{"id":12557,"date":"2025-03-27T12:30:52","date_gmt":"2025-03-27T16:30:52","guid":{"rendered":"https:\/\/blogs.mathworks.com\/cleve\/?p=12557"},"modified":"2025-03-29T11:10:19","modified_gmt":"2025-03-29T15:10:19","slug":"the-hat-a-tridecagon-aperiordic-monotile","status":"publish","type":"post","link":"https:\/\/blogs.mathworks.com\/cleve\/2025\/03\/27\/the-hat-a-tridecagon-aperiordic-monotile\/","title":{"rendered":"The Hat, a Tridecagon Aperiordic Monotile"},"content":{"rendered":"\n<div class=\"content\"><!--introduction-->\n<p>Two years ago, in March of 2023, an unlikely team of mathematical hobbyists announced the discovery of a remarkable 13-sided polygon that they nick-named the \"Hat\". Today, a Google search for the Hat's more formal name, \"Aperiodic Monotile\", yields dozens of links.<\/p>\n<p>This blog post is about the Hat and the resulting <tt>polyshape<\/tt> object.<\/p>\n<p>\n<img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"http:\/\/blogs.mathworks.com\/cleve\/files\/hat_.png\" alt=\"\"> <\/p>\n<!--\/introduction-->\n<h3>Contents<\/h3>\n<div>\n<ul>\n<li>\n<a href=\"#255f137f-1b1d-450c-b234-112acdcc41ef\">An Aperiodic Monotile<\/a>\n<\/li>\n<li>\n<a href=\"#b8324dec-00e7-4e81-9fb1-96d8ed96b16b\">Tilings<\/a>\n<\/li>\n<li>\n<a href=\"#3d2e57a3-9f03-4a57-92bc-9bc7f3a74704\">Tridecagons<\/a>\n<\/li>\n<li>\n<a href=\"#6473adc0-1dff-48cc-9014-2ae1a14756bf\">Penrose Tiling<\/a>\n<\/li>\n<li>\n<a href=\"#4700daf2-d018-4972-a84e-670ceb4712ec\">Reflections<\/a>\n<\/li>\n<li>\n<a href=\"#1c9ad6a5-741c-4d23-9984-e56d42e061d9\">Monotile Tiling<\/a>\n<\/li>\n<li>\n<a href=\"#f1eb4349-b7d3-4ed9-a300-578ceefe4b05\">Polyshapes<\/a>\n<\/li>\n<li>\n<a href=\"#7588ffc4-651d-40ed-a009-468cab0d7c3e\">Level 0<\/a>\n<\/li>\n<li>\n<a href=\"#6478bd2a-7ac4-46ca-a5e6-1c9ce4a44f77\">Level 1<\/a>\n<\/li>\n<li>\n<a href=\"#5126388f-0db4-4ccf-bb6a-bf58d28c3cea\">Level 2<\/a>\n<\/li>\n<li>\n<a href=\"#11ad4f77-fafe-410e-96e3-6def0f62b56f\">Level 3<\/a>\n<\/li>\n<li>\n<a href=\"#d5f07b6e-028b-4105-96e0-14f91bd5f510\">Numbers<\/a>\n<\/li>\n<li>\n<a href=\"#d40bb7b1-7e5b-477b-9f63-b873cb7d8f65\">Level Color<\/a>\n<\/li>\n<li>\n<a href=\"#1afb91ef-26bd-4ff8-bd6f-daa624de3d13\">Convex Hull<\/a>\n<\/li>\n<\/ul>\n<\/div>\n<h4>An Aperiodic Monotile<a name=\"255f137f-1b1d-450c-b234-112acdcc41ef\"><\/a>\n<\/h4>\n<p>The authors of the paper that announced the discovery of the Hat are <a href=\"https:\/\/the-orangery.weebly.com\" target=\"_blank\">David Smith<\/a>, <a href=\"https:\/\/www.polyomino.org.uk\" target=\"_blank\">Joseph Samuel Myers<\/a>, <a href=\"https:\/\/cs.uwaterloo.ca\/~csk\/\" target=\"_blank\">Craig S. Kaplan<\/a> and <a href=\"https:\/\/chaimgoodmanstrauss.com\/\" target=\"_blank\">Chaim Goodman-Strauss<\/a>. The announcement was made in an <tt>archive&gt;math<\/tt> preprint with the title <a href=\"https:\/\/arxiv.org\/abs\/2303.10798\" target=\"_blank\">\"An Aperiodic Monotile\"<\/a>.<\/p>\n<p>Within days of the announcement, articles like this one in <a href=\"https:\/\/www.sciencenews.org\/article\/mathematicians-discovered-einstein-tile\" target=\"_blank\">Science News<\/a> appeared. Two months later, Florentin Waligorski created an <a href=\"https:\/\/www.youtube.com\/watch?v=BoAx-rLo5P0\" target=\"_blank\">origami Hat<\/a>.