{"id":1306,"date":"2015-02-15T01:00:20","date_gmt":"2015-02-15T01:00:20","guid":{"rendered":"http:\/\/blogs.ams.org\/visualinsight\/?p=1306"},"modified":"2015-07-29T00:47:29","modified_gmt":"2015-07-29T00:47:29","slug":"pentagon-decagon-branched-covering","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/visualinsight\/2015\/02\/15\/pentagon-decagon-branched-covering\/","title":{"rendered":"Pentagon-Decagon Branched Covering"},"content":{"rendered":"<div align=\"center\">\n<div id=\"attachment_1305\" style=\"width: 406px\" class=\"wp-caption alignnone\"><a href=\"http:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch10.png\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1305\" src=\"http:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch10.png\" alt=\"Pentagon-Decagon Branched Covering (Stage 10) - Greg Egan\" width=\"400\" height=\"400\" class=\"size-full wp-image-1305\" srcset=\"https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch10.png 400w, https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch10-150x150.png 150w, https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch10-300x300.png 300w, https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch10-50x50.png 50w\" sizes=\"auto, (max-width: 400px) 100vw, 400px\" \/><\/a><p id=\"caption-attachment-1305\" class=\"wp-caption-text\">Pentagon-Decagon Branched Covering &#8211; Greg Egan<\/p><\/div>\n<\/div>\n<p>Two regular pentagons and a regular decagon fit snugly at a point: their interior angles sum to 360&deg;.  Despite this, you cannot tile the plane with regular pentagons and decagons:<\/p>\n<ul>\n<li>\n<a href=\"http:\/\/blogs.ams.org\/visualinsight\/2015\/02\/01\/pentagon-decagon-packing\/\">Pentagon-decagon packing<\/a>.\n<\/li>\n<\/ul>\n<p>However, there is a <a href=\"http:\/\/en.wikipedia.org\/wiki\/Branched_covering\">branched covering<\/a> of the plane tiled with pentagons and decagons that map to regular pentagons and decagons on the plane!<\/p>\n<p>In the following series of images by <a href=\"http:\/\/www.gregegan.net\">Greg Egan<\/a>, we see how to build up this branched covering.  The basic unit is a decagon with a pentagon attached to every other edge.  You can glue these units together along their edges, and end up with 10 pentagons meeting at a branch point (the black dot):<\/p>\n<div align=\"center\">\n<a href=\"http:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch1.png\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch1.png\" alt=\"Pentagon-Decagon Branched Covering (Stage 1) - Greg Egan\" width=\"250\" height=\"250\" class=\"size-full wp-image-1295\" srcset=\"https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch1.png 400w, https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch1-150x150.png 150w, https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch1-300x300.png 300w, https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch1-50x50.png 50w\" sizes=\"auto, (max-width: 250px) 100vw, 250px\" \/><\/a><\/p>\n<p>\n<a href=\"http:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch2.png\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch2.png\" alt=\"Pentagon-Decagon Branched Covering (Stage 2) - Greg Egan\" width=\"350\" height=\"350\" class=\"size-full wp-image-1296\" srcset=\"https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch2.png 400w, https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch2-150x150.png 150w, https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch2-300x300.png 300w, https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch2-50x50.png 50w\" sizes=\"auto, (max-width: 350px) 100vw, 350px\" \/><\/a> <\/p>\n<p>\n<a href=\"http:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch3.png\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch3.png\" alt=\"Pentagon-Decagon Branched Covering (Stage 3) - Greg Egan\" width=\"400\" height=\"400\" class=\"size-full wp-image-1298\" srcset=\"https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch3.png 400w, https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch3-150x150.png 150w, https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch3-300x300.png 300w, https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch3-50x50.png 50w\" sizes=\"auto, (max-width: 400px) 100vw, 400px\" \/><\/a> <\/p>\n<p>\n<a