{"id":622,"date":"2008-04-01T11:56:06","date_gmt":"2008-04-01T11:56:06","guid":{"rendered":"http:\/\/scientopia.org\/blogs\/goodmath\/2008\/04\/01\/understanding-non-euclidean-hyperbolic-spaces-with-yarn\/"},"modified":"2008-04-01T11:56:06","modified_gmt":"2008-04-01T11:56:06","slug":"understanding-non-euclidean-hyperbolic-spaces-with-yarn","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2008\/04\/01\/understanding-non-euclidean-hyperbolic-spaces-with-yarn\/","title":{"rendered":"Understanding Non-Euclidean Hyperbolic Spaces – With Yarn!"},"content":{"rendered":"

\"crochet_02.jpg\"<\/p>\n

One of my fellow ScienceBloggers, Andrew Bleiman from Zooilogix<\/a>, sent me an amusing link. If you’ve done things like study topology, then you’ll know about non-euclidean spaces. Non-euclidean spaces are often very strange, and with the exception of a few simple cases (like the surface of a sphere), getting a handle on just what a non-euclidean space looks like can be extremely difficult.<\/p>\n

One of the simple to define but hard to understand examples is called a hyperbolic space. The simplest definition of a hyperbolic space is a space
\nwhere if you take open spheres of increasing radius around a point, the amount of space in those open spheres increases exponentially.<\/p>\n

If you think of a sheet of paper, if you take a point, and you draw progressively larger circles around the point, the size of the circles increases
\nwith the square of the radius: for a circle with radius R, the amount of space inside the circle is proportional to R2<\/sup>. If you did it in three dimensions, the amount of space in the spheres would be proportional to R3<\/sup>. But it’s always a fixed exponent.<\/p>\n

In a hyperbolic space, you’ve got a constant N, which defines the “dimensionality” of the space – and the open spheres around it enclose a
\nquantity of space proportional to NR<\/sup>. The larger the open circle around
\na point, the higher<\/em> the exponent.<\/p>\n

What Andrew sent me is a link about how you can create models of hyperbolic
\nspaces using simple crochet.<\/a> And then you can get a sense of just how a hyperbolic space works by playing with the thing you crocheted!<\/p>\n

It’s absolutely brilliant. Once you see it, it’s totally obvious
\nthat this is a great model of a hyperbolic space, and just about anyone
\ncan make it, and then experiment with it to get an actual tactile<\/em> sense
\nof how it works!<\/p>\n

It just happens that right near where I live, there’s a great yarn shop whose owners my wife and I have become friends with. So if you’re interested in trying this out, you should go to their shop, Flying Fingers<\/a>, and buy yourself some yarn and crochet hooks, and crochet yourself some hyperbolic surfaces! And tell Elise and Kevin that I sent you!<\/p>\n","protected":false},"excerpt":{"rendered":"

One of my fellow ScienceBloggers, Andrew Bleiman from Zooilogix, sent me an amusing link. If you’ve done things like study topology, then you’ll know about non-euclidean spaces. Non-euclidean spaces are often very strange, and with the exception of a few simple cases (like the surface of a sphere), getting a handle on just what a […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[65],"tags":[],"class_list":["post-622","post","type-post","status-publish","format-standard","hentry","category-topology"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-a2","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/622","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=622"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/622\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=622"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=622"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=622"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}