Posts by gregeganSF@mathstodon.xyz
(DIR) Post #AxSXNtIsDzFxeTzwrQ by gregeganSF@mathstodon.xyz
2025-08-23T11:02:36Z
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@futurebird The hypercubes are all definitely convex. But the line that contains the two points (extended indefinitely beyond them, not the line segment that lies only between them) might intersect two sides of the hypercube that are not an opposite pair, just as the line through two points inside a square might intersect either two opposite edges, or two adjacent edges.
(DIR) Post #AxT6Qa2tmG1JBpXmTo by gregeganSF@mathstodon.xyz
2025-08-23T11:27:39Z
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Here’s some Gemini gaslighting:• giving a wrong answer to a puzzle• giving Python code that could test its claim• asserting it obtained results from the code supporting its claim• but when I ran the code myself, it showed the claim was false.https://g.co/gemini/share/2188cebf58cc
(DIR) Post #AxvfxHUIjXoo0kZF68 by gregeganSF@mathstodon.xyz
2025-09-05T23:11:16Z
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In a dangerous escalation in nominative determinism, several nations have acted to rename the bureaucracies governing their militaries “the Department of Victory”, “the Department of Unassailable Triumphs”, and “the Department of World Conquest Preordained by Manifest Destiny”.
(DIR) Post #Ayiz6my9QNCFm9IHGS by gregeganSF@mathstodon.xyz
2025-09-30T05:08:36Z
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2025: Hoorah, this *looks* perfect! The chatbots will do all the calculus for us!2030: Wait ... why did that bridge fall down?
(DIR) Post #AytcJcosJLTpr3fM4e by gregeganSF@mathstodon.xyz
2025-10-05T10:19:19Z
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In Newtonian gravity, if you drop an object down a borehole in a ball of uniform density it will oscillate at the same rate as an orbiting body grazing the surface.In General Relativity that perfect match is lost … unless you drop the object from higher up and raise the orbit.
(DIR) Post #AytcJlMqWvPKMjcb8S by gregeganSF@mathstodon.xyz
2025-10-05T10:22:48Z
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How does the proper time measured by clocks on the orbiting body and the radially oscillating body compare?The radially oscillating clock always measures slightly less proper time than the orbiting clock.
(DIR) Post #AytcJlMqWvPKMjcb8T by gregeganSF@mathstodon.xyz
2025-10-05T10:21:32Z
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There is an orbital radius that matches the period of the orbit to the period of the “vacuum+borehole” oscillation for a uniform ball of matter of any allowed radius (but R_ball/M must be greater than 9/4 for the pressure at the centre of the ball to be finite). The corresponding R_orbit/M is never less than 6, so that is always a physically possible, stable orbit.
(DIR) Post #AyttBwGro05H5QgCYa by gregeganSF@mathstodon.xyz
2025-10-05T13:35:59Z
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“Flow” was eerily beautiful, charming, sad, strange and hopeful.I must admit a part of my brain uncanny-valleyed a bit at the disparity between the lush, near-photorealistic landscapes and architecture versus the blocky palette in which the animals were rendered, but in the end I managed to shut that out and go with the story.
(DIR) Post #AyzAtzjmwrCjafbHyC by gregeganSF@mathstodon.xyz
2025-10-07T04:25:47Z
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An astonishing result, beautifully described in @QuantaMagazine — the mathematics linking scattering amplitudes in particle physics to a geometric object called the amplituhedron can be better understood by also mapping the amplituhedron to the mathematics of origami.https://www.quantamagazine.org/origami-patterns-solve-a-major-physics-riddle-20251006/
(DIR) Post #AzGRAj7OtPQ3vKxosS by gregeganSF@mathstodon.xyz
2025-10-15T17:58:57Z
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Gordon had been lost in the fog of Alzheimer’s, but then a new drug halts the progress of the disease, and a kind of neural pacemaker restores his ability to access memories and skills that seemed to have slipped away forever. But no new cure is perfect.“Spare Parts for the Mind”My new story in the Nov/Dec issue of Asimov’s SF.Read an excerpt here:https://asimovs.com/current-issue/story-excerpt2/
(DIR) Post #AzbIZoQNU05ELssyKe by gregeganSF@mathstodon.xyz
2025-10-26T10:37:48Z
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Oh, just a caterpillar that keeps all the husks it shed from its head when it was smaller as a kind of elaborate hat … because why wouldn’t you?Congratulations to Georgina Steytler, who just won a wildlife photography award for this extraordinary image!https://www.abc.net.au/news/2025-10-26/five-headed-caterpillar-wins-photo-award/105926608
(DIR) Post #AzvBsdQrrI44NKUn7Q by gregeganSF@mathstodon.xyz
2025-08-22T13:12:02Z
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Here’s a fun puzzle I heard from @octonion Pick two points uniformly at random in a square. What is the probability that the line that contains both points intersects two opposite edges of the square, rather than two adjacent edges?
(DIR) Post #AzvBseSK3Mp3Y8BT1c by gregeganSF@mathstodon.xyz
2025-08-22T13:13:09Z
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SOLUTION:Take the unit square [0,1] × [0,1], and consider the two horizontal edges.Parameterise the endpoints of a line joining those two edges as (α,0) and (β,1), then parameterise two points along that line as:(1-λ) (α,0) + λ (β,1)(1-μ) (α,0) + μ (β,1)where α, β, λ, μ all range from 0 to 1.So we have a map from the parameter space [0,1]^4 to the geometric space of pairs of points [0,1]^4:F(α,β,λ,μ) = (β λ + α (1-λ), λ, β μ + α (1-μ), μ)The absolute value of the Jacobian determinant of F is just:|J(F)| = |μ-y|The integral of this over [0,1]^4 in the parameter space gives the 4-volume in the geometric space:∫_[0,1]^4 |μ-y| = 1/3We multiply by two to account for the other case, where the pair of opposite edges are the two vertical edges.So P(opposite) = 2/3.I think this is a simpler approach than trying to specify the required subset of the geometric space in terms of a collection of inequalities in the Cartesian coordinates. My original intuition was that this subset ought to consist of a union of convex polytopes in R^4 ... but it doesn't, the boundaries are not hyperplanes!
