{"id":1565,"date":"2009-08-28T10:32:15","date_gmt":"2009-08-28T18:32:15","guid":{"rendered":"http:\/\/www.rudyrucker.com\/blog\/2009\/08\/28\/pseudospheres\/"},"modified":"2009-08-28T16:30:00","modified_gmt":"2009-08-29T00:30:00","slug":"pseudospheres","status":"publish","type":"post","link":"https:\/\/www.rudyrucker.com\/blog\/2009\/08\/28\/pseudospheres\/","title":{"rendered":"Pseudospheres"},"content":{"rendered":"

I recently acquired a copy of my old friend Clifford Pickover\u2019s new tome, The Math Book <\/a><\/em>, a really attractive and reasonably priced volume with 250 full page color illustrations, each illustration accompanied by a single-page description.<\/p>\n

In Pickover\u2019s words, \u201cMy goal in writing The Math Book <\/em>is to provide a wide audience with a brief guide to important mathematical ideas and thinkers, with entries short enough to digest in a few minutes.\u201d\u009d<\/p>\n


\n[Breather Pseudosphere, Copyright (C) 2006 by Paul Nylander. See a larger image on Paul\u2019s site<\/a>.]<\/em><\/p>\n

One of my favorite images in The Math Book <\/em>is Paul Nylander\u2019s rendition of the so-called Breather Pseudosphere. The idea behind this surface is that it has a constant curvature of -1, as opposed to a sphere, which might have a constant curvature of +1, and also as opposed to a plane, which has a constant curvature of 0. You\u2019re supposed to ignore the ribs, and you need to accept that the surface intersects itself along a circle, which is clearer in the image below, by Xah Lee. You can rotate this image on Xah\u2019s site.<\/p>\n


\n[Segment of a Breather Pseudosphere, Copyright 2006 by Xah Lee<\/a>.]<\/em><\/p>\n

From the arcane math references that I\u2019ve consulted\u2014see for instance the Wikipedia \u201cbreather<\/a>\u201d\u009d page\u2014I gather that the breather pseudosphere can in fact \u201cbreathe\u201d\u009d in the sense that, by diddling a certain parameter, someone (not me anymore) could create a sequence of images of it and then assemble these into a video in which this negatively curved object will pulsate like some omnivorous space squid from Dimension Z. If any of you ultra-math-and-CS maniacs out there has access to such a video\u2014or feels the urge to create one\u2014share the link with us via a comment on this post!<\/p>\n


\n[A traditional Beltrami pseudosphere, Copyright (C) 2006 by Richard Palais and the the 3DXM consortium.]<\/em><\/p>\n

As Pickover\u2019s book explains, the notion of pseudospheres was invented in 1868 by the mathematician Eugenio Beltrami, who formulated the more familiar \u201cdouble trumpet\u201d\u009d model, as shown above, created by the 3DXM Consortium<\/a> . (3DXM is a graphics program, now called 3D-XplorMath.)<\/p>\n

You can find further images of various kinds of pseudospheres (these images by Xah Lee, Luc Bernard and other members of the 3DXM consortium) on the Gallery of Pseudospherical surfaces <\/a>at the Virtual Math Museum. This page includes an essay \u201cAbout Pseudospherical Surfaces,\u201d\u009d which explains (amid much gnarly math) that, at least when depicted in our normal 3D space, any surface of constant curvature -1 will include \u201csingularities\u201d\u009d in the forms of self-intersections or cuspy lines where the surface has a crease in it—like those ribs in the breather pseudosphere or like the edge where Beltrami\u2019s two trumpets meet. But you can smoothly embed pseudospheres into 4D space, I believe.<\/p>\n

<\/p>\n

As I was discussing in an earlier post<\/a>, another way to create a negatively curved space is to start with a disk of some ductile material, and the keep stretching the disk all over, but without overly stretching the outer edge. The inside of the disk acquires more area than one would find in a regular flat disk. The extra room is there because the interior is now a negatively curved surface. And I think this surface is something like a pseudosphere. (A different approach is to stretch the outer edge of the disk to infinite length, and this is a different model of negatively curved space called the Poincar\u00c3\u00a9 plane.)<\/p>\n


\n[Copyright (C) 2009 by Vonda N. McIntyre . White hyperbolic anemone with red veins.] <\/em><\/p>\n

Some knitters and beaders have crafted physical objects like the edge-stretched pseudosphere, some of which appear among a wild Crochet Reef <\/a>show in 2009. Among the goodies on display was a bead construction of a pseudospherical sea anenome by no less a personage than the fantasy and SF writer Vonda N. McIntyre \u2014see the MathCrafts section<\/a> of her home page.<\/p>\n

<\/p>\n

When I was snorkeling near Palau <\/a>a few years ago, I noticed that many naturally occurring soft corals are indeed negatively curved surfaces.<\/p>\n

It\u2019s worth remembering, by the way, that if you\u2019re living in a very small house, it might be nice to have negatively curved space inside…<\/p>\n","protected":false},"excerpt":{"rendered":"

I recently acquired a copy of my old friend Clifford Pickover\u2019s new tome, The Math Book , a really attractive and reasonably priced volume with 250 full page color illustrations, each illustration accompanied by a single-page description. In Pickover\u2019s words, \u201cMy goal in writing The Math Book is to provide a wide audience with a […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-1565","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.rudyrucker.com\/blog\/wp-json\/wp\/v2\/posts\/1565","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.rudyrucker.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.rudyrucker.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.rudyrucker.com\/blog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.rudyrucker.com\/blog\/wp-json\/wp\/v2\/comments?post=1565"}],"version-history":[{"count":6,"href":"https:\/\/www.rudyrucker.com\/blog\/wp-json\/wp\/v2\/posts\/1565\/revisions"}],"predecessor-version":[{"id":1570,"href":"https:\/\/www.rudyrucker.com\/blog\/wp-json\/wp\/v2\/posts\/1565\/revisions\/1570"}],"wp:attachment":[{"href":"https:\/\/www.rudyrucker.com\/blog\/wp-json\/wp\/v2\/media?parent=1565"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.rudyrucker.com\/blog\/wp-json\/wp\/v2\/categories?post=1565"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.rudyrucker.com\/blog\/wp-json\/wp\/v2\/tags?post=1565"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}