Nostr notes by Greg Egan

Bacteria should have shared the 2025 chemistry Nobel Prize on ...

2026-10-28T11:00:04Z

Bacteria should have shared the 2025 chemistry Nobel Prize on metal-organic frameworks!

It’s been known for a while that there are “cable bacteria” that extrude conductive filaments up to 1 cm long, which penetrate sediments on the sea floor and allow them to draw electrons from buried sulphides to power their metabolism.

But it turns out the filaments made by Candidatus Electrothrix gigas are based on MOFs “strikingly similar to those used by chemists”. This was not a case of deliberate biomimicry; these MOFs were created in labs before they were discovered in nature.

https://www.chemistryworld.com/news/bacteria-discovered-to-be-synthesising-metal-organic-frameworks-with-extraordinary-conductivity/4024399.article

[You can register/subscribe at the Chemistry World site if you wish, but if you just close the box they still let you read this article.]

Today I bought a pack of assorted washers of various sizes, which ...

2026-10-27T11:15:44Z

Today I bought a pack of assorted washers of various sizes, which were described with dimensions like “28mm × 32mm”, which was simultaneously (A) perfectly comprehensible [inner diameter × outer diameter], (B) hilariously weird, and (C) an unforgivable transgression against the sacred dictates of mathematics and geometry.

Mathematica’s spellchecking dictionary contained the word ...

2026-10-24T18:42:01Z

In reply to nevent1q…qfpn
_________________________

Mathematica’s spellchecking dictionary contained the word “horocycle” but, amusingly, not the plural “horocycles”. [I have no idea how this dictionary was assembled.]

Thanks! It’s annoying that conventional search engines have ...

2026-10-24T13:27:15Z

In reply to nevent1q…aawk
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Thanks! It’s annoying that conventional search engines have deteriorated to the point where they apparently can’t find anyone else even mentioning this formula, but I’ll set aside my robot allergy and focus on the human-written paper itself, and take that as confirmation that I didn’t screw up the calculation.

Given two distinct points A and B in the hyperbolic plane, there ...

2026-10-24T10:29:06Z

Given two distinct points A and B in the hyperbolic plane, there are always two horocycles that contain both A and B.

A horocyle is a curve of constant geodesic curvature κ=1.

In the Poincaré disk model of the hyperbolic plane, a horocycle is represented by a circle that is tangent to the boundary of the disk (aka the circle of infinitely distant “ideal points”).

In the hyperboloid model of the hyperbolic plane, a horocycle is the intersection of the hyperboloid –t^2 + x^2 + y^2 = –1 in (2+1)-dimensional Minkowski spacetime and a plane with a null (aka “lightlike”) normal vector.

What is the area of the region bounded by the two horocycles that pass through two given points? This should only depend on the distance between the points, δ. The computations I performed in both the Poincaré disk model and the hyperboloid model give the same answer:

A(δ) = 4 sinh(δ/2) – 8 atan(tanh(δ/4))

That these two calculations agreed makes me fairly confident that this formula is correct ... but frustratingly I can’t locate anything on the web that says anything about this area.

I can’t believe this simple result isn’t stated *anywhere* on the internet. Even if the formula I’m claiming is wrong, whatever the correct result is, someone like Nikolai Lobachevsky or Eugenio Beltrami surely figured it out in the 19th century, so it must be mentioned somewhere.

So I’m wondering if anyone is aware of a reliable reference on hyperbolic geometry that gives a formula for the area between the two horocycles that contain two given points.

Suppose the first parabola is: y₁(x) = ½ a₁ (x–x₁)² so ...

2026-10-22T07:16:50Z

In reply to nevent1q…rckq
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Suppose the first parabola is:

y₁(x) = ½ a₁ (x–x₁)²

so it starts with a derivative of zero at x=x₁, y=0 and accelerates at a rate of a₁.

Suppose the second parabola is:

y₂(x) = Y₂ – ½ a₂ (x–x₂)²

so it ends up at x=x₂, y=Y₂ with a derivative of zero, after decelerating at a rate of a₂.