(DIR) Post #AzvBslZJblZjbqVeOe by gregeganSF@mathstodon.xyz
2025-08-23T05:25:31Z
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Here is a more geometrically intuitive solution.We assume the square has both x and y coordinates ranging from 0 to 1.Imagine sweeping a line across the square (red in the animation) so that it always joins the top and bottom edges, and one endpoint or the other is always in one of the corners of the square.If we call the two points (x₁,y₁) and (x₂,y₂), suppose we hold y₁ and y₂ fixed, and look at the values of x₁ and x₂ so that both points lie on the red line.In the (x₁, x₂) plane, their values sweep out a parallelogram with vertices at (0,0), (y₁,y₂), (1,1) and (1-y₁,1-y₂). The area of this parallelogram is:|y₁-y₂|So for any fixed y₁ and y₂, this is the probability that the line between the two points with uniformly random x₁ and x₂ joins the top and bottom edges of the square.What is the average of |y₁-y₂| across all values of y₁ and y₂? This is just the volume of two pyramids with equal-sized triangular bases, one with y₁ ≤ y₂ and one with y₁ ≥ y₂. The height of both pyramids is 1, since that is the maximum that |y₁-y₂| reaches in both triangles. So the volume of each pyramid is 1/6, for a total of 1/3.We multiply by two to account for the other case, where the pair of opposite edges are the left and right edges.So P(opposite) = 2/3.
(DIR) Post #B0R70jDuiH4abUvgSO by gregeganSF@mathstodon.xyz
2025-11-20T12:08:40Z
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@futurebird I lived without a washing machine for about 20 years, soaking/handwashing everything. That started when I was living in tiny rented flats with no laundry, and I was too lazy and/or poor to go to a laundromat, but once I was used to it, it was just part of the daily routine.
(DIR) Post #B0SulDt67scHwLx8MK by gregeganSF@mathstodon.xyz
2025-11-21T03:59:04Z
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A huge, beautiful library of tessellated materials in biology!https://tessellated-materials.mpikg.mpg.de/collectionAnd an interview in Science with the creators of this library:https://www.science.org/content/article/why-does-biology-keep-building-things-out-tiles
(DIR) Post #B0SulFA9NxqptWwGZs by gregeganSF@mathstodon.xyz
2025-11-21T09:00:22Z
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I learned about “mirror spiders” from here (despite living in Australia I’d never heard of them before), and here’s an open access paper in Nature that dives into the mechanism:“Animals precisely control the morphology and assembly of guanine crystals to produce diverse optical phenomena in coloration and vision.”https://www.nature.com/articles/s41467-023-35894-6
(DIR) Post #B1wj2RJXgyMI9mi9Bo by gregeganSF@mathstodon.xyz
2026-01-04T06:28:15Z
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A few days ago, I saw a nice approximation to the arcsine function by @eniko asin(x) ≈ a(x) = x + (π/2-x)(1-√[1-x^2])^2I won’t bother plotting a(x) against asin(x), because the two curves are indistinguishable to the eye, but the proportional error is plotted below.I spent some time trying to figure out if there’s an underlying *geometric* reason why this works so well ... in the same sense that there is for, say:asin(x) ≈ chord(x) = √[2(1-√[1-x^2])]the length of the chord that subtends the angle asin(x).There are some obvious nice features built into a(x): it clearly must agree with asin at x=0 and 1, and less obviously it will match derivatives at those points as well.But surely there had to be some special geometric relationship too, to make it work so well?If so, I never did find it. Maybe someone else will (or already has). But I found another approximation, roughly as simple and roughly as good:asin(x) ≈ b(x) = ½(π-4)x^2 + x + 1 - √[1-x^2]which also matches values and derivatives with asin(x) at x=0 and 1, and whose proportional error is the gold curve in the plot below. [Note that it matches values with asin exactly at an additional, intermediate point, asin(1/√2) = b(1/√2) = π/4.]That partly cured me of my conviction that there had to be a nice geometrical account for any approximation this good. Maybe all that’s really needed is a low-degree polynomial and one function, √[1-x^2], with an infinite derivative at x=1 the same as asin(x).
(DIR) Post #B1wj2YzHACttxFI4i8 by gregeganSF@mathstodon.xyz
2026-01-04T06:29:00Z
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The original post: https://mathstodon.xyz/@eniko@mastodon.gamedev.place/115809815616652881
(DIR) Post #B29edrabfX9IhRVtS4 by gregeganSF@mathstodon.xyz
2026-01-10T14:24:47Z
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I found this story equal parts interesting, silly, and downright creepy: scientists are trying to reconstruct Leonardo Da Vinci’s DNA, with a mixture of swabbing his artworks, tomb-raiding his ancestors, and tracking down living relatives.There’s a fair bit of weird celebrity-fetishism and gene-fetishism. Leonardo did many wonderful things, but all this effort to try to get as much of his genetic sequence as possible seems ... disproportionate.The most interesting thing I learned from the article was that he *might* have had atypically fine time resolution in his visual perception. But even there, surely we can discover much more about that whole subject by studying a large number of living people, rather than obsessing over one person’s genome.https://www.science.org/content/article/have-scientists-found-leonardo-da-vinci-s-dna