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 when the icosahedron program finishes running.

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 worldbuilding.stackexchange.com – in the form of a question, though I dislike *Jeopardy* for that gimmick.

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 flat 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.

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 . . )

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.

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 . . )

it’s in the literature

On a truncated icosahedron / buckyball / Telstar-style soccer ball, consider two adjacent hexagons and the two pentagons that are adjacent to both. These four faces can be removed, rotated by a right angle, and reattached, causing only a small change to the overall shape. Most fullerenes have at least one such patch.

If I ever get around to making more printable models of fullerenes, I would omit those that can be changed, by the above twist, into one of higher symmetry. I have a pretty good idea of how I’d go about listing the fullerenes and finding their siblings; but I do not have a grip on distinguishing symmetry groups of the same order – e.g., that of the regular tetrahedron versus that of a hexagonal prism – and a subgroup of one may not be a subgroup of the other.

So I got out *An Atlas of Fullerenes* in the hope of understanding how they did it – and happened to open to a chapter I had not looked at before, which covers the Stone-Wales transformation (for so it is named) and lists, up to C_{50} (15 hexagons), which fullerenes change with which.

The 812 smallest fullerenes are thus cut to 72 in 47 families. The biggest of these families has six remaining members, four with C2v symmetry (one axis of twofold rotation, and a reflection plane containing that axis) and two with C3 symmetry (chiral with one threefold axis). Their symmetry numbers are 4 and 3 respectively, but as C3 is not a subgroup of C2v I keep them all.

Surprisingly the ten families of C_{50} include two with no nontrivial symmetry at all.