href=\"http:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch4.png\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch4.png\" alt=\"Pentagon-Decagon Branched Covering (Stage 4) - Greg Egan\" width=\"400\" height=\"400\" class=\"size-full wp-image-1299\" srcset=\"https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch4.png 400w, https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch4-150x150.png 150w, https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch4-300x300.png 300w, https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch4-50x50.png 50w\" sizes=\"auto, (max-width: 400px) 100vw, 400px\" \/><\/a> <\/p>\n<p>\n<a href=\"http:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch5.png\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch5.png\" alt=\"pentagonDecagonBranch5\" width=\"400\" height=\"400\" class=\"alignnone size-full wp-image-1300\" srcset=\"https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch5.png 400w, https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch5-150x150.png 150w, https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch5-300x300.png 300w, https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch5-50x50.png 50w\" sizes=\"auto, (max-width: 400px) 100vw, 400px\" \/><\/a><\/p>\n<p>\n<a href=\"http:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch6.png\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch6.png\" alt=\"Pentagon-Decagon Branched Covering (Stage 6) - Greg Egan\" width=\"400\" height=\"400\" class=\"size-full wp-image-1301\" srcset=\"https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch6.png 400w, https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch6-150x150.png 150w, https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch6-300x300.png 300w, https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch6-50x50.png 50w\" sizes=\"auto, (max-width: 400px) 100vw, 400px\" \/><\/a> <\/p>\n<p>\n<a href=\"http:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch7.png\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch7.png\" alt=\"Pentagon-Decagon Branched Covering (Stage 7) - Greg Egan\" width=\"400\" height=\"400\" class=\"size-full wp-image-1302\" srcset=\"https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch7.png 400w, https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch7-150x150.png 150w, https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch7-300x300.png 300w, https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch7-50x50.png 50w\" sizes=\"auto, (max-width: 400px) 100vw, 400px\" \/><\/a><\/p>\n<p>\n<a href=\"http:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch8.png\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch8.png\" alt=\"Pentagon-Decagon Branched Covering (Stage 8) - Greg Egan\" width=\"400\" height=\"400\" class=\"size-full wp-image-1303\" srcset=\"https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch8.png 400w, https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch8-150x150.png 150w, https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch8-300x300.png 300w, https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch8-50x50.png 50w\" sizes=\"auto, (max-width: 400px) 100vw, 400px\" \/><\/a> <\/p>\n<p><a href=\"http:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch9.png\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch9.png\" alt=\"Pentagon-Decagon Branched Covering (Stage 9) - Greg Egan\" width=\"400\" height=\"400\" class=\"size-full wp-image-1304\" srcset=\"https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch9.png 400w, https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch9-150x150.png 150w, https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch9-300x300.png 300w, https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch9-50x50.png 50w\" sizes=\"auto, (max-width: 400px) 100vw, 400px\" \/><\/a><\/p>\n<p>\n<a href=\"http:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch10.png\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch10.png\" alt=\"Pentagon-Decagon Branched Covering (Stage 10) - Greg Egan\" width=\"400\" height=\"400\" class=\"size-full wp-image-1305\" srcset=\"https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch10.png 400w, https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch10-150x150.png 150w, https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch10-300x300.png 300w, https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch10-50x50.png 50w\" sizes=\"auto, (max-width: 400px) 100vw, 400px\" \/><\/a>\n<\/div>\n<p>Since the interior angle of a regular pentagon on the plane is \\( 3\\pi\/5 \\), the total interior angle at the branch point is<\/p>\n<p>$$ 10 \\times 3\\pi\/5 = 3 \\times 2 \\pi $$<\/p>\n<p>Thus we obtain a branch point of <a href=\"http:\/\/en.wikipedia.org\/wiki\/Branch_point#Algebraic_branch_points\">ramification index<\/a> 3.  