<\/p>\n<p>The official paper was published in June of 2024 in the journal <i>Combinatorial Theory<\/i>. It was also titled <a href=\"https:\/\/doi.org\/10.5070\/C64163843\" target=\"_blank\">\"An Aperiodic Monotile\"<\/a>.<\/p>\n<p>I first heard about the Hat is an article by Erica Klarreich in the magazine <i>Quanta<\/i>, <a href=\"https:\/\/www.quantamagazine.org\/hobbyist-finds-maths-elusive-einstein-tile-20230404\/\" target=\"_blank\">Hobbyist Finds Math&rsquo;s Elusive &lsquo;Einstein&rsquo; Tile<\/a>.<\/p>\n<h4>Tilings<a name=\"b8324dec-00e7-4e81-9fb1-96d8ed96b16b\"><\/a>\n<\/h4>\n<p>Quoting <a href=\"https:\/\/en.wikipedia.org\/wiki\/Tessellation\" target=\"_blank\">Wikipedia<\/a>:<\/p>\n<pre>  A \"tessellation\" or \"tiling\" is the covering of a surface,\n  often a plane, using one or more geometric shapes, called \"tiles\"\n  with no overlaps and no gaps.<\/pre>\n<pre>  A \"periodic tiling\" has a repeating pattern. Some special kinds\n  include regular tilings with regular polygonal tiles all of the\n  same shape, and semiregular tilings with regular tiles of more\n  than one shape and with every corner identically arranged.<\/pre>\n<pre>  A tiling that lacks a repeating pattern is called \"aperiodic\".\n  An aperiodic tiling uses a small set of tile shapes that\n  cannot form a repeating pattern.<\/pre>\n<h4>Tridecagons<a name=\"3d2e57a3-9f03-4a57-92bc-9bc7f3a74704\"><\/a>\n<\/h4>\n<p>A <i>tridecagon<\/i> is a polygon with 13 sides, like this gold coin from the Czech Republic. The sides of a <i>regular<\/i> <i>tridecagon<\/i> are all the same length. Any attempt to tile your floor with these coins inevitably has gaps. Regular tridecagons cannot tile the plane. The Hat is an <i>irregular<\/i> tridecagon that can tile the plane.<\/p>\n<p>\n<img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"http:\/\/blogs.mathworks.com\/cleve\/files\/20_CZK_.png\" alt=\"\"> <\/p>\n<h4>Penrose Tiling<a name=\"6473adc0-1dff-48cc-9014-2ae1a14756bf\"><\/a>\n<\/h4>\n<p>\n<a href=\"https:\/\/en.wikipedia.org\/wiki\/Roger_Penrose\" target=\"_blank\">Roger Penrose<\/a> is a <a href=\"https:\/\/www.google.com\/search?q=Nobel+prize\" target=\"_blank\">Nobel prize-winning<\/a> mathematician and physicist. Among his many achievements are the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Moore%E2%80%93Penrose_inverse\" target=\"_blank\">Moore-Penrose pseudoinverse<\/a> and the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Penrose_tiling\">Penrose tiling<\/a>. The Penrose tiling uses these regular quadrilateral tiles, the \"kite\" and the \"dart\", to produce an aperiodic tiling with two tiles. The Hat generates an aperiodic tiling with a <i>single<\/i> tile.<\/p>\n<p>\n<img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"http:\/\/blogs.mathworks.com\/cleve\/files\/kite_dart_.png\" alt=\"\"> <\/p>\n<p>\n<a href=\"https:\/\/steveeddins.com\/\" target=\"_blank\">Steve Eddins<\/a> is a <a href=\"https:\/\/blogs.mathworks.com\/cleve\/files\/foc_.png\">FOC-winning<\/a> French horn player. Among his many achievements is a <a href=\"https:\/\/blogs.mathworks.com\/cleve\/2018\/11\/26\/penrose-and-fourier-design-playing-cards\/\">Cleve's Corner<\/a> about this Penrose tiling.