Category Archives: mathematics

31 frets

Here is the neck of a guitar that I’d like to have made someday, if I should ever develop the dexterity to make it worthwhile. The blue stripes show where standard frets would be, for comparison.

The tuning is my tweaked version of meantone: compared to just intonation, each factor of 2 is sharp by 1/16 comma, each factor of 3 is flat by 1/8 comma, and each factor of 5 is sharp by 1/4 comma. (A comma is the difference between 64:81, the Pythagorean major third derived from compounding fifths, and the more harmonious 4:5.) This makes the thirds and sixths much truer than in equal temperament, and the fifths slightly truer than in traditional meantone, which puts all the error in the 3s.

This design has 31 frets in the first octave: 12 flats, 7 naturals, 12 sharps. The bent frets span the difference (~151:152) between 18 of my sharp octaves and 31 of my flat fifths.

To reduce crowding, the second octave has only three flats and three sharps. The bent frets span the difference (~50:51) between G♯ and A♭.

The charts below should look familiar to players, if you squint a little.

phase-packing

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.)

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.

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.

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 a 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 which fullerenes change with which, up to C50 (15 hexagons).

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 C50 include two with no nontrivial symmetry at all.