In other words, our surface maps to the Euclidean plane in a way that looks locally the same as the <a href=\"http:\/\/en.wikipedia.org\/wiki\/Riemann_surface\">Riemann surface<\/a> for the function \\( z^{1\/3} \\).  <\/p>\n<p>If we continue gluing on more units of type, we obtain a surface tiled by pentagons and decagons.  In this surface, each decagon touches 10 pentagons along its edges.  Each pentagon touches 2 decagons and 3 pentagons along its edges.  There are two kinds of vertices.  At some vertices, 2 pentagons and a decagon meet.  At others, 10 pentagons meet.<\/p>\n<p>This surface can be given a conformal structure, by defining angles between tangent vectors just as usual for points in the interiors of pentagons and decagons, or along their edges, or at vertices where 2 pentagons and a decagon meet&#8212;but rescaling angles at vertices where 10 pentagons meet, dividing by 3 to make the total angle $2 \\pi$.  <\/p>\n<p>This makes our surface into a Riemann surface, that is, a 1-dimensional <a href=\"http:\/\/en.wikipedia.org\/wiki\/Complex_manifold\">complex manifold<\/a>.  In fact, we can show that it is <a href=\"http:\/\/en.wikipedia.org\/wiki\/Biholomorphism\">biholomorphic<\/a> to the hyperbolic plane $\\mathcal{H}$, and its map to the Euclidean plane <\/p>\n<p>$$ p \\colon \\mathcal{H} \\to \\mathbb{C} $$<\/p>\n<p>is holomorphic, with infinitely many branch points, all of ramification index 3.  These branch points are dense in \\(\\mathbb{C}\\).<\/p>\n<p>Even better, there is an infinite discrete group acting as isometries of the Euclidean plane \\(\\mathbb{C}\\) and also of hyperbolic plane \\(\\mathcal{H}\\), with the property that \\(p\\) respects these symmetries:<\/p>\n<p>$$    p(g x) = g p(x)  $$<\/p>\n<p>for every \\(x \\in \\mathcal{H}\\) and group element \\(g\\).  This group is the <a href=\"http:\/\/en.wikipedia.org\/wiki\/Coxeter_group\">Coxeter group<\/a><\/p>\n<div align=\"center\">\n<b>o&#8212;5&#8212;o&#8212;10&#8212;o<\/b>\n<\/div>\n<p><\/br><\/p>\n<p>More or less by definition, this is the symmetry group of a tiling of the hyperbolic plane by equal-sized regular pentagons where 10 meet at each vertex.  <\/p>\n<p>Why are all these things true?  To see why, start by tiling \\(\\mathcal{H}\\) with equal-sized regular pentagons where 10 meet at each vertex.  This tiling, called the {5,10} tiling, is unique up to isometries of \\(\\mathcal{H}\\).  Note that the interior angles of <i>these<\/i> pentagons are not \\(3 \\pi \/ 5\\), but \\(\\pi\/5\\).<\/p>\n<div align=\"center\">\n<div id=\"attachment_1392\" style=\"width: 406px\" class=\"wp-caption alignnone\"><a href=\"http:\/\/blogs.ams.org\/visualinsight\/files\/2015\/02\/510_tiling.png\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1392\" src=\"http:\/\/blogs.ams.org\/visualinsight\/files\/2015\/02\/510_tiling.png\" alt=\"{5,10} Tiling of Hyperbolic Plane - Greg Egan\" width=\"400\" height=\"400\" class=\"size-full wp-image-1392\" \/><\/a><p id=\"caption-attachment-1392\" class=\"wp-caption-text\">{5,10} Tiling of Hyperbolic Plane &#8211; Greg Egan<\/p><\/div>\n<\/div>\n<p>Let \\(p : \\mathcal{H} \\to \\mathbb{C}\\) be a holomorphic map that sends each of these pentagons to a regular pentagon in the complex plane with its usual Euclidean metric.  This map is unique up to isometries of \\(\\mathbb{C}\\), and it has a branch point of order 3 at each pentagon vertex.  Thus, the interior angles of the pentagons in the hyperbolic plane get tripled when they are mapped via \\(p\\) to the Euclidean plane.