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Monday, 2021 January 11, 23:17 — me!me!me!

of our elaborate plans, the potential end

For three days I’ve had a fever – CV-negative, but likely my highest fever ever! – which provokes me to start this post that I had meant to do someday, listing some projects that I may never finish (or even begin).

Font-fitting. The idea is to design outline fonts to match certain old favorite bitmap fonts, dot for dot, using curves as graceful as possible. Two-thirds of my posts in the curve-fitting category relate to that project; I won’t recap here.

I have thrown up my hands in bafflement because the fittings often fail to converge. It would probably help if I could put progress on the screen, rather than in image files to examine afterward, but tutorials in that direction have bounced off my lazy old brain. (I’m willing to try more.)

Pseudo-fullerenes. I made some fullerene models using tables of coordinates found on the Web somewhere. (I think that material is gone now.) I found them unattractive, because the polygonal faces are often so far from planar that they’re hard to see as faces. So I’d like to do in software what I used to do with plastic tabbed polygons, back when I was first studying fullerene topologies on my own. (The company that distributed the kit is defunct.) That is, place some rigid polygons in space and have corresponding edges attract. To get started on this I need to know how to make a rotation matrix, given an axis (as a vector) and an angle, for which I have not found a comprehensible cookbook algo.

Tesseract sponge. Of the 28 convex uniform tilings of Euclidean 3-space, 11 can be generated by reflection in an irregular tetrahedron of mirrors. If you place a plane so that it cuts four of the tetrahedron’s six edges (there are three ways to do that), and apply Surface Evolver, sometimes the surface converges to an edge, but otherwise you get a triply periodic minimal surface.

How about in non-Euclidean space? The uniform 4-polytopes can be thought of as tilings of a hypersphere, and most of them can be generated in the same way. Sadly, in these tetrahedra Surface Evolver converges to an edge in every case. But I should get something pretty if the surface is constrained to cut the tetrahedron’s volume in half. It would help to find someone who has worked with Surface Evolver in curved space with a volume constraint!

I’d start with the symmetry system of the tesseract, which happens to be the only nonprismatic system in S³ with two different kinds of mirrors (four on the coordinate planes and twelve diagonal); in the others, each mirror takes on all four roles in different instances of the fundamental simplex. So my model, in stereographic projection, would look like a lentil, bounded by one mirror of each set.

3-tiling catalog. In 2006 I made pictures of the Euclidean tilings mentioned above, for Wikipedia. I would love to do the same for each of the Thurston geometries; there are I think 64 (plus two infinite families) in S³, about 230 known in H³, at least 18 (plus two infinite families) in S²×R, and an infinite number in H²×R (I have in mind a selection of 36). I have no idea whether the three more exotic geometries admit any uniform tilings at all. Need a non-Euclidean ray-tracer!

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