<\/p>\n<p>\n<img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"http:\/\/blogs.mathworks.com\/cleve\/files\/Penrose_.png\" alt=\"\"> <\/p>\n<h4>Reflections<a name=\"4700daf2-d018-4972-a84e-670ceb4712ec\"><\/a>\n<\/h4>\n<p>We say that the Hat on the left is reflected and the one on the right is not reflected.<\/p>\n<p>\n<img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"http:\/\/blogs.mathworks.com\/cleve\/files\/reflections_.png\" alt=\"\"> <\/p>\n<h4>Monotile Tiling<a name=\"1c9ad6a5-741c-4d23-9984-e56d42e061d9\"><\/a>\n<\/h4>\n<p>Here is (a finite piece of) an infinite, aperiodic tiling of the plane using only the Hat and its reflection. This figure is one half of figure 2.12 from the original preprint, <a href=\"https:\/\/doi.org\/10.5070\/C64163843\" target=\"_blank\">\"An Aperiodic Monotile\"<\/a>. Each dark blue tile is surrounded by three light blue tiles. The white tiles appear alone or in pairs. If you look carefully, you can see the grey tiles form filaments. The filaments are more apparent in the other half of figure 2.12 on page 20 of the preprint.<\/p>\n<p>The dark tiles are not reflected; all of the other tiles are reflected.<\/p>\n<p>\n<img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"http:\/\/blogs.mathworks.com\/cleve\/files\/monotile_tiling_.png\" alt=\"\"> <\/p>\n<h4>Polyshapes<a name=\"f1eb4349-b7d3-4ed9-a300-578ceefe4b05\"><\/a>\n<\/h4>\n<p>Quoting the MATLAB documentation for the <a href=\"https:\/\/www.mathworks.com\/help\/matlab\/ref\/polyshape.html\"><tt>polyshape object<\/tt><\/a>.<\/p>\n<pre>  The polyshape function creates polygon-like shapes from 2-D vertices.\n  However, unlike polygons, a polyshape can have discontiguous regions\n  and holes. The properties of a polyshape object describe its vertices,\n  solid regions, and holes.<\/pre>\n<p>All of the figures after this point in the blog post were made with <tt>polyshape\/plot<\/tt>.<\/p>\n<p>We begin with a pentagonal <tt>polyshape<\/tt> made from a portion of a regular hexagon.<\/p>\n<p>\n<img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"http:\/\/blogs.mathworks.com\/cleve\/files\/polyshapes0_.png\" alt=\"\"> <\/p>\n<p>The Hat itself is a <tt>polyshape<\/tt> formed from the <tt>union<\/tt> of four rotated and translated copies of the pentagon.<\/p>\n<p>\n<img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"http:\/\/blogs.mathworks.com\/cleve\/files\/polyshapes1_.png\" alt=\"\"> <\/p>\n<h4>Level 0<a name=\"7588ffc4-651d-40ed-a009-468cab0d7c3e\"><\/a>\n<\/h4>\n<p>We have experimented with a tiling created by expanding rings of unreflected hats centered around a single reflected hat. We stop after three rings because additional reflected hats are needed to continue.<\/p>\n<p>The zeroth level is a single reflected hat.<\/p>\n<p>\n<img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"http:\/\/blogs.mathworks.com\/cleve\/files\/hats_0_.gif\" alt=\"\"> <\/p>\n<h4>Level 1<a name=\"6478bd2a-7ac4-46ca-a5e6-1c9ce4a44f77\"><\/a>\n<\/h4>\n<p>The first level adds a ring of three hats.<\/p>\n<p>\n<img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"http:\/\/blogs.mathworks.com\/cleve\/files\/hats_1_.gif\" alt=\"\"> <\/p>\n<h4>Level 2<a name=\"5126388f-0db4-4ccf-bb6a-bf58d28c3cea\"><\/a>\n<\/h4>\n<p>The second ring has nine more hats.