
Dessert Survivorhttp://www.blogger.com/profile/04616064444288249273noreply@blogger.com0tag:blogger.com,1999:blog-4026157408460542489.post-40169649122138865402018-04-20T08:31:00.000-07:002018-04-20T08:31:20.437-07:00Tessellating wine glassesGoblets, wine glasses, or chalices are easy to tessellate and if you search the Internet for images you should be able to find at least two ways in which the shape fits into a tiling pattern. Below is one of these ways.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg9P_AXfwxiOfGuWAakZkuO_QRV2Py0Ol1KAUt3nHbirsK3zprDCUafRTBz8ekdZ7C-8v7pDcEc6vXEunuM0GONPSCNBgwJbClHFTgHduziODA98qLtuvD0TlpbSXXcZODjhATnquSxC1A/s1600/Wine5.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="282" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg9P_AXfwxiOfGuWAakZkuO_QRV2Py0Ol1KAUt3nHbirsK3zprDCUafRTBz8ekdZ7C-8v7pDcEc6vXEunuM0GONPSCNBgwJbClHFTgHduziODA98qLtuvD0TlpbSXXcZODjhATnquSxC1A/s320/Wine5.jpg" width="320" /></a></div>
The five-edged goblets are an example of isohedral class IH26. IH26 is the result of bisecting isohedral class IH17. If the straight bottom is considered an edge of central rotation, the pattern fits Heesch type TCTCC. If it is an edge of reflection, it is not a Heesch type.<br />
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If we slide alternate rows, each glass has six neighbors rather than the five in the pattern above.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiTXQ3oh_D8J1-1Rp4wlQEMA9ZCROr_7j-y2taUTjgLaj3LD_tF-KLhS9Aegskxgg8kPz2mz9WtcuOWwdjJQu4wyLaIV9BLfkd23F_3GhiYyPW6EYteOXbbVZhBDZ-hpzgLHHujz7RJyRM/s1600/Wine2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="293" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiTXQ3oh_D8J1-1Rp4wlQEMA9ZCROr_7j-y2taUTjgLaj3LD_tF-KLhS9Aegskxgg8kPz2mz9WtcuOWwdjJQu4wyLaIV9BLfkd23F_3GhiYyPW6EYteOXbbVZhBDZ-hpzgLHHujz7RJyRM/s320/Wine2.jpg" width="320" /></a></div>
When the six-edged goblets are centered over each other as in the image above, we have an example of isohedral class IH15, which simultaneously fits Heesch types TCCTCC, TCCTGG, and TG1G1TG2G2.<br />
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The other easy way to tessellate this shape is to stack them in the pattern below.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6YuNgA-1MLDLhGmYHskutbXzTgIKBTRyKAjFay6qI7Vk-uuOjKy3JQzXD5lj6RKCnnm_-Q3SOTmNz8aPQEspCm4sBwKEekHZHhxzssNzF4UWqOF-z8mIkw8waWbTak6ZDxH-VH4pJ20A/s1600/Wine3.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6YuNgA-1MLDLhGmYHskutbXzTgIKBTRyKAjFay6qI7Vk-uuOjKy3JQzXD5lj6RKCnnm_-Q3SOTmNz8aPQEspCm4sBwKEekHZHhxzssNzF4UWqOF-z8mIkw8waWbTak6ZDxH-VH4pJ20A/s1600/Wine3.jpg" /></a></div>
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This arrangement is an example of isohedral class IH12 that simultaneously fits Heesch types TTTTTT and TG1G1TG2G2. Four of the edges are identically shaped.<br />
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All six edges can be identically shaped and the image still resembles a chalice or wine glass.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg58XPVOIp9-5WfQB79muOModlST8R8Al_SlPyWovWRPslpIVsMgO8PbAf_LZYU873fe8MpUbLI5yLbzMiGc6TqTJyAcmpk0A2bjIPS5_uKYafM7n-ovF4bUolyqBjZhroB1IluJxpaLtg/s1600/Wine4.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg58XPVOIp9-5WfQB79muOModlST8R8Al_SlPyWovWRPslpIVsMgO8PbAf_LZYU873fe8MpUbLI5yLbzMiGc6TqTJyAcmpk0A2bjIPS5_uKYafM7n-ovF4bUolyqBjZhroB1IluJxpaLtg/s1600/Wine4.jpg" /></a></div>
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However, this shape can also be fit together in a different way, one that I have not found on the Internet.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgVeWdLK7CBTydWZZgaIVaOMbzoWO0beYTT410AnLyBjYWc2cOnH4nAI4o-YyOySIoVzR09m7_iQPpteQXTfI-ssTLnZZn3ef9m5r25Ad_-jxH-otYq7ddfUe9d40GDxG-s-CkCjtdLdV8/s1600/Wine1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgVeWdLK7CBTydWZZgaIVaOMbzoWO0beYTT410AnLyBjYWc2cOnH4nAI4o-YyOySIoVzR09m7_iQPpteQXTfI-ssTLnZZn3ef9m5r25Ad_-jxH-otYq7ddfUe9d40GDxG-s-CkCjtdLdV8/s1600/Wine1.jpg" /></a></div>
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It is an example of Heesch type TG1G1TG2G2 but not of TTTTTT. Can you find the TT pair?Dessert Survivorhttp://www.blogger.com/profile/04616064444288249273noreply@blogger.com0tag:blogger.com,1999:blog-4026157408460542489.post-47895054340142526492017-10-04T20:23:00.000-07:002017-10-04T20:23:02.440-07:00Isohedral classificationI have finally begun to understand the main thrust of Grünbaum and Shepard's classification of isohedral tilings, though there are details that still elude me. They begin with the eleven Laves tilings, tilings in which the angles at each vertex are equal. In the tiling below there are vertices with three line converging, so all the angles at these vertices are 120 degrees. There are also vertices with six lines converging, so the angles here must be 60 degrees.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj16WL-7RGzBr8riohbz7OE0RPt0K6WmA1ULqHc-ci0EhQtaDVAXyvU_ThWuPfH3YkWoR_uCm20-nCsnPaDt7_vy1eakHof-qBqPgb7RiVyHutUOUCilJSpgo-z3mQq2f0OnghikGbel00/s1600/final1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj16WL-7RGzBr8riohbz7OE0RPt0K6WmA1ULqHc-ci0EhQtaDVAXyvU_ThWuPfH3YkWoR_uCm20-nCsnPaDt7_vy1eakHof-qBqPgb7RiVyHutUOUCilJSpgo-z3mQq2f0OnghikGbel00/s1600/final1.jpg" /></a></div>
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Grünbaum and Shepard then consider the various rotational and reflectional symmetries that each of the tiles can have. There are ten for the regular hexagon, with twofold, threefold, and sixfold rotational symmetry, plus reflection over one side, three sides, one diagonal, three diagonals, reflection over one diagonal and one edge, reflection over three sides and three diagonals, and no symmetry at all. They then see how each of these possibilities can fit together consistently. Each of those ten yield at least one isohedral class; the one with no symmetry yields the seven hexagonal Heesch types.<br />
