![]() And I think this surface is something like a pseudosphere. The extra room is there because the interior is now a negatively curved surface. The inside of the disk acquires more area than one would find in a regular flat disk. But you can smoothly embed pseudospheres into 4D space, I believe.Īs I was discussing in an earlier post, 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. This page includes an essay “About Pseudospherical Surfaces,” which explains (amid much gnarly math) that, at least when depicted in our normal 3D space, any surface of constant curvature -1 will include “singularities” 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’s two trumpets meet. 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 at the Virtual Math Museum. (3DXM is a graphics program, now called 3D-XplorMath.) If any of you ultra-math-and-CS maniacs out there has access to such a video-or feels the urge to create one-share the link with us via a comment on this post!Īs Pickover’s book explains, the notion of pseudospheres was invented in 1868 by the mathematician Eugenio Beltrami, who formulated the more familiar “double trumpet” model, as shown above, created by the 3DXM Consortium. You can rotate this image on Xah’s site.įrom the arcane math references that I’ve consulted-see for instance the Wikipedia “ breather” page-I gather that the breather pseudosphere can in fact “breathe” 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. You’re 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. 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. One of my favorite images in The Math Book is Paul Nylander’s rendition of the so-called Breather Pseudosphere. In Pickover’s words, “My goal in writing The Math Book 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.” I recently acquired a copy of my old friend Clifford Pickover’s 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.
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