<\/p>\n<p>\n<img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"http:\/\/blogs.mathworks.com\/cleve\/files\/hats_2_.gif\" alt=\"\"> <\/p>\n<h4>Level 3<a name=\"11ad4f77-fafe-410e-96e3-6def0f62b56f\"><\/a>\n<\/h4>\n<p>The level 3 ring has 18 hats.<\/p>\n<p>\n<img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"http:\/\/blogs.mathworks.com\/cleve\/files\/hats_3_.gif\" alt=\"\"> <\/p>\n<h4>Numbers<a name=\"d5f07b6e-028b-4105-96e0-14f91bd5f510\"><\/a>\n<\/h4>\n<p>Our <tt>Hats<\/tt> program allows you to move a hat around the screen with your mouse. When you get close to another hat, numbers appear to guide your final approach. Here are vertices 8, 9 and 10 on hat number 6 near vertices 12, 11 and 10 on hat number 2.<\/p>\n<p>\n<img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"http:\/\/blogs.mathworks.com\/cleve\/files\/target_.png\" alt=\"\"> <\/p>\n<p>This crowded figure shows all the hat numbers and all the vertex indices at level 2.<\/p>\n<p>\n<img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"http:\/\/blogs.mathworks.com\/cleve\/files\/numbers_.png\" alt=\"\"> <\/p>\n<h4>Level Color<a name=\"d40bb7b1-7e5b-477b-9f63-b873cb7d8f65\"><\/a>\n<\/h4>\n<p>Each level has a single color.<\/p>\n<div>\n<ul>\n<li>0 Dark blue<\/li>\n<li>1 Light blue<\/li>\n<li>2 White<\/li>\n<li>3 Grey<\/li>\n<\/ul>\n<\/div>\n<p>Compare this with the detail in figure 2.12 of <a href=\"https:\/\/arxiv.org\/abs\/2303.10798\" target=\"_blank\">\"An Aperiodic Monotile\"<\/a>.<\/p>\n<p>\n<img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"http:\/\/blogs.mathworks.com\/cleve\/files\/levelcolor_.png\" alt=\"\"> <\/p>\n<h4>Convex Hull<a name=\"1afb91ef-26bd-4ff8-bd6f-daa624de3d13\"><\/a>\n<\/h4>\n<p>Convex hull is one of many other methods available for <a href=\"https:\/\/www.mathworks.com\/help\/matlab\/ref\/polyshape.html\"><tt>polyshape objects<\/tt><\/a>.<\/p>\n<p>\n<img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"http:\/\/blogs.mathworks.com\/cleve\/files\/convexhull_.png\" alt=\"\"> <\/p>\n<script language=\"JavaScript\"> <!-- \n    function grabCode_fa96bca28308424ab70012eb3ed11708() {\n        \/\/ Remember the title so we can use it in the new page\n        title = document.title;\n\n        \/\/ Break up these strings so that their presence\n        \/\/ in the Javascript doesn't mess up the search for\n        \/\/ the MATLAB code.\n        t1='fa96bca28308424ab70012eb3ed11708 ' + '##### ' + 'SOURCE BEGIN' + ' #####';\n        t2='##### ' + 'SOURCE END' + ' #####' + ' fa96bca28308424ab70012eb3ed11708';\n    \n        b=document.getElementsByTagName('body')[0];\n        i1=b.innerHTML.indexOf(t1)+t1.length;\n        i2=b.innerHTML.indexOf(t2);\n \n        code_string = b.innerHTML.substring(i1, i2);\n        code_string = code_string.replace(\/REPLACE_WITH_DASH_DASH\/g,'--');\n\n        \/\/ Use \/x3C\/g instead of the less-than character to avoid errors \n        \/\/ in the XML parser.