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Can there be a class formed from reflectional symmetry in the above figure? The answer is, "No". If we bisect the tile, we get the result on the bottom in the figure. Notice that some of the tiles have five neighbors and some have only four. The symmetry line cuts the base of the tile but the base of the tile never abuts another base. It would need to do so for it to make fit into its own isohedral class. The only isohedral class that the tiling above yields is IH21, which is Heesch type CC3C3C6C6.Dessert Survivorhttp://www.blogger.com/profile/04616064444288249273noreply@blogger.com0tag:blogger.com,1999:blog-4026157408460542489.post-69999806499617613182017-06-06T19:50:00.004-07:002017-06-06T19:50:52.309-07:00A link to an note about tessellationsHere is a link to a short note of mine in the <i>Mathematical Gazette </i>that was discussed in an <a href="http://mazepuzzles.blogspot.com/2016/11/fun-with-puzzle-pieces.html">earlier post</a>:<br />
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<a href="https://goo.gl/LnJREO">https://goo.gl/LnJREO</a><br />
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The note asks how many different shapes formed on a square template using identical, asymmetric edges will tessellate. In simpler language, how many different four-edged puzzle pieces are there that will tessellate if all edges are shaped in the same way? If all edges are asymmetric but identical, there are only fifteen distinct shapes with two "out" edges and two "in" edges and all tessellate.<br />
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A more difficult question is how many of the different shapes formed based on the template of a regular hexagon using identical, asymmetric edges will tessellate. A total of 34 tile isohedrally as Heesch types and another 9 tile anisohedrally. The demonstration of this result is in <i><a href="http://ingrimayne.com/mazes/exploringtessellations.htm">Exploring Tessellations: A Journey through Heesch Types And Beyond. </a></i>Dessert Survivorhttp://www.blogger.com/profile/04616064444288249273noreply@blogger.com0tag:blogger.com,1999:blog-4026157408460542489.post-27949367166626942292017-01-08T10:22:00.000-08:002017-01-23T08:59:53.376-08:00New for 2017With the new year I am adding a new title, <i style="text-align: start;">Tessellating Animals Activity Book</i><span style="text-align: start;">, to my books available on <a href="https://www.amazon.com/dp/1541173449">Amazon</a> and <a href="https://www.createspace.com/6801808">CreateSpace</a>.
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<span style="text-align: start;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi8X1suzE25DoOv6NlDueQGzlAUX_Uv_1tXlGxCXm9694ZSdPJb72ekDaCYSBTKZofp-NIMvU9t4mzdU9vemD_EF2fa7ixsVhfg0IYFslEu9XKo4MmFXFuYiZYLThVWduJwnCslVTk4ttY/s1600/Screen+Shot+2016-12-29+at+10.54.32+PM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi8X1suzE25DoOv6NlDueQGzlAUX_Uv_1tXlGxCXm9694ZSdPJb72ekDaCYSBTKZofp-NIMvU9t4mzdU9vemD_EF2fa7ixsVhfg0IYFslEu9XKo4MmFXFuYiZYLThVWduJwnCslVTk4ttY/s1600/Screen+Shot+2016-12-29+at+10.54.32+PM.png" style="cursor: move;" /></a></span></div>
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All but one of my previous books have been either <a href="http://ingrimayne.com/mazes/mazeindex.htm">maze</a> or <a href="http://ingrimayne.com/mazes/coloringindex.htm">coloring books</a> and this new one contains a little of each plus much more. It is a combination of a traditional activity book and a book about tessellations.<br />
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I am not sure where the idea for this book came from. I was toying with the possibility of doing a maze book with only animal tessellations, but for reasons I no longer remember, I changed course and opted to expand the content to include more than mazes. The final draft of the book contains 21 mazes and ten coloring pages, all of them illustrating very short fables, mostly from Aesop.<br />
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I was not planning to develop new tessellations for the book but I found gaps between what I already had and what would fit well in the book. One of the additions was an attempt to do a scorpion tessellation that is used for a maze. It is not very realistic but I like the stinger part.<br />
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<span style="text-align: start;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiIS1zgjE-7aw-Q5lg80fyoWiaK1FZyCibifhXCcN4epyt0JGnSqbcWRszgieT35abDwzUIsHCovxJn8k2DOuit2IYeDHJJ157erESOHFprr4HLpvHLEDkZNjIF1Q926m7S1UAdr7T5KCg/s1600/Screen+Shot+2016-12-29+at+10.58.26+PM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiIS1zgjE-7aw-Q5lg80fyoWiaK1FZyCibifhXCcN4epyt0JGnSqbcWRszgieT35abDwzUIsHCovxJn8k2DOuit2IYeDHJJ157erESOHFprr4HLpvHLEDkZNjIF1Q926m7S1UAdr7T5KCg/s1600/Screen+Shot+2016-12-29+at+10.58.26+PM.png" /></a></span></div>
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What else fits into a tessellation activity book? Matching and identification problems apply short explanations of topics such as symmetry, translation, rotation, and valence. These activities fill 19 pages and use almost half of tessellations designs that are in the book.<br />
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Mazes led me to tessellations because both use grids. I looked for other puzzle types that might fit a grid. Word searches were an obvious possibility, and there are ten pages of this type of puzzle. Below is the corner of one of them showing how tessellating turtles are used to frame the puzzle.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPD2CNN3EMKE40giXjnbDbiMU2wHaFE04gXhqJHxR35LIA0x-f3ktUUmDaIqWhNWNWKyHse6rqHHccg_g4X_H3G4BxKP1nhsNupHGh2lVhiwiQ18ABhUhQsn9PDs1SMmd1WvBYtEif4Fw/s1600/Screen+Shot+2016-12-29+at+11.41.48+PM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPD2CNN3EMKE40giXjnbDbiMU2wHaFE04gXhqJHxR35LIA0x-f3ktUUmDaIqWhNWNWKyHse6rqHHccg_g4X_H3G4BxKP1nhsNupHGh2lVhiwiQ18ABhUhQsn9PDs1SMmd1WvBYtEif4Fw/s400/Screen+Shot+2016-12-29+at+11.41.48+PM.png" /></a></div>