You can find the x value where they intersect by setting:

y₁(x) = y₂(x)

which gives a quadratic equation in x that is solved by the first of two roots:

xᵢ = (a₁x₁+a₂x₂–√[2(a₁+a₂)Y₂–a₁a₂(x₁–x₂)²]) / (a₁+a₂)

In order for the two parabolas to join up smoothly, their derivatives at x=xᵢ must be equal:

y₁'(xᵢ) = y₂'(xᵢ)

This condition simplifies to give:

2(a₁+a₂)Y₂ = a₁a₂(x₁–x₂)²

This also means the quadratic we got earlier actually has only one root, i.e.the two parabolas intersect at a single point rather than two points. If you think about the geometry that makes sense, because the two parabolas then are tangent to each other at the point where they meet.

You can solve this condition to find x₂ in terms of Y₂, which is how I think your problem is set up:

x₂ = x₁ + √[2(a₁+a₂)Y₂ / (a₁+a₂)]

and then you also have:

xᵢ = (a₁x₁+a₂x₂) / (a₁+a₂)

where xᵢ is the value for x that gives a smooth transition between the parabolas.

Fishing boat crew rescue kangaroo found 3km offshore ...

2026-10-21T11:15:57Z

Fishing boat crew rescue kangaroo found 3km offshore

https://www.abc.net.au/news/2026-10-21/kangaroo-rescued-from-sea-boston-bay-port-lincoln/107291116

I have a new story, “Osculation Point”, in the anthology ...

2026-10-20T08:18:15Z

I have a new story, “Osculation Point”, in the anthology VASTER THAN EMPIRES

https://www.amazon.com/Vaster-than-Empires-Speculative-Ephemeral/dp/0262054876

Also, “Spare Parts For the Mind” (originally in Asimov’s SF) is reprinted in THINK WEIRDER Volume 2:

https://thinkweirder.com

I had to look that up, but I should have guessed that either ...

2026-10-16T12:45:57Z

In reply to nevent1q…l63u
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I had to look that up, but I should have guessed that either Terry Pratchett or Neal Stephenson would have come up with a similar proposal at some point.

Now that we’ve abolished leap seconds, and won’t tweak ...

2026-10-16T12:36:58Z

Now that we’ve abolished leap seconds, and won’t tweak standard time to match the rotation of the Earth until the discrepancy reaches 1 hour, we should establish a generation-spanning secret society that shuffles objects around to make Earth’s spin conform to atomic time instead.

Can I ask what is the insignia? It makes me think of carnivorous ...

2026-10-15T12:25:56Z

In reply to nevent1q…c2ed
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Can I ask what is the insignia? It makes me think of carnivorous plants, minor internal organs and an anchor, so along with the E H it’s all very mysterious.

Today I had my first ever diffusion-weighted MRI scan, Some ...

2026-10-12T13:44:11Z

Today I had my first ever diffusion-weighted MRI scan, Some people dislike the noises from MRI machines, but all the beeping and buzzing sounded quite hypnotic to me, like a kind of minimalist ambient techno music. Maybe the devices are actually silent, and the manufacturers secretly hired Brian Eno to compose a soundtrack that would make them seem more mysterious and impressive.

But MRI is impressive enough on its own. The hand-wavy explanation is that nuclear spins lined up in a magnetic field are made to precess by applying an oscillating, orthogonal magnetic field, and then as the excited spins relax they emit radio waves at a frequency that’s highly sensitive to the magnetic field at the nucleus, which includes contributions from its particular chemical environment.

The details are quite subtle; the spins do not relax by spontaneous emission, but from interactions between all the nuclei.

[See https://en.wikipedia.org/wiki/Spin–lattice_relaxation and https://en.wikipedia.org/wiki/Spin–spin_relaxation ]

Gradients in the imposed magnetic field make it possible to identify where in the body the signal originates. But if a water molecule moves through this gradient while its spin is relaxing, that modifies the signal. Diffusion-weighted MRI exploits this, to discriminate between, say, water inside a cell and in various extracellular environments.

The fanciest version, diffusion tensor imaging, can determine the *direction* water is diffusing along neurons, mapping connectivity between brain regions!

But the purpose of my scan was to measure the “cellularity” of my bone marrow, to see where, and to what degree, the intracellular vs extracellular water ratio has been elevated by a pesky clone of over-reproducing plasma cells.

It’s now possible to be overwhelmed, not by the vast quantity ...