\n        \/\/ Use '\\x26#60;' instead of '<' so that the XML parser\n        \/\/ doesn't go ahead and substitute the less-than character. \n        code_string = code_string.replace(\/\\x3C\/g, '\\x26#60;');\n\n        copyright = 'Copyright 2025 The MathWorks, Inc.';\n\n        w = window.open();\n        d = w.document;\n        d.write('<pre>\\n');\n        d.write(code_string);\n\n        \/\/ Add copyright line at the bottom if specified.\n        if (copyright.length > 0) {\n            d.writeln('');\n            d.writeln('%%');\n            if (copyright.length > 0) {\n                d.writeln('% _' + copyright + '_');\n            }\n        }\n\n        d.write('<\/pre>\\n');\n\n        d.title = title + ' (MATLAB code)';\n        d.close();\n    }   \n     --> <\/script>\n<p style=\"text-align: right; font-size: xx-small; font-weight:lighter;   font-style: italic; color: gray\">\n<br>\n<a href=\"javascript:grabCode_fa96bca28308424ab70012eb3ed11708()\"><span style=\"font-size: x-small;        font-style: italic;\">Get \n      the MATLAB code <noscript>(requires JavaScript)<\/noscript>\n<\/span><\/a>\n<br>\n<br>\n      Published with MATLAB&reg; R2024b<br>\n<\/p>\n<\/div>\n<!--\nfa96bca28308424ab70012eb3ed11708 ##### SOURCE BEGIN #####\n%% The Hat, a Tridecagon Aperiodic Monotile\n% Two years ago, in March of 2023, an unlikely team of\n% mathematical hobbyists announced the discovery of a remarkable \n% 13-sided polygon that they nick-named the \"Hat\".\n% Today, a Google search for the Hat's more formal name,\n% \"Aperiodic Monotile\", yields dozens of links.\n%\n% This blog post is about the Hat and the resulting |polyshape| object.\n% \n% <<hat_.png>>\n%\n\n%% An Aperiodic Monotile\n% The authors of the paper that announced the discovery of the Hat are\n% <https:\/\/the-orangery.weebly.com David Smith>, \n% <https:\/\/www.polyomino.org.uk Joseph Samuel Myers>, \n% <https:\/\/cs.uwaterloo.ca\/~csk\/ Craig S. Kaplan> and \n% <https:\/\/chaimgoodmanstrauss.com\/ Chaim Goodman-Strauss>.\n% The announcement was made in an |archive>math| preprint with the title\n% <https:\/\/arxiv.org\/abs\/2303.10798 \"An Aperiodic Monotile\">.\n%\n% Within days of the announcement, articles like this one in\n% <https:\/\/www.sciencenews.org\/article\/mathematicians-discovered-einstein-tile\n% Science News> appeared.\n% Two months later, Florentin Waligorski created an\n% <https:\/\/www.youtube.com\/watch?v=BoAx-rLo5P0 origami Hat>.\n%\n% The official paper was published in June of 2024 in the journal\n% _Combinatorial Theory_.  It was also titled \n% <https:\/\/doi.org\/10.5070\/C64163843 \"An Aperiodic Monotile\">.\n%\n% I first heard about the Hat is an article by Erica Klarreich\n% in the magazine _Quanta_,\n% <https:\/\/www.quantamagazine.org\/hobbyist-finds-maths-elusive-einstein-tile-20230404\/\n% Hobbyist Finds Math\u2019s Elusive \u2018Einstein\u2019 Tile>.\n%\n\n%% Tilings\n% Quoting <https:\/\/en.wikipedia.org\/wiki\/Tessellation Wikipedia>:\n%\n%    A \"tessellation\" or \"tiling\" is the covering of a surface, \n%    often a plane, using one or more geometric shapes, called \"tiles\"\n%    with no overlaps and no gaps.\n%\n%    A \"periodic tiling\" has a repeating pattern. Some special kinds \n%    include regular tilings with regular polygonal tiles all of the \n%    same shape, and semiregular tilings with regular tiles of more\n%    than one shape and with every corner identically arranged.