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In addition to visual mazes, I have long been interested in what I call <a href="http://mazepuzzles.blogspot.com/2013/02/really-this-is-maze.html">hidden-path</a> mazes in which the challenge is to discover the maze. This type of puzzle occupies 13 pages. Below is a corner of one in which the path is on the ponies with letters that have mirror symmetry and the walls are ponies with letters that do not have mirror symmetry.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj0wpRiqolg3WLxc_rzOupRlS9dc4glBQyzH27PWmjcOpBncEyCgQFkXn8T8vf8Tka47DMkPUny_NP_K0mrCeBhOo-n0V-1pGR9N22EZolBxiFIYjnzK_igRfgwX99Ydv4mWkQaPbEsIsY/s1600/Screen+Shot+2016-12-29+at+11.43.06+PM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj0wpRiqolg3WLxc_rzOupRlS9dc4glBQyzH27PWmjcOpBncEyCgQFkXn8T8vf8Tka47DMkPUny_NP_K0mrCeBhOo-n0V-1pGR9N22EZolBxiFIYjnzK_igRfgwX99Ydv4mWkQaPbEsIsY/s400/Screen+Shot+2016-12-29+at+11.43.06+PM.png" /></a></div>
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Sudoku puzzles are grid based but I thought the 9-by-9 variety might be too complex for the book, so I settled for mini-Sudoku puzzles that are based on grids of 16 and 36 cells. There are seven pages of them, with two per page. I used the extra space on some of these to point out a few features of tessellations. Below is part of a six-by-six puzzle framed with a design of tessellating elephants that I did for the book.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgHPXsuUQ6AfyXRilcogw9jD4P2TK3fIEkJxupoGLdYTjwP85Y0YpHCIB-UAWn91lAMyDngKjgMN-HXTLmyklyiaThmW8HXdnquuE4F2YgAhrZSxi0hj8lmHfsb1-FWeLwA413FK9uixaQ/s1600/Screen+Shot+2016-12-29+at+10.57.08+PM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgHPXsuUQ6AfyXRilcogw9jD4P2TK3fIEkJxupoGLdYTjwP85Y0YpHCIB-UAWn91lAMyDngKjgMN-HXTLmyklyiaThmW8HXdnquuE4F2YgAhrZSxi0hj8lmHfsb1-FWeLwA413FK9uixaQ/s400/Screen+Shot+2016-12-29+at+10.57.08+PM.png" /></a></div>
Decoder puzzles do not need a grid but can be put into one, as can word scrambles in which the order of letters is altered. There are eight pages of the former and three of the latter. Below is the corner of a decoder puzzle that contains two messages that are mixed together, one contained in letters that have rotational symmetry and the other in the rest of the letters. To decode, you must move back or forward from the given letter, with the dots telling you how far and in which direction. (This is the most complex of the decoder puzzles in the book.)<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgZuAKpFStV0846bIrA4Mu3y1cPc57SjnSwIYLYOZJXakh-8imyUQqC04WMuXeVNfmBBqAuL9-HbGO1CeFwlX_RrF-f_DlggaxAJuJIkFvtUpOgl9dVc4rG12pocEAbzGWdTnoiA8fY6fg/s1600/Screen+Shot+2016-12-29+at+11.45.28+PM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgZuAKpFStV0846bIrA4Mu3y1cPc57SjnSwIYLYOZJXakh-8imyUQqC04WMuXeVNfmBBqAuL9-HbGO1CeFwlX_RrF-f_DlggaxAJuJIkFvtUpOgl9dVc4rG12pocEAbzGWdTnoiA8fY6fg/s400/Screen+Shot+2016-12-29+at+11.45.28+PM.png" /></a></div>
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Finally, there is one dot-to-dot puzzle making a total of 92 pages of puzzles and explanations using about 160 different tessellations patterns. More than half of the tessellations are of birds because I find them by far the easiest animal to tessellate. The final 14 pages of the book give solutions to the puzzles.</div>
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Searching through Amazon for something similar turned up one short book and I am not sure how similar it actually is. I suspect that the reason there is so little that is similar is that very few people find tessellations as nearly as interesting as I have found them. The suggested audience is anyone who enjoys tessellations and that may be a small group.<br />
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The main reason I designed this book is because it was fun. It would be nice if the book would also earn a bit of money, but at least it will not lose money thanks to on-demand printing.</span>Dessert Survivorhttp://www.blogger.com/profile/04616064444288249273noreply@blogger.com0tag:blogger.com,1999:blog-4026157408460542489.post-81987747105913098792017-01-05T10:00:00.000-08:002017-01-05T10:00:06.180-08:00Questions no one is asking part four(Part three is <a href="http://mazepuzzles.blogspot.com/2017/01/questions-no-one-is-asking-part-three.html">here</a>. It begins the exercise of examining tilings with tiles that have all edges translated.)<br />
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Moving to the hexagonal types, a TTTTTT tiling by definition has translated edges.<br />
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Isohedral class IH8 fits both the TTTTTT type and the TCCTCC type. The tile has twofold rotational symmetry.<br />
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Isohedral class IH12 is simultaneously TG1G1TG2G2 and TTTTTT. IH12 reflects over the midpoint of its TT edges.<br />
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Isohedral class IH10 is both type C3C3C3C3C3C3 and TTTTTT. All edges are identical and each is rotated 120º to form the adjacent edge. (Also fitting both of these types are isohedral classes IH11 and IH18. IH11 is formed with identical edges that reflect over their midpoints and IH18 with identical edges of central rotation.)<br />
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Isohedral class IH14 fits both types TG1G2TG2G1 and TTTTTT but it has a pair of straight, unshaped edges that give the tile reflective symmetry over a diagonal. If the straight edges are replaced with asymmetric edges, the reflective symmetry of the tile is lost but the tile can still fit types TTTTTT and TG1G2TG2G1, though no longer simultaneously. As a TG1G2TG2G1 type it has a translation block of two.<br />