2026-10-09T08:18:41Z

It’s now possible to be overwhelmed, not by the vast quantity of claimed proofs and counterexamples recently disgorged by OpenAI (most of which would be over my head), but merely by trying to read all the reactions of expert mathematicians to the vast quantity of claimed proofs and counterexamples recently disgorged by OpenAI.

“100+ reactions to 100+ solutions”

https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/

Is it just me, or does “Firmus” sound like an inept, ...

2026-10-08T07:17:58Z

Is it just me, or does “Firmus” sound like an inept, narcissistic villain from Futurama or Rick and Morty?

https://www.abc.net.au/news/2026-10-08/firmus-ipo-underwhelms-before-it-hits-the-market/107241548

Though this proposal has been ruled out, I wouldn’t give up ...

2026-10-08T01:59:42Z

In reply to nevent1q…v4lt
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Though this proposal has been ruled out, I wouldn’t give up hope that we could eventually construct something with a much larger cross-section for neutrino interactions than ordinary matter.

In my story “Wang’s Carpets”, which became a part of DIASPORA, there is some hand-waving about excited nuclear states with this property [and which are kept from decaying too soon to be useful by employing the quantum Zeno effect].

LUX–ZEPLIN dark matter detector: 7 tonnes of ultra-pure liquid ...

2026-10-07T12:12:28Z

LUX–ZEPLIN dark matter detector: 7 tonnes of ultra-pure liquid xenon

IceCube Neutrino observatory: a cubic kilometre of ancient, pristine Antarctic ice

KM3NeT: a cubic kilometre of the Mediterranean. May contain plankton, fish poo and Vibrio cholerae.

That’s a shame!

2026-10-07T11:24:23Z

In reply to nevent1q…l3wy
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That’s a shame!

They still haven’t resolved the square-free status of Fermat ...

2026-10-07T07:32:48Z

In reply to nevent1q…xrr7
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They still haven’t resolved the square-free status of Fermat numbers. My novel where this takes more than 800,000 years remains possible!

Head of climate action residents’ group: "We cannot dig up ...

2026-10-07T05:21:02Z

Head of climate action residents’ group: "We cannot dig up coal, ship it overseas, watch it drive climate change, and then pretend the consequences have nothing to do with us and won't be felt by us."

3/5 judges in Australia’s High Court just agreed.

https://www.abc.net.au/news/2026-10-07/mount-pleasant-coal-mine-high-court-ruling/107233650

Me in GP’s office: I think I have neutrino cholera. GP: Can’t ...

2026-10-06T12:07:53Z

In reply to nevent1q…wmfp
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Me in GP’s office: I think I have neutrino cholera.
GP: Can’t you cut down on your neutrino exposure?

I hate to break it to you, but that was part of the joke.

2026-10-06T12:05:05Z

In reply to nevent1q…w4e0
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I hate to break it to you, but that was part of the joke.

This is more about enhancing neutrino emission, but I think it ...

2026-10-06T11:56:48Z

In reply to nevent1q…aucw
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This is more about enhancing neutrino emission, but I think it also opens up some possibilities for enhanced detection as well.

https://news.mit.edu/2025/physicists-devise-idea-lasers-shoot-beams-neutrinos-0908

Don’t you hate it when someone mixes up the Nobel Prizes and ...

2026-10-06T11:30:46Z

Don’t you hate it when someone mixes up the Nobel Prizes and inserts ion channels in your neurons that open when exposed to neutrinos instead of light, and your brain just goes pffzzz ....

If you pick 3 points at random in the unit disk, the probability ...

2026-10-04T13:36:29Z

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

If you listen to the latest “Joy of Why” podcast, where the ...

2026-10-04T10:04:52Z

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 Quanta Magazine (npub10v3…3j5v) article on Miller’s work on Seifert surfaces:

https://quantamagazine.org/special-surfaces-remain-distinct-in-four-dimensions-20220616/

It includes an image of two surfaces whose boundary is the same knot, but which cannot be deformed into each other when embedded in either 3 or 4 dimensions.

I don’t know what cheesy LLM these scammers are using, but the ...

2026-10-04T02:47:03Z

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.]

RE: https://mstdn.social/@QuantaMagazine/117359884077976693 ...

2026-09-30T12:17:07Z

RE: https://mstdn.social/@QuantaMagazine/117359884077976693

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?