\n%\n%    A tiling that lacks a repeating pattern is called \"aperiodic\". \n%    An aperiodic tiling uses a small set of tile shapes that\n%    cannot form a repeating pattern.\n\n%% Tridecagons\n% A _tridecagon_ is a polygon with 13 sides, like this gold coin from\n% the Czech Republic. The sides of a _regular_ _tridecagon_ are all\n% the same length.  Any attempt to tile your floor with these coins\n% inevitably has gaps.  Regular tridecagons cannot tile the plane.\n% The Hat is an _irregular_ tridecagon that can tile the plane. \n%\n% <<20_CZK_.png>>\n%\n\n%% Penrose Tiling\n% <https:\/\/en.wikipedia.org\/wiki\/Roger_Penrose Roger Penrose> is a\n% <https:\/\/www.google.com\/search?q=Nobel+prize Nobel prize-winning>\n% mathematician and physicist.\n% Among his many achievements are the\n% <https:\/\/en.wikipedia.org\/wiki\/Moore%E2%80%93Penrose_inverse\n% Moore-Penrose pseudoinverse> and the\n% <https:\/\/en.wikipedia.org\/wiki\/Penrose_tiling Penrose tiling>.\n% The Penrose tiling uses these regular quadrilateral tiles,\n% the \"kite\" and the \"dart\", to produce an aperiodic tiling with\n% two tiles.\n% The Hat generates an aperiodic tiling with a _single_ tile.\n%\n% <<kite_dart_.png>>\n%\n% <https:\/\/steveeddins.com\/ Steve Eddins> is a \n% <https:\/\/blogs.mathworks.com\/cleve\/files\/foc_.png FOC-winning>\n% French horn player.\n% Among his many achievements is a \n% <https:\/\/blogs.mathworks.com\/cleve\/2018\/11\/26\/penrose-and-fourier-design-playing-cards\/\n% Cleve's Corner> about this Penrose tiling.\n%\n% <<Penrose_.png>>\n% \n\n%% Reflections\n% We say that the Hat on the left is reflected and the one on the right\n% is not reflected.\n%\n% <<reflections_.png>>\n%\n\n%% Monotile Tiling\n% Here is (a finite piece of) an infinite, aperiodic tiling of the plane\n% using only the Hat and its reflection.  \n% This figure is one half of figure 2.12 from the original preprint,\n% <https:\/\/doi.org\/10.5070\/C64163843 \"An Aperiodic Monotile\">.\n% Each dark blue tile is surrounded by three light blue tiles.\n% The white tiles appear alone or in pairs.\n% If you look carefully, you can see the grey tiles form filaments.\n% The filaments are more apparent in the other half of figure 2.12\n% on page 20 of the preprint.\n%\n% The dark tiles are not reflected; all of the other tiles are reflected.\n%\n% <<monotile_tiling_.png>>\n%\n\n%% Polyshapes\n% Quoting the MATLAB documentation for the\n% <https:\/\/www.mathworks.com\/help\/matlab\/ref\/polyshape.html\n% |polyshape object|>.\n%\n%    The polyshape function creates polygon-like shapes from 2-D vertices.\n%    However, unlike polygons, a polyshape can have discontiguous regions\n%    and holes. The properties of a polyshape object describe its vertices,\n%    solid regions, and holes.\n\n%%\n% All of the figures after this point in the blog post were made with\n% |polyshape\/plot|.\n\n%%\n% We begin with a pentagonal |polyshape| made from a portion of a \n% regular hexagon.\n%\n% <<polyshapes0_.png>>\n%\n\n%%\n% The Hat itself is a |polyshape| formed from the |union| of four rotated\n% and translated copies of the pentagon.