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Below is a tiling that fits type CG1CG2G1G2 and also type TCCTGG.<br />
For both types a C edge must be paired with a G edge for the edges to translate so these pairs must be formed with center-point rotation. In the case of CG1CG2G1G2, the G1 edges are the edges that reflect over their midpoints. Because of symmetry, the translation block for both types is reduced to two.<br />
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In the beginning of this exercise I stated that I would use an asymmetric edge when possible. If an asymmetric edge is used for the TCCTGG type, the tile has no symmetry and the result is a translation block of four rather than the two in the above figure.</div>
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<br />Dessert Survivorhttp://www.blogger.com/profile/04616064444288249273noreply@blogger.com0tag:blogger.com,1999:blog-4026157408460542489.post-71465300784885083352017-01-02T07:30:00.000-08:002017-01-02T07:30:17.348-08:00Questions no one is asking part threeTwo previous posts (<a href="http://mazepuzzles.blogspot.com/2016/12/questions-no-one-is-asking-part-1.html">here</a> and <a href="http://mazepuzzles.blogspot.com/2016/12/qustions-no-one-is-asking-part-two.html">here</a>) explored tessellations formed using only glided edges. This post looks at tessellations formed with only translated edges. Like glided edges, translated edges must be paired and the pairs must be equally long, which limits them to quadrilateral and hexagonal types. Glided edges, however, can be aligned in several ways while translated edges must be opposite and parallel to each other. Hence, the possibilities when using tiles with only translated edges are even more limited than those when using only glided edges.<br />
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In what follows tilings that have a translation block of one are presented when possible. If asymmetric edges are possible, they are used, then edges with reflective symmetry over their midpoints, with edges formed using center-point rotation only as a last resort.<br />
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First, a TTTT type.<br />
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Isohedral class IH68 is both a TTTT type and a G1G1G2G2 type. All edges are identical and the tile mirrors over a diagonal.<br />
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In isohedral class IH57 edges are formed with center-point rotation and opposite edges translate. It fits three types: TTTT, CCCC, and TCTC.<br />
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IH62 is similar to IH57 but it requires all four edges to be identical. Each edge is rotated 90º to form the adjacent edge. It fits type C4C4C4C4 in addition to the three types that IH57 fits.<br />
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Isohedral class IH74 is another tile that uses identical edges formed with central rotation. The tile mirrors over both diagonals. In addition to types TTTT and CCGG, it fits types CCCC, TCTC, and G1G1G2G2.<br />
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Although the topic of this exercise is translated edges, the last three examples highlight the flexibility of edges with center-point rotation. If we want an all-translated version of C4C4C4C4 formed with asymmetric edges, it will have a translation block of four. (In this and in some other cases below, it should be obvious how the tile can arranged as a TTTT pattern with a translation block of one.)<br />
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Similarly, if we want to use an asymmetric edge for the TT pair of the TCTC type, the tiling will have a translation block of two.<br />
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A TGTG tile with translated edges also has a translation block of two. Edges can be both glided and translated only when they mirror over their midpoints.<br />
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A CGCG type with all edges translated results in another tiling with a translation block of two.<br />
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An all-translated G1G2G1G2 type has a translation block of four.<br />
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Below is a C3C3C6C6 type in which the tiles have translated edges. For the edges to both translate to opposite edges and rotate to form adjacent edges, they must be formed with reflection over their midpoints. The tile is symmetrical over its long diagonal and the tiling fits isohedral class IH68.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgL7SwK0qVPDGZ7JzDYUfVGttj0H95sDW5z8h_fEG5ecAjQuX6p4k7zOjQkR18jPZqfxZ0csLJqLrrFtiF-T5vrJPCBYTzglFVtIKjrf6E0DMwwVl_ZniQ7N1Fp4jFLzHe2zpsdVRhmQwY/s1600/Screen+Shot+2016-11-17+at+8.39.17+PM.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgL7SwK0qVPDGZ7JzDYUfVGttj0H95sDW5z8h_fEG5ecAjQuX6p4k7zOjQkR18jPZqfxZ0csLJqLrrFtiF-T5vrJPCBYTzglFVtIKjrf6E0DMwwVl_ZniQ7N1Fp4jFLzHe2zpsdVRhmQwY/s1600/Screen+Shot+2016-11-17+at+8.39.17+PM.jpg" /></a></div>
The figure below shows the tile from the above figure arranged as a TTTT type. In this arrangement it fits isohedral class IH68.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhJws5ADP4xlMhoJhOKEPRxrGF_yv8OjGyVsSj8jplkfLBVuu6fnsySdw1hGs1m3St-m7YwpFthR4cPDx_XJqy1IJRRAD75HkNDxGjQ6TdRKazJCEWNvwHb1wkC8AU2AhR7ph_od_rkYKI/s1600/Screen+Shot+2016-11-17+at+8.39.30+PM.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhJws5ADP4xlMhoJhOKEPRxrGF_yv8OjGyVsSj8jplkfLBVuu6fnsySdw1hGs1m3St-m7YwpFthR4cPDx_XJqy1IJRRAD75HkNDxGjQ6TdRKazJCEWNvwHb1wkC8AU2AhR7ph_od_rkYKI/s1600/Screen+Shot+2016-11-17+at+8.39.30+PM.jpg" /></a></div>
