Mathematician. Reader. Fiftysomething. Cruciverbalist. Not very good at thinking of witty things to put in short bios.

I wasn't on Twitter before that nice Mr Musk suggested that everyone go to Mastodon, and I don't expect to be very active here, but who knows?

Last Notes

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My reading of that thread is that this isn't a _nicer counterexample_ but a _nicer description of the same counterexample_. Have I misunderstood?
(It is very likely that I have. I am no algebraic geometer.)

2026-07-18 22:13:55 CEST

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Superficial observation that might help somehow. (I suspect from Oscar's remark that he has made the same observation.)
Let A,B,C be the centres of the three half-size S. triangles making up the big S. triangle. For each 𝑛≥0, let 𝑋ₙ be the random variable (random one of A,B,C) - (random one of A,B,C), all 𝑋ₙ independent. Then the difference between any two points in the S. triangle is ∑2⁻ⁿ𝑋ₙ.
(So the expectation of the distance _squared_ is twice the expectation of |𝑋₀|² or twice (1/3)² so the RMS distance is √2/3, as per Oscar's remark.)
It feels like someone who knows more than I do about random vectors in euclidean space might find it pretty easy to use this to find the answer, or at least to find an efficient way to approximate it. Unfortunately, I know only exactly as much as I do, which isn't very much.

2026-07-18 19:42:06 CEST

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And, looking more carefully, my extrapolations do seem like they're closer to one another and to the value claimed on that page than they are to 41√3/168.
(Logs of differences: last Aitken / last Wynn -16.4, last Aitken / second-last Wynn -16.3, last two Wynn -19.9; last Aitken / claim -16.4, last Wynn / claim -24.0, second-last Wynn / claim -19.9.)
So I retract what I said before: it seems _very unlikely_ that 41√3/168 is the true value, and the available evidence suggests that the programming contest problem's claimed 0.4227021810348385578571 could well be correct.

2026-07-18 19:22:47 CEST

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Some numerology:
Write 𝐷ₙ for the average distance between points in the 𝑛th-order approximation to the S. triangle, where the 1st order is just the vertices and the (𝑛+1)th order consists of three scaled copies of the 𝑛th order. (Note in particular that some points appear multiple times. I think this is OK.)
Then, aside from mere accumulated floating-point calculation errors, the first ten values of 𝐷ₙ are:
0.6666666666666667
0.4986704301902867
0.44520689777765476
0.4289567387204627
0.4243722628662094
0.42313749521839766
0.4228139755506694
0.4227306238555088
0.42270937415127596
Writing 𝑔 for the conjectured 41√3/168, the logs of the differences 𝐷ₙ−𝑔 are: -1.41, -2.58, -3.79, -5.07, -6.40, -7.74, -9.11, -10.49, -11.94 which seem to be decreasing at a pretty consistent rate.
It seems plausible that the 𝐷ₙ might converge in a clean enough way to try standard convergence-acceleration techniques. Aitken Δ² extrapolation gives things for which the logs of the differences are: -6.01, -7.08, -8.93, -10.79, -12.49, -13.65, -14.07 which are more-negative than the originals and seem to be decreasing nicely. Wynn ε extrapolation, which is a bit like an iterated version of Aitken, produces broadly similar results.
All of this seems _very consistent_ with your proposed answer being correct. None of it, of course, comes within a light-year of proving anything.

2026-04-28 00:23:18 CEST

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This may well be an important and very correct and necessary book, but I really really dislike the title.
By and large, the actual nerds of Silicon Valley -- the engineers and programmers and whatnot -- aren't fascist or authoritarian at all. (Even, I _think_, the ones who work for the most blatantly evil tech companies: even a lot of employees at bloody _Palantir_ turn out to have been worried by its embracing of fascism.)
The people who are disproportionately fascist (or fascist-adjacent, or out past fascism and on the other side, or otherwise anti-democratic and generally awful) are not the nerds but the venture capitalists and big-tech-company CEOs. Some of them _started out_ as nerdy engineering types, but that's not what they are any more. (And most of them seem to have been idealistic liberals back when they were nerds. They became fascists when they _stopped_ being nerds and started being Masters of the Universe.)
It's a conspiracy of billionaires and their hangers-on, not of nerds. Nerds get enough hate and contempt without encouraging people to blur the lines between them and the bastards who own the companies some of them work for.
For the avoidance of doubt: yes, some individual nerds (e.g., Curtis Yarvin) are appalling authoritarians; yes, lots of nerds are working for companies that do evil and you can make a good case that that makes them complicit. But the people who deserve that word "Reich" attached to them are not generally the nerds.
(Declaration of interest: I am myself by any reasonable definition a nerd, I have never worked in Silicon Valley, I despise authoritarianism, and part of what bothers me is seeing People Like Me unfairly associated with fascism.)

2025-10-09 14:49:30 CEST

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Note that the gravitational field of _any_ spherically-symmetrical thing is the same (up to a scaling of the total mass) _everywhere_ outside the sphere containing it. So it doesn't particularly matter whether the sample points are all on a single sphere.
Of course, that's what's true for an _actually literally spherically symmetrical_ mass density, and what you've got here is just an approximation to that. I don't think I'd have predicted ahead of time that the determinant would be so very small ... though maybe what's going on isn't that there are _incredibly small eigenvectors_ but that there are _quite a lot of fairly small eigenvectors_.
Further conjecture: if you move your sample points further out, the determinant will get smaller; if you move them further in, it will get larger. (On the grounds that I expect the spherically-symmetrical approximation to get better at larger distances.)

2025-10-09 14:22:50 CEST

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Have you looked to see what the almost-nullspace looks like? That is, what not-very-short vectors are mapped almost-to-zero? Or, to put it differently: your empirical finding shows that some distributions of mass have almost the same effect on your 83 red dots as others; what are they?
My best guess, which I don't trust much: your 83 red dots are all at a single distance, so any distribution of mass that looks "enough" like a spherical shell of any size smaller than that will look (up to scaling the masses) exactly like any other, and perhaps your nearly-nullspace vectors look like linear combinations of spherical shells.

2025-09-02 13:48:27 CEST

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It's too bad it can't be A000001. Though it wouldn't by any means be the first known-to-be-finite OEIS sequence; for instance A100000 is another.

2025-08-02 12:30:57 CEST

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Pedantic note: I think you mean not "The Protecting Veil", a famous Tavener piece but only ~45 minutes long, but "The Veil of the Temple", a newer piece by the same composer that is, indeed, 8 hours long.

2025-07-22 02:04:07 CEST

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This is a slightly atypical case because usually when something's named after X it's because someone else discovered the thing first and then X wrote about it later[1], whereas in this case IIUC Moessner really did get there first but didn't _prove_ it.
[1] Though my actual favourite example is Pell's equation, which so far as I can tell Pell never worked on _at all_.