Since an initial breakthrough in 2022, three groups of young mathematicians have proved* this for different regimes: large subsets (as a fraction of p), small subsets, and finally those of intermediate size, covering all cases.

*Like many results in combinatorics, the proofs only hold for sufficiently large p.

quoting
note1xrj…kkxp

“It’s interesting, in fact, that many mathematicians and computer scientists have an interest in juggling,” the late mathematician Ronald Graham said in a 1980 television interview. “I think it’s the search for patterns and structure that is responsible for this.” https://www.quantamagazine.org/mathematicians-harness-randomness-to-crack-a-55-year-old-conjecture-20260928/

L’adaptation en bande dessinée de la nouvelle Le Coffre-Fort ...

2026-09-29T04:24:06Z

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…

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.

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.

http://www.fnac.com/a23000856/Clement-C-Fabre-Un-jour-j-ecrirai-mon-nom

Oh, Hostinger, have you no shame? ...

2026-09-28T13:23:35Z

Oh, Hostinger, have you no shame?

I was aware of Google announcing an implementation of Shor’s ...

2026-09-27T11:28:21Z

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 ... 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!

https://www.scientificamerican.com/article/why-mathematicians-are-hiding-research-results-in-zero-knowledge-proofs/

RE: https://infosec.exchange/@Prometheus/117337083492569840 ...

2026-09-26T11:47:40Z

In reply to nevent1q…vj5z
_________________________

RE: https://infosec.exchange/@Prometheus/117337083492569840

Sudo Sudo Sudo ⒶⒺⓋ (npub1xnj…2vu3)/117337083524644151">https://mathstodon.xyz/Sudo Sudo Sudo ⒶⒺⓋ (npub1xnj…2vu3)/117337083524644151

quoting
note1qxu…vd7e

Well where is the link to a trailer we can watch then? Hold my coffee while I go find one... Well its all over YouTube anyway... But I found one that wasn't.
https://www.imdb.com/video/vi1686358553/

Greg Egan (npub12cu…d38p) https://mathstodon.xyz/@gregeganSF/117335724327675904

“So much for free speech. The studio releasing Alex Gibney’s ...

2026-09-26T05:37:47Z

“So much for free speech.

The studio releasing Alex Gibney’s new Elon Musk documentary Musk is finding itself unable to buy an ad on the biggest social platforms.”

https://www.hollywoodreporter.com/business/digital/elon-musk-documentary-trailer-rejected-tiktok-youtube-1236710259/

Let’s hope that it doesn’t turn out that there’s a ...

2026-09-23T08:22:11Z

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.

https://en.wikipedia.org/wiki/Pseudopseudohypoparathyroidism

Breaking: Tilly Norwood, Max Headroom quizzed on their positions ...

2026-09-22T12:01:25Z

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

How does a single amino-acid substitution render the diphtheria ...

2026-09-22T05:37:56Z

In reply to nevent1q…fv8g
_________________________

How does a single amino-acid substitution render the diphtheria toxin non-toxic?

https://pmc.ncbi.nlm.nih.gov/articles/PMC3325714/

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.

I just had the pneumococcus vaccine [free for people 65+ in ...

2026-09-22T05:27:04Z

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.

“A square symbol ☐ denotes that the member belongs to our ...

2026-09-21T08:56:12Z

In reply to nevent1q…dp4n
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“A square symbol ☐ denotes that the member belongs to our AI-free caucus. These members do not use AI models in their research.”

So, currently 424 out of 645.

Since Peter Scholze (and doubtless many other highly talented ...

2026-09-20T11:57:45Z

In reply to nevent1q…fffq
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Since Peter Scholze (and doubtless many other highly talented people) has eschewed the use of AI completely, I expect there will be many great mathematical theorems proved in the future without the use of AI.

If each linear combination of the four h_i gives a different ...

2026-09-20T11:47:03Z

In reply to nevent1q…hz0c
_________________________

If each linear combination of the four h_i gives a different orbit under SL(9), and the union of these orbits is the whole space of trivectors, is there a straightforward way to take an arbitrary trivector and compute four SL(9)-invariant scalars that correspond to the coefficients of the h_i in the linear combination a_i h_i that is a representative of the same orbit?

n=9 is the first value when n^2 (the dimension of GL(n,C)) is ...