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Type C3C3C6C6 cannot be formed with translated edges because the template cannot be a parallelogram.</div>
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Part four will continue with hexagonal types.</div>
Dessert Survivorhttp://www.blogger.com/profile/04616064444288249273noreply@blogger.com0tag:blogger.com,1999:blog-4026157408460542489.post-61020284848051099442016-12-19T10:30:00.000-08:002016-12-20T15:03:42.724-08:00Qustions no one is asking part two(This post continues <a href="http://mazepuzzles.blogspot.com/2016/12/questions-no-one-is-asking-part-1.html">a previous post</a>.)<br />
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Moving on to hexagonal types, the way to form an all-glided type TTTTTT using as many asymmetric edges as possible is with isohedral class IH12, which is also type TG1G1TG2G2 with mirroring over the midpoint of the TT edges. The mirroring requires that the G1 and G2 edges be identical.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjXdYnYs4F9QS0h_aPj7evYRxGO8vV7LPg_DD-FTz-h8gt_0_LlX3ua0SM8jZqZ5GR2QwF-6cGWWpaTY6pDX3A6GKZOGN6hAnr0B7NuQGh-E663Gb55qOOKetDEy4DSu3Qz2UooxHCI3GU/s1600/Screen+Shot+2016-11-17+at+2.46.43+PM.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="248" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjXdYnYs4F9QS0h_aPj7evYRxGO8vV7LPg_DD-FTz-h8gt_0_LlX3ua0SM8jZqZ5GR2QwF-6cGWWpaTY6pDX3A6GKZOGN6hAnr0B7NuQGh-E663Gb55qOOKetDEy4DSu3Qz2UooxHCI3GU/s320/Screen+Shot+2016-11-17+at+2.46.43+PM.jpg" width="320" /></a></div>
The all-glided type TG1G1TG2G2 below is similar to the tiling above but uses a different shape for the G1 and G2 edges. It no longer fits TTTTTT.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEggk5eNBtsN34tZMEZIDrWX49glz_4pLM8RXcgntvPEBeaelRc1TUNVSnhyVqvL8wZyBEmKXqboSjtTGz0bAM8j58olTgYCtNjs3YPCqOYxwN-s9rA_oukwF04TGk-vEpgI83ktaGCK7dE/s1600/Screen+Shot+2016-11-17+at+2.46.54+PM.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEggk5eNBtsN34tZMEZIDrWX49glz_4pLM8RXcgntvPEBeaelRc1TUNVSnhyVqvL8wZyBEmKXqboSjtTGz0bAM8j58olTgYCtNjs3YPCqOYxwN-s9rA_oukwF04TGk-vEpgI83ktaGCK7dE/s320/Screen+Shot+2016-11-17+at+2.46.54+PM.jpg" width="310" /></a></div>
Below is an example of TG1G2TG2G1 with the TT edges glided with reflective symmetry.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjFOfg843RJLBM7fT4DL4L_aEZwTYjHWtHuIcQVGTPhTYGD42ORhqIWpf2PNUuhTBeZKRw28aj-puzF3HBJLvyeKkRwNHdo0pFH8sMzY07n2OYj4YRTZbIgMlqrXAniyoG3Ntf1tID9mz0/s1600/Screen+Shot+2016-11-17+at+2.47.14+PM.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjFOfg843RJLBM7fT4DL4L_aEZwTYjHWtHuIcQVGTPhTYGD42ORhqIWpf2PNUuhTBeZKRw28aj-puzF3HBJLvyeKkRwNHdo0pFH8sMzY07n2OYj4YRTZbIgMlqrXAniyoG3Ntf1tID9mz0/s320/Screen+Shot+2016-11-17+at+2.47.14+PM.jpg" width="315" /></a></div>
The type TCCTCC example below formed with glided edges also fits type TG1G1TG2G2. If the translated edges were straight, it would be an example of isohedral class IH15.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhYqnPVlsSnTlBZglhxQcgyXLGRfn4XI4h-JN3dwilQ9kFwZecAhhBqNqWhAdm7GmaMzud7CBoE6UMLAV2z-aBz9I7oNRMWrHJPOKCoR3SY6t4hh6LhcvYzFqxY02ZLETenseKqkOGhQ7k/s1600/Screen+Shot+2016-11-17+at+2.47.24+PM.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="316" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhYqnPVlsSnTlBZglhxQcgyXLGRfn4XI4h-JN3dwilQ9kFwZecAhhBqNqWhAdm7GmaMzud7CBoE6UMLAV2z-aBz9I7oNRMWrHJPOKCoR3SY6t4hh6LhcvYzFqxY02ZLETenseKqkOGhQ7k/s320/Screen+Shot+2016-11-17+at+2.47.24+PM.jpg" width="320" /></a></div>
An alternative way of forming a glided TCCTCC will fit type TG1G2TG2G1 and is an example of IH13. As such, it also fits CG1CG2G1G2.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgeaI9qUeaJ7YyeDipJyFGirZKaZ-KcH2zti4KOvsrdYDqQvVqZePsa_lOQPiDjDTRXJaHFs1oWoQTSylDC8at6bqHa9qP1TUL0dVE6EKTkaSBJq8SAICnvneU55olOUVkRMEU2OC-docE/s1600/Screen+Shot+2016-11-17+at+2.47.43+PM.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgeaI9qUeaJ7YyeDipJyFGirZKaZ-KcH2zti4KOvsrdYDqQvVqZePsa_lOQPiDjDTRXJaHFs1oWoQTSylDC8at6bqHa9qP1TUL0dVE6EKTkaSBJq8SAICnvneU55olOUVkRMEU2OC-docE/s320/Screen+Shot+2016-11-17+at+2.47.43+PM.jpg" width="309" /></a></div>
TCCTGG has a translation unit of 4x1. This example uses all three types of edges that can be glided: asymmetric edges, edges with reflection over the midpoint, and edges with center-point rotation.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjmF5lxB-k3retAbgvEMTHFqFukSteKUuGCfYkYDmK11rfLxd4bM3KK0DIJ-lBzAwzXOyFDmZXTFxuo_KmLAuyIGlDFUIJFxo_e5eZ5uCilbfo8YC5yXjcXCSg6_T7o0AyB9VhcQUpJbUA/s1600/Screen+Shot+2016-11-17+at+2.47.53+PM.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjmF5lxB-k3retAbgvEMTHFqFukSteKUuGCfYkYDmK11rfLxd4bM3KK0DIJ-lBzAwzXOyFDmZXTFxuo_KmLAuyIGlDFUIJFxo_e5eZ5uCilbfo8YC5yXjcXCSg6_T7o0AyB9VhcQUpJbUA/s1600/Screen+Shot+2016-11-17+at+2.47.53+PM.jpg" /></a></div>
Because the tile used to form this all-glide version of TCCTGG has the CC part glided, it will also tile as TG1G1TG2G2.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj0bsRWdB2M34Qh4Oxy8sLk193hlqjMv-rEFkkPEJ-_Vh55hw7DpDU8pTXG2UFMROrBaHlfDdbA9oTfTVUXW4AFzNG5QnWGPr1PZJCHxLiXvXAkqL-8to0HyczILb9yQTgvZ6kkaR6wg9U/s1600/Screen+Shot+2016-11-17+at+2.48.05+PM.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="311" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj0bsRWdB2M34Qh4Oxy8sLk193hlqjMv-rEFkkPEJ-_Vh55hw7DpDU8pTXG2UFMROrBaHlfDdbA9oTfTVUXW4AFzNG5QnWGPr1PZJCHxLiXvXAkqL-8to0HyczILb9yQTgvZ6kkaR6wg9U/s320/Screen+Shot+2016-11-17+at+2.48.05+PM.jpg" width="320" /></a></div>
To form an all-glided version of type C3C3C3C3C3C3, the edges are formed with edges of mirror reflection arranged as G1G1G2G2G3G3.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhNip_EazP6Pjspm7OmPEKmxFmnUaVXsDYdpCEkd1mFOn1wCtwCY3YqG7HXyEq1NK9K-qdMpsFtx_IIKldRC43hhUBFcjacPKGeOuJHzYCikqzArhdi-JulbWKdgoJiWBPqePms_G1LWeM/s1600/Screen+Shot+2016-11-17+at+2.49.04+PM.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhNip_EazP6Pjspm7OmPEKmxFmnUaVXsDYdpCEkd1mFOn1wCtwCY3YqG7HXyEq1NK9K-qdMpsFtx_IIKldRC43hhUBFcjacPKGeOuJHzYCikqzArhdi-JulbWKdgoJiWBPqePms_G1LWeM/s1600/Screen+Shot+2016-11-17+at+2.49.04+PM.jpg" /></a></div>
