{"id":3726,"date":"2017-02-08T20:20:03","date_gmt":"2017-02-09T04:20:03","guid":{"rendered":"http:\/\/bendwavy.org\/wp\/?p=3726"},"modified":"2019-08-14T19:58:38","modified_gmt":"2019-08-15T03:58:38","slug":"klein-bagel-mark-n","status":"publish","type":"post","link":"https:\/\/bendwavy.org\/wp\/?p=3726","title":{"rendered":"Klein bagel, mark N"},"content":{"rendered":"


\nI’ve had other designs made in steel but not this one. (The sintering leaves the steel highly porous, so liquid bronze is brought in by capillary action to fill it; the result is about three parts steel to two parts bronze – if I understand right. Hence the color.) (Later:<\/strong> I was mistaken: the steel powder is not sintered but glued; I guess the bronze burns away the glue.)<\/p>\n

While it was on its way to me, I thought of some improvements. <\/p>\n

One subtle practical change<\/strong><\/p>\n

The model is defined by its surface, which consists of thousands of triangles. All of my designs so far are made of bent rods; I place the vertices of the triangles along a helix. Previously I’ve used an odd number of vertices in two turns.
\n
\nFor metal, Shapeways wants a flat area 1mm square (I guess to attach sprues for the bronze), and also they want the rod 1.5mm thick rather than 1.0mm as for plastic; so I did this:
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\nThe flat top and bottom are more appropriate anyway, since the model represents a surface<\/em>. But oops, the parallelograms are split on their long diagonal; this means that on the outside of each curve the inevitable concavities are longer and deeper than necessary. I’ve corrected that for my next designs.<\/p>\n

One radical change for my sense of mathematical purity<\/strong><\/p>\n

I prefer to use conformal<\/em> mappings<\/a> wherever possible: these are geometric transformations that preserve local angles. When the figure-eight is swept around the circle, the inner part moves less than the outer, and I wanted the inner lobe to shrink in the same proportion. The obvious solution was the exponential function in the complex plane. (Every function that has a well-defined derivative is conformal, except where that derivative is zero.)
\n
\nObvious and wrong. The mapping within this plane<\/em> is conformal, but I want scaling to be proportional to distance from the axis of rotation, not from the center point. I began to suspect there’s no way to do it right.<\/p>\n

But then I think of Clifford torus<\/a> and I get idea, haha!<\/em><\/p>\n

A Clifford torus is a surface in four dimensions whose points’ coordinates (w,x,y,z) fit two equations: w² + x² = a² and y² + z² = b², for some constants a,b. (Some restrict the definition to the special case a = b = 1\/√2.) You can add the two equations, w² + x² + y² + z² = a² + b²; this implies that a Clifford torus lies on the three-dimensional hypersurface of a hypersphere, a space known as S³.<\/p>\n

A torus in Euclidean 3-space (E³) has positive (spherelike) curvature on its outside and negative (saddle-like) curvature on its inside. A Clifford torus, in contrast, has the same curvature everywhere. To an inhabitant of S³, it looks like an endless cylinder whose axis is a great circle<\/a> (the straightest possible arc in that space).<\/p>\n

A similar uniformity applies to any figure in S³ swept along a great circle orthogonal to its own plane. I can then bring it into E³ by stereographic projection<\/a>, which is conformal in any number of dimensions.<\/p>\n

I doubt that I’ll be able to see a difference, but this correction makes me happy.<\/p>\n

One change for popular understanding<\/strong><\/p>\n

It was important to me at first that the crossing be at right angles (because the concept was provoked by designs like this<\/a>, consisting of two Möbius strips in the form of ladders penetrating each other). But I find that people typically see my Klein bagel as a round tube making a double coil. So in my next print the crossing will be at 60°, and maybe then it will be clearer. \u2014 Later:<\/strong> It isn’t; and I don’t like that it feels less round to my fingers. Instead, I’ll make the grid more open, so that it passes more visibly through the crossing. \u2014 That did help!<\/p>\n","protected":false},"excerpt":{"rendered":"

I’ve had other designs made in steel but not this one. (The sintering leaves the steel highly porous, so liquid bronze is brought in by capillary action to fill it; the result is about three parts steel to two parts … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[70],"tags":[],"class_list":["post-3726","post","type-post","status-publish","format-standard","hentry","category-merch"],"_links":{"self":[{"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=\/wp\/v2\/posts\/3726","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=3726"}],"version-history":[{"count":25,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=\/wp\/v2\/posts\/3726\/revisions"}],"predecessor-version":[{"id":4033,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=\/wp\/v2\/posts\/3726\/revisions\/4033"}],"wp:attachment":[{"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3726"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3726"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3726"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}