2026-09-19T13:46:03Z

In reply to nevent1q…qmfy
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n=9 is the first value when n^2 (the dimension of GL(n,C)) is greater than Binomial(n,3) (the dimension of the space of trivectors).

But 9^2 = 81 and Binomial(8,3) = 84, so naively I would have thought the space of orbits would be 3-dimensional, not 4-dimensional. I guess the action of GL(9,C) on the trivectors must produce 80-dimensional orbits rather than 81-dimensional.

“But when k = 3, n < 9” No, that’s stated in a list of ...

2026-09-19T11:20:21Z

In reply to nevent1q…4rqz
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“But when k = 3, n < 9”

No, that’s stated in a list of cases where the number of orbits is finite. The interesting case k=3 n=9 is interesting because there are an infinite number of different orbits.

Laurie Metcalf as Phyllis Gardner in “The Dropout”: ...

2026-09-18T13:27:48Z

Laurie Metcalf as Phyllis Gardner in “The Dropout”:

“Don’t ever quote Yoda to anybody ever again.”

I just Googled “Six Birds emergence calculus” and it .. did ...

2026-09-18T12:30:18Z

In reply to nevent1q…nyh4
_________________________

I just Googled “Six Birds emergence calculus” and it .. did not inspire confidence.

Now more than ever we need a Saturday Night Live sketch with ...

2026-09-15T13:14:18Z

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

From the linked paper: “Finally, it is well known that the ...

2026-09-14T09:01:06Z

In reply to nevent1q…f7zx
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From the linked paper:

“Finally, it is well known that the aurora creates intense ELF and VLF radio waves that reach the ground. We think that it is quite possible that the many reports of audible aurora by high-latitude, indigenous peoples are, in fact, true. Hopefully, this paper will inspire research on this topic.”

TIL from ...

2026-09-14T07:26:08Z

TIL from
[@dandkuniverse.bsky.social](https://bsky.brid.gy/r/https://bsky.app/profile/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:

“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. 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. 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. 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

https://agupubs.onlinelibrary.wiley.com/doi/full/10.1002/2017GL072911

Breaking: a team of human mathematicians working in secret over ...

2026-09-11T13:39:36Z

Breaking: a team of human mathematicians working in secret over the last three years, and at least two frontier AI labs with seven-figure budgets for their own month-long projects, are believed to be neck-and-neck in a race towards an answer to the fiendishly difficult question of whether the Birch-Swinnerton-Dyer conjecture was posed by three people, or just two.

That passage about vorticity and forcing from El Pais seems much ...

2026-09-10T12:41:29Z

In reply to nevent1q…zs6j
_________________________

That passage about vorticity and forcing from El Pais seems much more scientifically literate than what you’d read in most English-language newspapers!

Steven Strogatz posted an English translation on X: ...

2026-09-10T12:09:02Z

In reply to nevent1q…glqa
_________________________

Steven Strogatz posted an English translation on X:

https://x.com/stevenstrogatz/status/2098015128578904069

https://x.com/stevenstrogatz/status/2098015365175386474

One Year of Work by Two Spanish Mathematicians Versus 88 Hours ...

2026-09-10T11:58:09Z

One Year of Work by Two Spanish Mathematicians Versus 88 Hours and OpenAI’s €15 Million: “Without Our Idea, AI Wouldn’t Have Solved It”

https://elpais.com/ciencia/2026-09-10/un-ano-de-trabajo-de-dos-matematicos-espanoles-frente-a-88-horas-y-15-millones-de-openai-sin-nuestra-idea-la-ia-no-lo-habria-resuelto.html

The details of the competition are here: ...

2026-09-10T10:22:38Z

In reply to nevent1q…rmq8
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The details of the competition are here:

https://mathathonchallenge.com

Open Letter about the Mathathon This open letter calling for the ...

2026-09-10T05:52:07Z

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.]

“On October 30th, 2026, Caltech will begin hosting the Caltech Mathathon, a math hackathon in which participants are encouraged to solve open research problems by prompting LLMs. 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.”

https://docs.google.com/forms/d/e/1FAIpQLSf6dBeDwjRP4UkchGSpLwLQjiG8qLX5_wW_3-xoC4dImTPp6g/viewform

I think I learned about this from Wikipedia. In 2019 I posted ...