Notice how the two CC edges are mirrored in this all-glided version of CG1CG2G1G2.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjcv90sjNTlZ-ElNrN8zrgKL9SN3iX0YA_D2qN_QEGWc0w3vpALc7Qp971Is1IIphfFfuay8eohv1el1UdeVHTGW66jcG_ePtiblB1HWMN68aMCE0484MCVJEEHvmTNoz95A3LH1lzn7QE/s1600/Screen+Shot+2016-11-17+at+2.49.18+PM.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjcv90sjNTlZ-ElNrN8zrgKL9SN3iX0YA_D2qN_QEGWc0w3vpALc7Qp971Is1IIphfFfuay8eohv1el1UdeVHTGW66jcG_ePtiblB1HWMN68aMCE0484MCVJEEHvmTNoz95A3LH1lzn7QE/s1600/Screen+Shot+2016-11-17+at+2.49.18+PM.jpg" /></a></div>
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We have finished looking at tiles with all edges formed with glides. We will continue in part three to look at another question no one is asking, how many of the Heesch types can be formed with tiles in which all edges are translated.</div>
Dessert Survivorhttp://www.blogger.com/profile/04616064444288249273noreply@blogger.com0tag:blogger.com,1999:blog-4026157408460542489.post-34779204084980570392016-12-07T18:56:00.000-08:002016-12-20T15:03:15.691-08:00Questions no one is asking, part 1<div class="separator" style="clear: both;">
A <a href="http://mazepuzzles.blogspot.com/2016/11/fun-with-puzzle-pieces.html">previous post</a> mentioned that 25 of the 28 Heesch types could be formed using only identical edges of central rotation. The other three can also be formed using edges of central rotation, but they require at least two sizes of edges. Edges formed with center-point rotation can serve not just as C edges in the Heesch classification of types but also as T, G, C3, C4, and C6 edges, that is, any type of edge.</div>
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Mirroring an edge formed with central rotation is equivalent to gliding it. Translating an edge that mirrors over its midpoint is also equivalent to gliding it. Hence, some edges that are translated or mirrored can be seen as glided. How many of the Heesch types can be formed with all edges that are glided whether or not the edge fits as a glided edge?</div>
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Glided edges have less flexibility than edges formed with center-point rotation partly because they must be paired and not all edges pair in types formed on triangular and pentagonal templates. The closest we can get to an all-glided version of a triangular type, for example, is shown below. Notice that it can satisfy the CGG type in two ways, but not three. This tiling satisfies CCC and CC6C6 as well as CGG.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj0u6GEeF4Lic_ma3m8dudNasbo02IUT3BiSF1UkjB0nfVClgxfFqfPoXGWxlAtM34FMGVmub2ykVdoiNYQ_chyAS24VUl2SsJswk5N98eUJ1dAxPe1bzYxBwb4KskkZRH-0nAHIfljgAk/s1600/Screen+Shot+2016-11-17+at+2.43.23+PM.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj0u6GEeF4Lic_ma3m8dudNasbo02IUT3BiSF1UkjB0nfVClgxfFqfPoXGWxlAtM34FMGVmub2ykVdoiNYQ_chyAS24VUl2SsJswk5N98eUJ1dAxPe1bzYxBwb4KskkZRH-0nAHIfljgAk/s1600/Screen+Shot+2016-11-17+at+2.43.23+PM.jpg" /></a></div>
However, all edges of quadrilateral and hexagonal types can have edges paired and all of these can be formed with edges that are glided. In constructing them below, I have used asymmetric edges when the edges serve as G edges in the Heesch type and whenever else they can be used. When the type calls for C edges, I have used an edge with center-point rotation and mirrored it. All other edges have mirror symmetry over their midpoints. When there two or more pairs of the same type, different shapes are used to differentiate them if it is possible.<br />
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Let us start with the two quadrilateral types that must be formed with glided edges, G1G1G2G2 and G1G2G1G2. Below is an example of G1G1G2G2 with matching pairs differentiated.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhFWWJo1p-ixpena3sRQQUoiJM2WbNhMMTzFUdg7toaVSxWH0iIE0JzEnYyVE64eY3veXttREBGBP7i68f-r-w2lTF96kITjK9StQAf1VCmlhz4ZGUIzT6Llrho2qx_1dl_zME0rMXbmtg/s1600/Screen+Shot+2016-11-17+at+2.43.50+PM.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhFWWJo1p-ixpena3sRQQUoiJM2WbNhMMTzFUdg7toaVSxWH0iIE0JzEnYyVE64eY3veXttREBGBP7i68f-r-w2lTF96kITjK9StQAf1VCmlhz4ZGUIzT6Llrho2qx_1dl_zME0rMXbmtg/s1600/Screen+Shot+2016-11-17+at+2.43.50+PM.jpg" /></a></div>
Next is an example of G1G2G1G2.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh3wvAyqwIm_hYVMKCu_x-VP4Ve7-_d0M6QmZI9XJwnR0FrYVOxsl4nwHBpnqoPgYtDJA3N29FCNRT4yodcV4hn7kZPPA47SF5JLwyKCa8ImXsZag9Yn5OX24vaLrVAGKAaDFYuvSujYJo/s1600/Screen+Shot+2016-11-17+at+2.44.00+PM.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh3wvAyqwIm_hYVMKCu_x-VP4Ve7-_d0M6QmZI9XJwnR0FrYVOxsl4nwHBpnqoPgYtDJA3N29FCNRT4yodcV4hn7kZPPA47SF5JLwyKCa8ImXsZag9Yn5OX24vaLrVAGKAaDFYuvSujYJo/s1600/Screen+Shot+2016-11-17+at+2.44.00+PM.jpg" /></a></div>
Some of the other quadrilateral types formed with glided edges also fit either G1G1G2G2 or G1G2G1G2. To form TGTG with only glided edges, the TT pair must be formed with reflective symmetry over the midpoint. It also fits type G1G2G1G2.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgiMo-iJA974yB4sqzX6-uY4FOqSO_-NMcO-M5KdCKkxj0GsZEzPSAnRZ_oGICZ3b7iI0_GMDQ_ezrj33IytGW0izfNNhCO_LhmzXX0o42f3iculcK0NIA_olvHUZDwyO0u6XHQTmP2DdA/s1600/Screen+Shot+2016-11-17+at+2.44.08+PM.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgiMo-iJA974yB4sqzX6-uY4FOqSO_-NMcO-M5KdCKkxj0GsZEzPSAnRZ_oGICZ3b7iI0_GMDQ_ezrj33IytGW0izfNNhCO_LhmzXX0o42f3iculcK0NIA_olvHUZDwyO0u6XHQTmP2DdA/s1600/Screen+Shot+2016-11-17+at+2.44.08+PM.jpg" /></a></div>
