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No, the red curve takes the value 1024 / (525 π²) at κ = 0. There is no step; the probability that the 3 points lie *exactly* on a straight line is zero, but the probability that they lie on a circle with any specified exact curvature is also zero. The probability density is non-zero for all κ≥0.

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The red curve I can only compute numerically, apart from the zero-curvature limit. I have a closed form expression for the blue curve.

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If you pick 3 points at random in the unit disk, the probability that they lie on a perfectly straight line is zero. However, the probability that the curvature of the circle they lie on is less than ε is, in the limit of small ε, equal to: 1024 ε / (525 π²) ≈ 0.2 ε

Roughly bell-shaped curve starts at (0,0.2), peaks at (1.24,0.822), long tail

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… is the same knot, but which cannot be deformed into each other when embedded in either 3 or 4 dimensions.

Two different surfaces whose boundaries are the same complicated knot.

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Special Surfaces Remain Distinct in Four Dimensions | Quanta Magazine

If you listen to the latest “Joy of Why” podcast, where the guest is topologist Maggie Miller, it will help a lot if you first read this @QuantaMagazine article on Miller’s work on Seifert surfaces: quantamagazine.org/special-surf... It includes an image of two surfaces whose boundary …

quantamagazine.org

For decades mathematicians have searched for a specific pair of surfaces that can’t be transformed into each other in four-dimensional space. Now they’ve found them.

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Greg Egan @gregegansf.bsky.social · 30/09/2026

I don’t know what cheesy LLM these scammers are using, but the glitches can be hilarious. [Though I have written a novel called “Morphotrophic”, “Morphological Regency” sounds like a token midway between that and the average of all titles on Smashwords.]

Hello Greg,

A strong book deserves more than simply being available for readers to find, particularly when its ideas invite readers to spend time thinking, questioning, and discussing what they have read. Your work in hard science fiction, mathematical speculation, and quantum theory creates exactly that kind of reading experience, where the scientific concepts and larger questions can become part of an active reader conversation. I’m Justin Brierley, a member of Authors Tables, where I partner with the group’s moderator to bring selected books into our Reading Challenge Cycles.

I believe Morphological Regency could create a particularly engaging discussion among readers interested in hard science fiction and intellectually demanding speculative fiction. The Reading Challenge Cycle gives readers an opportunity to discover the book, read it together, discuss its themes and ideas, and engage with the author through the reading experience. The focus is not on manufactured praise or predetermined reviews, but on putting the book in front of readers who are willing to engage with its scientific and speculative ideas thoughtfully.

I’d like to discuss the possibility of featuring Morphological Regency in an upcoming Reading Challenge Cycle and showing you how the cycle could give the book a stronger reader-facing opportunity. Would you be open to seeing how it works?

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Since an initial breakthrough in 2022, three groups of young mathematicians have proved* this for different regimes: large subsets (compared to p), small subsets, and those of intermediate size, covering all cases. *Like many results in combinatorics, the proofs only hold for sufficiently large p.

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Greg Egan @gregegansf.bsky.social · 30/09/2026

Graham’s conjecture (1971): Modulo some prime p, can you find a way to order any subset of {1,...,p-1} so that its successive partial sums are all distinct? www.quantamagazine.org/mathematicia...

quantamagazine.org

Mathematicians Harness Randomness To Crack a 55-Year-Old Conjecture | Quanta Magazine

After a long hiatus, the problem, which was likely inspired by juggling, has finally been resolved by a group of young mathematicians.

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Greg Egan @gregegansf.bsky.social · 29/09/2026

Pour comprendre qui il est et retrouver son véritable nom, il doit reconstituer les fragments de son existence. Une mystérieuse quête d’identité entre présent et souvenirs. www.fnac.com/a23000856/Cl...

fnac.com

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Greg Egan @gregegansf.bsky.social · 29/09/2026

Chaque matin, un nouveau corps, une seule mémoire. Mais une question : qui est-il vraiment ? Chaque jour, "il" se réveille dans un corps différent, avec les souvenirs de toutes ses vies passées.

fnac.com

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Greg Egan @gregegansf.bsky.social · 29/09/2026

L’adaptation en bande dessinée de la nouvelle Le Coffre-Fort de Greg Egan, sublimée par le graphisme singulier de Clément C. Fabre. Une histoire fascinante autour de l’identité, de la mémoire et des vies que l’on traverse…

Image of a man, with the words "Un jour, j'écrirai mon nom" replacing his face. At the top of the image, Clément C. Fabre, d'aprés Greg Egan.