2026-09-09T12:43:14Z

In reply to nevent1q…acja
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I think I learned about this from Wikipedia. In 2019 I posted this on Twitter:

Here’s my favourite immersion of Klein’s bottle. The region just inside each loop of the figure 8 is continuous with that just outside the other loop, so a Möbius-like-twist connects the two sides.

Then by splitting the surface in two (locally), you get an immersion of a torus!

The correct answer is “North of the Antarctic Circle”. Note ...

2026-09-08T06:28:17Z

In reply to nevent1q…hgyc
_________________________

The correct answer is “North of the Antarctic Circle”.

Note that the paths must be counterclockwise, or Alice’s bearing would be west of north, not east.

It’s tempting to think the cases are related by a symmetry, but Alice is flying into a circle, Bob is flying *out* of one.

Alice flies along a path the size & shape of the Arctic ...

2026-09-07T06:33:14Z

Alice flies along a path the size & shape of the Arctic Circle—but whose centre lies *on* the Arctic Circle.

As she crosses the Arctic Circle on a bearing ~29° east of north, Bob crosses the Antarctic circle with the same bearing, on a path that’s identical except its centre is:

In the free group generated by four letters, if you remove any ...

2026-09-06T05:36:19Z

In the free group generated by four letters, if you remove any two letters (and their inverse) from this product of 16 elements:

𝑎𝑏𝑎⁻¹𝑏⁻¹𝑐𝑑𝑏𝑎𝑑⁻¹𝑐⁻¹𝑑𝑐𝑏⁻¹𝑎⁻¹𝑐⁻¹𝑑⁻¹

it simplifies to 1, but dropping any single letter does not give 1.

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 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:

https://arxiv.org/abs/2605.27617

that I read about in Scientific American:

https://www.scientificamerican.com/article/mathematicians-discover-the-worst-way-to-hang-a-painting/

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}).

My favourite map of the world is ... a globe. My family was not ...

2026-09-05T00:15:04Z

My favourite map of the world is ... a globe.

My family was not wealthy, but when I was a child we had, not just a world atlas with its flat maps, but a world globe made of cardboard (from Reader’s Digest, I think). So I did not grow up confused about the size of Greenland versus Africa.

So, let there be more cheap, cardboard world globes. And online, let there be more animated and interactive images of the world as it actually is: spherical. I’m as fascinated as anyone by the merits of different map projections, but most people are not 15th century navigators having to choose a projection for their flat paper charts.

https://www.abc.net.au/news/2026-09-05/africa-bigger-us-smaller-united-nations-adopts-new-world-map/107119646

I’ve walked past this door (in Sir Charles Gairdner Hospital in ...

2026-09-02T06:13:21Z

I’ve walked past this door (in Sir Charles Gairdner Hospital in Perth) a few times, but today I finally gave in to the urge to take a picture, because ... physics!

[I once worked in the Medical Physics dept of a different hospital, so I guess this is the same kind of dept, though I’m not actually sure.]

RE: https://mathstodon.xyz/@robinhouston/117190244310733633 “My ...

2026-08-31T13:36:10Z

RE: https://mathstodon.xyz/@robinhouston/117190244310733633

“My failed attempts to prove the θ(p_c)=0 conjecture generated dozens of ideas that I later repurposed in other contexts, leading to discoveries I would never have imagined making.”

quoting
note1p47…yhfz

I thought Hugo Duminil-Copin's post yesterday was interesting too. https://proofsandprompts.com/2026/08/30/care-for-a-little-more-ai/

“If we want to make sure that mathematics survives the AI era, ...

2026-08-31T12:51:57Z

“If we want to make sure that mathematics survives the AI era, and if we want our field to continue to attract human talent and humans hoping to build a better and stronger community, then we need to heavily invest in humans, and vociferously celebrate – more than ever – the human contributions by the great people that have built our community piece by piece”

— Álvaro Lozano-Robledo,

https://proofsandprompts.com/2026/08/31/invest-in-human-mathematicians/

Greetings, fellow genuine humans! Text smooth write can I with ...

2026-08-30T00:24:21Z

Greetings, fellow genuine humans! Text smooth write can I with simple splicing tools even not LLMs! Scams much cheaper Completely and also convincing!