Type TCTC can be formed with glided edges if the TT pair of edges reflects over their midpoints and the CC pair of edges is identical and mirrors. The tile has symmetry over the translated edges and fits isohedral class IH66. It also fits G1G2G1G2.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjU2dDPQUknHOiJoeU6jVGKjA8xczH8k4cnXHAsE17YkPNFCnv7wlilrPdMJL-ElPbVv73_M4fVsJZ6DTdDYl99gnaDxBo0OGgELym8Pt7jlV6Y_SnU5AHHMmlHhEyg_Rezf8SI0Fxsbt0/s1600/Screen+Shot+2016-11-17+at+2.44.21+PM.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjU2dDPQUknHOiJoeU6jVGKjA8xczH8k4cnXHAsE17YkPNFCnv7wlilrPdMJL-ElPbVv73_M4fVsJZ6DTdDYl99gnaDxBo0OGgELym8Pt7jlV6Y_SnU5AHHMmlHhEyg_Rezf8SI0Fxsbt0/s1600/Screen+Shot+2016-11-17+at+2.44.21+PM.jpg" /></a></div>
The CGCG type has the CC pair of edges formed identically and mirrored rather than translated. Like the previous two, it also fits G1G2G1G2.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqAr3aoBuiJrQkqqxQf29S0JKT3bdSq4tZBptG_oCkuVdPIEG6HSylD6UXch5K9r8yhDcERwJ-LIoyXT-Lz9NEajL0P-yFQ6d9837y5_PuCCYKabzwcvDyhD2-FCNSINgfjE-IDZlSfRM/s1600/Screen+Shot+2016-11-17+at+2.44.27+PM.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqAr3aoBuiJrQkqqxQf29S0JKT3bdSq4tZBptG_oCkuVdPIEG6HSylD6UXch5K9r8yhDcERwJ-LIoyXT-Lz9NEajL0P-yFQ6d9837y5_PuCCYKabzwcvDyhD2-FCNSINgfjE-IDZlSfRM/s1600/Screen+Shot+2016-11-17+at+2.44.27+PM.jpg" /></a></div>
The CC edges can also be mirrored when they are adjacent, as this example of a CCGG type shows. The tiling also fits G1G1G2G2.<br />
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A CCCC type that uses glided edges can reflect over the diagonal and fit isohedral class IH69. It is type G1G1G2G2 formed with edges of central rotation.<br />
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Alternatively, the edges can mirror as opposite edges in which case it is also type G1G2G1G2.<br />
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Two of the eleven quadrilateral Heesch types can be formed with asymmetric edges that will simultaneously fit either G1G2G1G2 or G1G1G2G2. Isohedral class IH68 is simultaneously TTTT and G1G1G2G2. All edges are shaped identically and there is mirroring over one diagonal<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjtmp14e0nus1gAV5vZkyHQbOtN6I8wuvB1-UXEgxpV3gQYmZg2rupZcLyO9qgzjhm0KrwKtWlG4j7bw7IShOXTFdNwSFyoMtmYmP_K7MY-j9Nf5DBat9uqdngElwqwGLYsLY8trKYe4sk/s1600/Screen+Shot+2016-11-17+at+2.45.13+PM.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjtmp14e0nus1gAV5vZkyHQbOtN6I8wuvB1-UXEgxpV3gQYmZg2rupZcLyO9qgzjhm0KrwKtWlG4j7bw7IShOXTFdNwSFyoMtmYmP_K7MY-j9Nf5DBat9uqdngElwqwGLYsLY8trKYe4sk/s1600/Screen+Shot+2016-11-17+at+2.45.13+PM.jpg" /></a></div>
Isohedral class IH71 can be seen as either G1G2G1G2 or as C4C4C4C4. All edges must be identically formed and there is mirroring over one diagonal.<br />
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An alternative way of getting a glided C4C4C4C4 tiling is with an arrangement that is simultaneously G1G1G2G2 and C4C4C4C4.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgzRN4WdRqghohXRX6SEMhh709HXGPV6yFqUkVZxM-vL90mhVt22WvZutv5jwgK-kDY87O7-uYemdERwpQku5_cbQ_EGq0j2TzKqGx0WbDMRQMX9253BUWUMQ5lT9ZrtA_fpMY63bzdCJQ/s1600/Screen+Shot+2016-11-17+at+2.45.35+PM.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgzRN4WdRqghohXRX6SEMhh709HXGPV6yFqUkVZxM-vL90mhVt22WvZutv5jwgK-kDY87O7-uYemdERwpQku5_cbQ_EGq0j2TzKqGx0WbDMRQMX9253BUWUMQ5lT9ZrtA_fpMY63bzdCJQ/s1600/Screen+Shot+2016-11-17+at+2.45.35+PM.jpg" /></a></div>
(By accident I discovered that this shape will tile anisohedrally.)<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgV6RU2GA-Wh9wasYgO0QGLAoQmVXEan1haJwpfvtl5d2WQdOlzoqyM4tEk9LnIbdl5wxXhirgAFTG8Zw8FgEppvtQYJ4ybv0tN2aLqUIuPvCp2z5STp65iqpb33n4xdKX-gAzoHo1kRuc/s1600/Screen+Shot+2016-11-17+at+2.45.43+PM.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgV6RU2GA-Wh9wasYgO0QGLAoQmVXEan1haJwpfvtl5d2WQdOlzoqyM4tEk9LnIbdl5wxXhirgAFTG8Zw8FgEppvtQYJ4ybv0tN2aLqUIuPvCp2z5STp65iqpb33n4xdKX-gAzoHo1kRuc/s1600/Screen+Shot+2016-11-17+at+2.45.43+PM.jpg" /></a></div>
Finally, two of the eleven quadrilateral Heesch types can be formed with glided edges but the tilings are not G1G1G2G2 or G1G2G1G2. To form type C3C3C3C3 with glided edges, the edges must reflect over their midpoints.<br />
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The tile in the tiling above can be arranged in a G1G1G2G2 tiling.<br />
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As with type C3C3C3C3, to form type C3C3C6C6 with glided edges, the edges must reflect over their midpoints.<br />
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The tile can also be arranged in a G1G1G2G2 pattern as illustrated below.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPzGX2fef0Ot2CZruOR-iyj3R4kwjbQ8F0Af1NvdOomArQWmFMyXLr2d8Opz7chKLzLWnhxXNzMshq5h89K3TATnIVvZTbt_7y_QMnDIimvDIwyBWlTR2Q500uTh7KZdQrVFw6cxZJhkU/s1600/Screen+Shot+2016-11-17+at+2.46.32+PM.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPzGX2fef0Ot2CZruOR-iyj3R4kwjbQ8F0Af1NvdOomArQWmFMyXLr2d8Opz7chKLzLWnhxXNzMshq5h89K3TATnIVvZTbt_7y_QMnDIimvDIwyBWlTR2Q500uTh7KZdQrVFw6cxZJhkU/s1600/Screen+Shot+2016-11-17+at+2.46.32+PM.jpg" /></a></div>
Part two will consider the<a href="http://qustions%20no%20one%20is%20asking%20part%20two/"> hexagonal types</a>.Dessert Survivorhttp://www.blogger.com/profile/04616064444288249273noreply@blogger.com0tag:blogger.com,1999:blog-4026157408460542489.post-76697030418396044172016-11-15T08:06:00.000-08:002016-11-15T08:06:00.714-08:00Fun with puzzle piecesIf all edges placed on a square template are identical with mirror symmetry over the midpoint, there are two distinct shapes that will tessellate, one with two “outs” that are adjacent and one with two “outs” that are opposite. Note that the shape must have two “out” edges and two “in” edges; if the number of “outs” is not equal to the number of “ins”, the pieces will not tessellate. Below is an illustration of the two possibilities with an edge that forms puzzle pieces.<br />