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Greg Egan @gregegansf.bsky.social · 28/09/2026

Oh, Hostinger, have you no shame?

Superintelligence – SI – is what everyone's suddenly calling AI. And as luck would have it, .si has been a real domain all along, just waiting for its moment.
We got it live. Grab yours before the good names are gone.

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Carl Zimmer @carlzimmer.com · 27/09/2026

Four days ago, a scientist was surprised by the news that Anthropic discovered new virus genes for making DNA. He says he’s been studying them for years—and feeding his data to Anthropic’s AI as part of his own research. Coincidence? Here’s my story. nyti.ms/3VSWayc

nyti.ms

Did Anthropic’s A.I. Really Make a Scientific Discovery on Its Own? (Gift Article)

An expert at the University of Copenhagen said his team had been sharing its research with the company’s A.I. model, Claude, and that its new finding matched their work.

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Greg Egan @gregegansf.bsky.social · 27/09/2026

… and also the bit where it turned out there was a bug in the ZK protocol implementation, which allowed someone to troll them by “proving” *they* had a quantum circuit design that could do the same with zero Toffoli gates!

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Greg Egan @gregegansf.bsky.social · 27/09/2026

I was aware of Google announcing an implementation of Shor’s algorithm that needed substantially fewer qubits than previous methods, but I missed the bit where they released a Zero Knowledge Proof of their claim ... www.scientificamerican.com/article/why-...

scientificamerican.com

Why mathematicians are suddenly hiding research results

Google made a major breakthrough in quantum computing. To prevent the results from being misused, the company kept them secret. That backfired

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Greg Egan @gregegansf.bsky.social · 23/09/2026

Let’s hope that it doesn’t turn out that there’s a condition that closely resembles this, to the point where it could easily be mistaken for it, while actually being quite different. en.wikipedia.org/wiki/Pseudop...

en.wikipedia.org

Pseudopseudohypoparathyroidism - Wikipedia

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Greg Egan @gregegansf.bsky.social · 22/09/2026

Breaking: Tilly Norwood, Max Headroom quizzed on their positions re destroying all humanity in Senate confirmation hearings for AI Czar.

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Greg Egan @gregegansf.bsky.social · 22/09/2026

How does a single amino-acid substitution render the diphtheria toxin non-toxic? pmc.ncbi.nlm.nih.gov/articles/PMC... I have only skimmed this paper, but the short answer seems to be that the two proteins fold into almost identical shapes, but CRM197 is more flexible at a crucial binding site.

pmc.ncbi.nlm.nih.gov

Structural basis for lack of toxicity of the diphtheria toxin mutant CRM197

CRM197 is an enzymatically inactive and nontoxic form of diphtheria toxin that contains a single amino acid substitution (G52E). Being naturally nontoxic, CRM197 is an ideal carrier protein for conjug...

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Greg Egan @gregegansf.bsky.social · 22/09/2026

I just had the pneumococcus vaccine [free for people 65+ in Australia]. It contains sugars from 21 types of pneumococcus, linked to CRM197 — a version of diphtheria toxin that is highly immunogenic, but rendered non-toxic by a single mutation, altering just 1 of 535 amino acids.

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Michael Busch @michael-w-busch.bsky.social · 19/09/2026

The large main-belt asteroid 216 #Kleopatra and its moons will have an occultation along a swath from Ireland to Ukraine on 2026 September 22: cloud.occultwatcher.net/event/2090-2... . For anyone who would like to chase the shadows. (It may also be barely visible from Greenland, but at dusk.)

cloud.occultwatcher.net

OccultWatcher Cloud

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Greg Egan @gregegansf.bsky.social · 18/09/2026

Laurie Metcalf as Phyllis Gardner in “The Dropout”: “Don’t ever quote Yoda to anybody ever again.”

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Corey S. Powell @coreyspowell.bsky.social · 16/09/2026

Wow: Giant new impact crater spotted on the Moon! An asteroid the size of an apartment building hit the lunar surface in early 2024 & created 220-meter (730-foot) wide McGetchin crater. It's "the largest contemporary impact crater found in the Solar System." 🧪🔭 science.nasa.gov/solar-system...

Officially named McGetchin after pioneering lunar scientist Tom McGetchin, the crater formed on the Moon’s eastern edge sometime between April 11 and May 22, 2024, after a comet or asteroid the size of a three- to six-story building hit the surface.

The crash left a crater, 728 feet wide, that spans the length of two football fields. And at 141 feet deep, the crater could fit three vertically stacked yellow school buses. Scientists estimate that an impact of this magnitude happens on the Moon about once in a century or even longer.