“We are currently planning our upcoming Author Spotlight series, and we would love to feature Perihelion Summer. Your book caught our attention because A highly technical sci-fi cli-fi handbook standardizing rapid orbital disruption dataset tracking metrics, thermodynamic parameters, and localized macro-collapse parameters., which aligns perfectly with the deep, reflective discussions our community enjoys.”

“We handle all production and coordination. To help cover these administrative setup costs, we request a standard contribution, though we are always happy to accommodate different budgets.”

Every time they played “Nine to Five” on the radio this week ...

2026-08-28T08:01:41Z

Every time they played “Nine to Five” on the radio this week it turned into an unstable earworm that kept slipping into the adjacent attractor in music-space for “Heart of Glass”.

But now I just learned that Dolly Parton did a *cover* of “Heart of Glass” in 2023 featuring Debbie Harry, and after listening to *that* the two attractors just gave up their separate existence and decided to merge completely.

An amazing result giving the first improvement in the “Komlós ...

2026-08-24T10:24:25Z

An amazing result giving the first improvement in the “Komlós combinatorial discrepancy problem” in almost 30 years (and it wasn’t done with AI).

Given N unit-length vectors in R^d, you want to partition them into two subsets, A and B, so the sum of the vectors in A and those in B differ in all of their components by as little as possible.

The new result, from Nikhil Bansal and Haotian Jiang, describes an algorithm that bounds the difference by O((log N)^{1/4}), improving on a previous result of O((log N)^{1/2}). János Komlós conjectured that the true bound is independent of N.

Great Quanta article covering the history of the problem:
https://www.quantamagazine.org/huge-breakthrough-in-the-math-of-imbalance-20260821/

Paper from Bansal and Jiang:
https://arxiv.org/abs/2508.03961

Thanks!

2026-08-20T07:04:29Z

In reply to nevent1q…du97
_________________________

Thanks!

I love the black hole at the centre of the galaxy so much that I ...

2026-08-20T05:52:21Z

In reply to nevent1q…kadc
_________________________

I love the black hole at the centre of the galaxy so much that I wrote a novel about aliens learning General Relativity from the experience of orbiting it:

https://gregegan.net/INCANDESCENCE/Incandescence.html

Wow, not only have astronomers found a star that comes so close ...

2026-08-20T04:00:14Z

Wow, not only have astronomers found a star that comes so close to the black hole at the centre of our galaxy that a detailed study of its orbit over the next decade should be able to quantify the black hole’s spin, there is a chance that future spectroscopic analysis will show that it was once part of a binary pair with a known hypervelocity star moving away from the galaxy’s centre, providing some evidence for the capture/eject Hills mechanism, explaining how stars end up so close to a black hole in the first place.

https://www.science.org/content/article/fastest-known-star-could-reveal-spin-our-galaxy-s-giant-black-hole

Just had my 1st dose of shingles vaccine. A few relatives have ...

2026-08-20T03:06:58Z

Just had my 1st dose of shingles vaccine. A few relatives have had shingles in the last few years, and it was *very* unpleasant. In Australia, the Shingrix vaccine is now free for 65+, so if you don’t want your childhood chickenpox infection waking up to torment you again ...

Yes. I’ve edited the post now to make that explicit. You can ...

2026-08-19T11:09:28Z

In reply to nevent1q…t9d4
_________________________

Yes. I’ve edited the post now to make that explicit.

You can also compute the average geodesic curvature, which for a circle with radius R [in spherical geometry with Gaussian curvature 1] is cot(R). For example, R=π/2 gives a great circle, with geodesic curvature of zero. The average geodesic curvature for circles passing through 3 random points is 1/2.

I am measuring the radius of the circle on a sphere by the length ...

2026-08-19T10:56:27Z

In reply to nevent1q…77c4
_________________________

I am measuring the radius of the circle on a sphere by the length of the geodesic on the sphere joining a point on the circle to the (nearest) centre of the circle; this is larger than the radius of the circle as a subset of R^3.

BTW, if you want a laugh, here is one of the results offered up ...

2026-08-19T10:31:05Z

In reply to nevent1q…5f7t
_________________________

BTW, if you want a laugh, here is one of the results offered up by the clown car that used to be Google Search.

The MathOverflow post referenced here gives a correct argument for why R diverges, but says nothing about κ.

