Thursday, 2021 October 21, 18:55 — mathematics


I just thought of a new kind of packing problem, a mutant extension of the Thomson problem. In this version, each particle has coordinates in two independent spaces; in each it is confined to a sphere (of some dimension). In each space, pairs of particles repel each other with a force inversely proportional to the square of their distance in the other space.

(I was imagining a head with striped hairs, as one does, and considered making each hair’s phase anti-correlated with those of its neighbors, and what that would mean.)

Friday, 2021 October 8, 11:07 — mathematics

O fairest of randomizers

On most numbered dice, opposite sides are complementary; on a cube, for example, they add to 7. As a result, if you have the skill to throw a die so that the {1,2,3} corner lands on the table, the upward face must be at least 4.

I would prefer to design dice so that, if numbers are considered as masses, the center of mass coincides with the geometric center. I think this is equivalent to saying: the sum of the numbers in any hemisphere must be equal.

You can’t do that with a tetrahedron, cube, D10 or regular dodecahedron; but I found three solutions for the octahedron, and 876 for the rhombic dodecahedron. (At least I see no obvious way to reduce that number further with symmetries.)

For a cube the best you can do is put pairs of adjacent numbers on opposite faces.
The twelve best arrangements on the regular dodecahedron all have 0 and 11 opposite each other, 1–5 around 11, and 6–10 around 0.
The two best arrangements for D10 are 0285364179 and 0582367149 (reading around the fivefold axis, alternating upper and lower faces).
(Add 1 to each number if you don’t like zero-based indexing; it doesn’t affect the math.)

I’ll update here if I come up with an approach to the icosahedron problem that won’t take thousands of years.

Thursday, 2021 September 16, 19:58 — sciences

shining eyes

Could an animal have eyes like a reflecting telescope, rather than with a lens? The back of the eyeball is a paraboloid mirror, and the retina is a small body on its focal plane.

Because the retina must be small, such an eye would have poorer resolution than a vertebrate eye of similar size.

Are there any organic mirrors in the real world? How smooth is the reflective layer behind a cat’s retina?

Perhaps I’ll inflict this idea on – in the form of a question, though I dislike Jeopardy for that gimmick.

Thursday, 2021 August 26, 19:02 — astronomy, mathematics

witness on Whidbey

I watched Behind the Curve (2018), a documentary about the Flat Earth movement. In the beginning, Mark Sargent says (I paraphrase), “I know the Earth is not round because I can see Seattle from here [Whidbey Island].”

If I knew the distance from the Space Needle to Sargent’s house, the altitude of that house and the altitude of the lowest part of the Space Needle visible from there, I could put an upper bound on the curvature.

Tuesday, 2021 March 9, 23:03 — futures, medicine

dot product of Cupid’s arrows

The backstory of Methuselah’s Children, by Heinlein, involves a foundation to promote human longevity. One thing it does is study natural long-lifers by paying a bounty for marriages between people whose grandparents all lived 100 years or more.

Now here’s a stack of wacky ideas of mine. ( . . more . . )

Thursday, 2020 April 23, 06:43 — medicine, politics

hope you don’t mind if I sit this one out

Looks like I’m staying home alone until a vaccine comes; it’s what I mostly do anyway, though I miss the weekly card games. As a libertarian, I do not presume to know what’s best for others. So, lucky me, I need not obsess about policy.

Monday, 2020 March 30, 10:35 — curve-fitting

Scribbles: The Ensmoothening, Part III

Many of the curves in this chart have some unsightly wiggles. That’s because, when a function of degree 2 or higher tries to approximate a piecewise constant, it tends to go back and forth across the target. So here instead I fitted each such function not to the piecewise constant directly but to the fit of the next lower degree.
( . . more . . )

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