<\/p>\n<p>Subdivide each pentagon in the Euclidean plane as follows:<\/p>\n<div align=\"center\">\n<div id=\"attachment_1332\" style=\"width: 606px\" class=\"wp-caption alignnone\"><a href=\"http:\/\/blogs.ams.org\/visualinsight\/files\/2015\/02\/PentagonSubstitution1.png\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1332\" src=\"http:\/\/blogs.ams.org\/visualinsight\/files\/2015\/02\/PentagonSubstitution1.png\" alt=\"Pentagons, each Subdivided into a Decagon and 10 Half-Pentagons - Greg Egan\" width=\"600\" height=\"300\" class=\"size-full wp-image-1332\" srcset=\"https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/02\/PentagonSubstitution1.png 600w, https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/02\/PentagonSubstitution1-300x150.png 300w\" sizes=\"auto, (max-width: 600px) 100vw, 600px\" \/><\/a><p id=\"caption-attachment-1332\" class=\"wp-caption-text\">Pentagons, Each Subdivided Into a Decagon and 10 Half-Pentagons &#8211; Greg Egan<\/p><\/div>\n<\/div>\n<p>Lifting this subdivision to a subdivision of the pentagons in \\(\\mathcal{H}\\), we obtain a way of tiling \\(\\mathcal{H}\\) by regions conformally equivalent to decagons and pentagons.  (Beware: the edges of these regions in \\(\\mathcal{H}\\) are not geodesics, unlike the larger pentagons they were created from.  Only their images in \\(\\mathbb{C}\\) have straight lines as edges: these are regular decagons and pentagons in the plane.)<\/p>\n<p>Since we have now constructed the map <\/p>\n<p>$$ p \\colon \\mathcal{H} \\to \\mathbb{C} $$<\/p>\n<p>starting from just the tiling of \\(\\mathcal{H}\\) by equal-sized regular pentagons with 10 meeting at each vertex, without making any choices that break the symmetry, we can now conclude that the Coxeter group <\/p>\n<div align=\"center\">\n<b>o&#8212;5&#8212;o&#8212;10&#8212;o<\/b>\n<\/div>\n<p><\/p>\n<p>acts as isometries of the Euclidean plane \\(\\mathbb{C}\\) and also of hyperbolic plane \\(\\mathcal{H}\\), and that<\/p>\n<p>$$    p(g x) = g p(x).  $$<\/p>\n<hr \/>\n<p><i>Visual Insight<\/i> is a place to share striking images that help explain advanced topics in mathematics. I\u2019m always looking for truly beautiful images, so if you know about one, please drop a comment <a href=\"http:\/\/blogs.ams.org\/visualinsight\/about-visual-insight\/\">here<\/a> and let me know!<\/p>\n<div style=\"margin-top: 0px; margin-bottom: 0px;\" class=\"sharethis-inline-share-buttons\" ><\/div>","protected":false},"excerpt":{"rendered":"<p>Two regular pentagons and a regular decagon fit snugly at a point: their interior angles sum to 360&deg;.  Despite this, you cannot tile the plane with regular pentagons and decagons.   However, there is a <a href=\"http:\/\/en.wikipedia.org\/wiki\/Branched_covering\">branched covering<\/a> of the plane tiled with pentagons and decagons, which map to regular pentagons and decagons on the plane.  Here  <a href=\"http:\/\/www.gregegan.net\">Greg Egan<\/a> has drawn a portion of this branched covering.<\/p>\n<div style=\"margin-top: 0px; margin-bottom: 0px;\" class=\"sharethis-inline-share-buttons\" data-url=https:\/\/blogs.ams.org\/visualinsight\/2015\/02\/15\/pentagon-decagon-branched-covering\/><\/div>\n","protected":false},"author":66,"featured_media":1305,"comment_status":"open","ping_status":"closed","sticky":true,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[11,2,7],"tags":[],"class_list":["post-1306","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-complex-analysis","category-images-library","category-tilings"],"jetpack_featured_media_url":"https:\/\/blogs.ams.org\/visualinsight\/files\/2015\/01\/pentagonDecagonBranch10.png","jetpack_shortlink":"https:\/\/wp.me\/p42Vmc-l4","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/posts\/1306","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/users\/66"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/comments?post=1306"}],"version-history":[{"count":43,"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/posts\/1306\/revisions"}],"predecessor-version":[{"id":1451,"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/posts\/1306\/revisions\/1451"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/media\/1305"}],"wp:attachment":[{"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/media?parent=1306"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/categories?post=1306"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/tags?post=1306"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}