\n%\n%\n% <<polyshapes1_.png>>\n%\n\n%% Level 0\n% We have experimented with a tiling created by expanding rings of\n% unreflected hats centered around a single reflected hat.\n% We stop after three rings because additional reflected hats\n% are needed to continue.\n\n%%\n% The zeroth level is a single reflected hat.\n%\n% <<hats_0_.gif>>\n%\n\n%% Level 1\n% The first level adds a ring of three hats.\n%\n% <<hats_1_.gif>>\n%\n\n%% Level 2\n% The second ring has nine more hats.\n%\n% <<hats_2_.gif>>\n%\n\n%% Level 3\n% The level 3 ring has 18 hats.\n%\n% <<hats_3_.gif>>\n%\n\n%% Numbers\n% Our |Hats| program allows you to move a hat around the screen with\n% your mouse.  When you get close to another hat, numbers appear\n% to guide your final approach.  Here are vertices 8, 9 and 10 on\n% hat number 6 near vertices 12, 11 and 10 on hat number 2. \n%\n% <<target_.png>>\n\n%%\n% This crowded figure shows all the hat numbers and all the vertex\n% indices at level 2.\n%   \n% <<numbers_.png>>\n%\n\n%% Level Color\n% Each level has a single color.\n%\n% * 0  Dark blue\n% * 1  Light blue\n% * 2  White\n% * 3  Grey\n%\n% Compare this with the detail in figure 2.12 of\n% <https:\/\/arxiv.org\/abs\/2303.10798 \"An Aperiodic Monotile\">. \n%\n% <<levelcolor_.png>>\n%\n\n%% Convex Hull\n% Convex hull is one of many other methods available for\n% <https:\/\/www.mathworks.com\/help\/matlab\/ref\/polyshape.html\n% |polyshape objects|>.\n%\n% <<convexhull_.png>>\n%\n\n##### SOURCE END ##### fa96bca28308424ab70012eb3ed11708\n-->\n","protected":false},"excerpt":{"rendered":"<div class=\"overview-image\"><img src=\"https:\/\/blogs.mathworks.com\/cleve\/files\/polyhat1.png\" class=\"img-responsive attachment-post-thumbnail size-post-thumbnail wp-post-image\" alt=\"\" decoding=\"async\" loading=\"lazy\" \/><\/div><!--introduction-->\n<p>Two years ago, in March of 2023, an unlikely team of mathematical hobbyists announced the discovery of a remarkable 13-sided polygon that they nick-named the \"Hat\". Today, a Google search for the Hat's more formal name, \"Aperiodic Monotile\", yields dozens of links.... <a class=\"read-more\" href=\"https:\/\/blogs.mathworks.com\/cleve\/2025\/03\/27\/the-hat-a-tridecagon-aperiordic-monotile\/\">read more >><\/a><\/p>","protected":false},"author":78,"featured_media":12623,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[32,23,4,59,56],"tags":[],"_links":{"self":[{"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/posts\/12557"}],"collection":[{"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/users\/78"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/comments?post=12557"}],"version-history":[{"count":8,"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/posts\/12557\/revisions"}],"predecessor-version":[{"id":12636,"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/posts\/12557\/revisions\/12636"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/media\/12623"}],"wp:attachment":[{"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/media?parent=12557"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/categories?post=12557"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/tags?post=12557"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}