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Suppose that the edge does not have the bump centered in the middle but rather offset to one side. How many distinctly different shapes with these identical asymmetric edges will tile the plane?<br />
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I answer this question in a <a href="http://dx.doi.org/10.1017/mag.2016.121">note published</a> in the November 2016 issue (Vol 100 Issue 549, pp 511-516) of the <i>Mathematical Gazette:</i> there are 15 distinctly different shapes and all will tile the plane. Thirteen will tile the plane as Heesch types and two will tile it in a non-isohedral pattern.<br />
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The note in the <i>Mathematical Gazette</i> is limited to the square template. <i><a href="http://ingrimayne.com/mazes/exploringtessellations.htm">Exploring Tessellations: A Journey Through Heesch Types and Beyond</a></i> extends the analysis. If the template is a rhombus or diamond (a square squashed), there are 30 distinct shapes of which 20 will tile the plane. If the template is a regular hexagon, there are 108 distinct shapes that have three “ins” and three “outs” when identical, asymmetric edges are fitted to the template. Thirty-four tile isohedrally as Heesch types and another nine tile anisohedrally. Sixty-five will not tile.<br />
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<i>Exploring Tessellations: A Journey Through Heesch Types and Beyond</i> also considers equilateral templates fitted with edges that have central rotation and some of these results have been mentioned in past posts on this blog. There are <a href="http://mazepuzzles.blogspot.com/2015/12/fabtwo.html">two distinct shapes</a> if the template is an equilateral triangle, <a href="http://mazepuzzles.blogspot.com/2015/11/the-fabfours-family-of-typefaces.html">four for a square</a>, <a href="http://mazepuzzles.blogspot.com/search?updated-max=2016-01-08T19:51:00-08:00&max-results=7">seven for a rhombus</a>, and nine for a regular hexagon. One of the nine for the regular hexagon will not tessellate. Also, all Heesch types except three that will not fit as equilateral polygons (CC3C3, CC4C4, and C3C3C6C6) can be formed with identical edges of central rotation.Dessert Survivorhttp://www.blogger.com/profile/04616064444288249273noreply@blogger.com0tag:blogger.com,1999:blog-4026157408460542489.post-79124049628323269692016-11-02T12:00:00.000-07:002016-11-06T07:46:58.654-08:00More Tesselmaniac fun<div class="separator" style="clear: both; text-align: center;">
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A couple posts from October, 2015 (<a href="http://mazepuzzles.blogspot.com/2015/10/restricted-tessellation-types-part-1.html">here</a> and <a href="http://mazepuzzles.blogspot.com/2015/10/restricted-tesselation-types-part-2.html">here</a>) examined ways of using <i>Tesselmaniac!</i> to construct isohedral class tilings that that did not have a template in <i>Tesselmaniac!.</i> Three hexagonal tilings with opposite straight edges were not included and all three of these can be easily constructed without removing interior lines.<br />
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First is isohedral class IH14. It has four edges that are shaped the same. Each edge is glided and reflected, and the result is that all three opposite pairs are translated. It fits two Heesch types, TG1G2TG2G1 and TTTTTT. It also will tile as a non-Heesch (and non-isohedral) type with flips over the straight edges. It has cm symmetry. It can formed in <i>Tesselmaniac!</i> in the IH68 template by positioning a point to create a straight edge. Obviously, if that line is reduced to zero, the tiling will be an IH68 type, a mirrored G1G1G2G2 type.<br />
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Isohedral class IH15 is another mirrored tiling with two adjacent edges formed identically with center-point rotation and mirrored and the other two adjacent edges on the other side of the two straight edges also formed identically and mirrored. It satisfies the condition Heesch type TCCTCC with the straight edges serving as the translated pair. Because mirroring an edge formed with central rotation is the same as flipping it, IH15 also satisfies types TG1G1TG2G2, and TCCTGG. Finally, because the tiles can be flipped over their straight edges, it also tiles as a non-Heesch (and non isohedral) type. IH15 has pmg symmetry.<br />
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In <i>Tesselmaniac!</i> IH15 can be constructed with the mirrored C*CC*C or IH69 template by positioning a point to create a straight edge.<br />
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Isohedral class IH17 is a special case of both IH14 and IH15 (as well as IH8, IH9, IH12, and IH13). Two opposite edges are unshaped, straight lines. The other four edges are all shaped with identical center-point rotation and each is reflected (which is the same as gliding) both vertically and horizontally. In addition to tiling in a non-Heesch manner with flips over the straight edges, it satisfies the conditions of six of the seven hexagonal Heesch types, everything but the C3C3C3C3C3C3 type. It has cmm symmetry. Notice the the shape of the tile is restored when it is rotated 180 degrees, when it is flipped over its horizontal midpoint, and when it is flipped over its vertical midpoint.<br />
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In <i>Tesselmaniac!</i> IH17 can be constructed with the mirrored C*C*C*C *or IH74 template by positioning a point to create a straight edge.<br />
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The tiles in IH14 and IH15 (and also IH16) have symmetry over one diagonal. The tiles in IH17 have symmetry over one diagonal and one edge, which also gives them twofold rotational symmetry.Dessert Survivorhttp://www.blogger.com/profile/04616064444288249273noreply@blogger.com0