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Greg Egan @gregegansf.bsky.social · 15/09/2026

Now more than ever we need a Saturday Night Live sketch with Matthew Rhys playing Dario Amodei.

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Greg Egan @gregegansf.bsky.social · 15/09/2026

My email client has an easy-to-manage blocklist, so scammers & crackpots go straight on the list to be blissfully forgotten. Occasionally I see one of their emails appear in my inbox for a second before being deleted … including a weird persistent scam telling me to pay “filing fees” in California.

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The phrase “the electric field maps to the E region” seems to refer to some process where a component of the electric field at the meteor itself will also end up being present all along a magnetic field line, all the way to the E region. But I don’t really follow the mechanism.

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At first I thought it was some oddly ungrammatical reference to electric fields, but no ... en.wikipedia.org/wiki/Ionosph...

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agupubs.onlinelibrary.wiley.com

Electrophonic conversion to sound efficiency then needs to be only 0.1% to explain why humans can hear and see meteors nearly concurrently.” “On the electrophonic generation of audio frequency sound by meteors” Michael C. Kelley, Colin Price agupubs.onlinelibrary.wiley.com/doi/full/10....

On the electrophonic generation of audio frequency sound by meteors

Meteors can create audible sound at the ground that is nearly concurrent with visual detection. ELF radiation from the ambipolar electric field mapping to the E region causes electrophonic sound. A...

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This localized electric field maps to the E region then drives a large Hall current that launches the electromagnetic wave. Using antenna theory and following, a power flux of over 10^{−8} W/m^2 at the ground is found.

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Coma ions around the meteor head can easily travel across magnetic field lines up to ~120 km. The electrons, however, are tied to magnetic field lines, since they must gyrate around the field above ~75 km. A large ambipolar electric field must be generated to conserve charge neutrality.

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“Recorded for centuries, people can hear and see meteors nearly concurrently. Electromagnetic energy clearly propagates at the speed of light and converts to sound (called electrophonics) when coupled to metals. An explanation for the electromagnetic energy source is suggested.

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Michael Busch @michael-w-busch.bsky.social · 14/09/2026

TIL from @dandkuniverse.bsky.social that you can sometimes hear meteors burning up in the atmosphere *at the same time as you see them*, despite the discrepancy in the speed of light and the speed of sound. How? Here is an abstract from a relevant paper:

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Reposted by Greg Egan

When an orbit is eccentric enough that the seasons from eccentricity win over those from obliquity.

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Reposted by Greg Egan

David Ho @davidho.bsky.social · 12/09/2026

I wrote this quickly at a bus stop and didn't realize it would go viral, or else I would have worded it better. But the point is that the media is too credulous about AI killing everyone in 10 years (how?) when climate change is killing people now and will only get worse (we know exactly how).

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Greg Egan @gregegansf.bsky.social · 11/09/2026

Breaking: a team of human mathematicians working in secret over the last 3 years, and at least two frontier AI labs with 7-figure budgets, are believed to be neck-and-neck in a race to determine whether the Birch-Swinnerton-Dyer conjecture was posed by three people, or just two.

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Greg Egan @gregegansf.bsky.social · 10/09/2026

… Two prominent AI companies, Anthropic and OpenAI, will supply participants with 2 million dollars in AI credits. This event is likely to have destructive impacts for the mathematical community.” docs.google.com/forms/d/e/1F...

docs.google.com

Open Letter about the Mathathon

The full text of the open letter is below. By signing this open letter, you consent to your name, title, and institutional affiliation being publicly displayed beneath this open letter. You can access...

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Greg Egan @gregegansf.bsky.social · 10/09/2026

Open Letter about the Mathathon This open letter calling for the cancellation of the Caltech “Mathathon”, has received 471 signatures so far. [Only members of the Caltech community or mathematicians at PhD student level or beyond are eligible to sign.]

docs.google.com

Open Letter about the Mathathon

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From the paper: Demaine et al.’s general solution gave a word of length O(N^{1561600}) [yeow!] Wästlund improved this to O(N^{7.004}).

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… cause the picture to fall, but removing fewer than K nails will leave the picture supported. The result above is from recent work by Tom Verhoeff: arxiv.org/abs/2605.27617 that I read about in Scientific American: www.scientificamerican.com/article/math...

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There are no shorter products with this property. This is an example of the famous “picture hanging puzzle”: if you have N nails in the wall, what are the least complicated ways to wind a string holding up a picture around the nails so that removing any K of the nails will …

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