{"id":205,"date":"2006-11-07T08:04:32","date_gmt":"2006-11-07T08:04:32","guid":{"rendered":"http:\/\/scientopia.org\/blogs\/goodmath\/2006\/11\/07\/building-interesting-shapes-by-gluing\/"},"modified":"2006-11-07T08:04:32","modified_gmt":"2006-11-07T08:04:32","slug":"building-interesting-shapes-by-gluing","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2006\/11\/07\/building-interesting-shapes-by-gluing\/","title":{"rendered":"Building Interesting Shapes by Gluing"},"content":{"rendered":"

I thought it would be fun to do a couple of strange shapes to show you the interesting things that you can do with a a bit of glue in topology. There are a couple of standard *strange* manifolds, and I’m going to walk through some simple gluing constructions of them.<\/p>\n


\nLet’s start by building a Torus. It’s not strange, but it’s useful as an example of interesting gluing. We can make a torus out of simple rectangular manifolds quite easily. We start by building a cylinder by doing the circle construction using rectangles instead of just line segments. So we take four squares; curve each one into a half cylinder, and then overlap them, like in the following picture. The picture shows the
\nfour squares; each of the edges are colored, and each has a dotted line down the middle, which is
\nwhere the two matching-color edges of the neighboring rectangles will meet. That gives us a cylinder.
\n\"cylinder-glue.jpg\"
\nNow, take four cylinders, and using the cylinders instead of line-segments, repeat the same process we used for the circle. So each cylinder is curved into a half-torus, and they’re overlapped. And we have a torus, by gluing. (Sorry, but I can’t figure out a good way to draw toruses with color bands to show the overlaps using my software!)
\n\"torus-glue.jpg\"
\nOk.. That was an easy one.
\nTo make some of the harder ones a bit easier, here’s a simple trick; if a manifold has edges, those edges can be glued together in the same way as edges of *different* manifolds. This is subject to the exact same kinds of constraints as gluing together multiple manifolds – you need to have reversable
\ncontinuous functions for the overlaps.
\nUsing that, we can create a Mobius strip very easily. Take a rectangle, like the following image:
\n\"mobius.jpg\"
\nTake the colored edges, and glue them together so that the colors match. To do that, you’ll need to put a twist into it. The result is a mobius strip: a manifold with only one edge.
\nNow, take two mobius strips: one with a right-handed twist, and one with a left-handed twist. Put the two side by side, and glue their edges together. What you’ll wind up with is a fascinating shape called *Klein bottle*: a manifold with no boundaries. A true Klein bottle can’t really be represented in three dimensions, because any three dimensional embedding will require the manifold to *cross through* itself, which it doesn’t do. But an approximate image of a Klein bottle (from Wikipedia) look like:
\n
\nAnother way of making a Klein bottle is to take a square, and glue it so that the edges all match in the following diagram.
\n\"klein-glue.jpg\"
\nWhat’s cool in an extremely geeky way is that there are people who make *almost* Klein bottles out of glass or paper. They’re fascinating things – a bottle with *no inside* and *no outside*. There are even [Klein bottle *beer mugs*!](http:\/\/www.kleinbottle.com\/drinking_mug_klein_bottle.htm). (If any readers ever wanted to make me very happy, just send me one of these. I’ve never been able to convince myself to spend that much on a mug.)<\/p>\n","protected":false},"excerpt":{"rendered":"

I thought it would be fun to do a couple of strange shapes to show you the interesting things that you can do with a a bit of glue in topology. There are a couple of standard *strange* manifolds, and I’m going to walk through some simple gluing constructions of them.<\/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-205","post","type-post","status-publish","format-standard","hentry","category-topology"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-3j","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/205","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=205"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/205\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=205"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=205"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=205"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}