No doubt some LLMs could answer this correctly, but what’s dismaying is that Google’s default now is not to do a proper search, but to waste resources generating its own wrong “answer”.

If you pick 3 points at random on a unit sphere, the average ...

2026-08-19T10:29:49Z

If you pick 3 points at random on a unit sphere, the average radius, R, of the circle that contains them is 7/6.

For 3 points on a flat disk of radius ρ, mean(R) diverges. However, the curvature, κ = 1/R, has a finite mean which can be computed exactly:

16 (3 – 16/π^2) / (15ρ)

Although it’s easy to write down the radius of the circle through any 3 points, given their coordinates, integrating the reciprocal of this over all triples of points inside a disk is unwieldy.

It’s easier to switch to a coordinate system where R is one of the coordinates, along with the location of the centre of the circle and the positions of the 3 points around the circle.

Details at https://www.gregegan.net/SCIENCE/ThreePointCurvature/ThreePointCurvature.html

TIL about “cancellous bone”. I’d previously misunderstood ...

2026-08-18T07:16:32Z

TIL about “cancellous bone”. I’d previously misunderstood what the solid sample taken in a bone marrow biopsy was composed of; I thought it was part of the outer, “cortical” bone. But today the doctor doing the biopsy showed me the sample and explained exactly what it was.

I had a naïve picture of marrow being soft & pliable as toothpaste, from memories of sucking it out of the bones in lamb stew. But the interior of living bones is a network of rods & plates, with the squishy bits, like hematopoietic stem cells, in the pores of this solid sponge.

Thanks for the encouragement!

2026-08-18T05:38:42Z

In reply to nevent1q…tq60
_________________________

Thanks for the encouragement!

His book-length proof looks like it’s beautifully written. ...

2026-08-16T21:53:09Z

In reply to nevent1q…av4c
_________________________

His book-length proof looks like it’s beautifully written. Shorter than “War and Peace”, and probably more interesting.

https://arxiv.org/abs/1501.05438

I would be happy to include a credit, though if anyone does ...

2026-08-16T12:56:09Z

In reply to nevent1q…uexk
_________________________

I would be happy to include a credit, though if anyone does resolve the square-free status of Fermat numbers, I think they will get much more fame from the AI and mathematical communities than anything I can offer by mentioning that they made me change the plot of a science fiction novel.

I’m not really qualified to say this, but my hunch is that this question is orders of magnitude harder than anything that LLMs have resolved so far.

“more people say over 700 will be solved than 600-699.” The ...

2026-08-16T12:21:24Z

In reply to nevent1q…973f
_________________________

“more people say over 700 will be solved than 600-699.”

The chart says 26% of people say over 700 will be solved, and 23%+27%+9% = 59% say 600-699.

Did you mean one of the smaller categories within 600-699?

In the novel I’m currently working on, which starts 800,000 ...

2026-08-16T12:05:38Z

In reply to nevent1q…973f
_________________________

In the novel I’m currently working on, which starts 800,000 years in the future, people still don’t know whether all Fermat numbers are square-free, and there is a subplot in which two of the characters devote an astronomically huge amount of resources to see if they can find a proof or a counterexample.

So if anyone wants to punish me for my LLM-scepticism they should try to settle this question before the book is published, forcing me to rewrite it to make that subplot about a different problem.

Dave Bacon likes to point out that if the whole universe is ...

2026-08-13T22:38:37Z

In reply to nevent1q…jac5
_________________________

Dave Bacon likes to point out that if the whole universe is described by a single state vector in a high-dimensional Hilbert space (which is what most many-worlds proponents believe) then there is a *vast* orthogonal complement to that vector that describes things that do not happen to anyone in any “world”.

So it is definitely not true that “anything you can imagine happens to someone in some world”.

AIs hallucinating that the patients whose medical notes they are ...

2026-08-13T22:03:17Z

AIs hallucinating that the patients whose medical notes they are transcribing used illegal drugs. What could possibly go wrong?

https://www.abc.net.au/news/2026-08-14/ai-medical-scribe-error-leaves-patient-devastated/107031672

I have refugee friends who don’t know their true dates of ...

2026-08-12T12:21:45Z

In reply to nevent1q…j62v
_________________________

I have refugee friends who don’t know their true dates of birth, and who have been forced to either invent dates or have invented dates foisted upon them, and then end up ensnared by the “inconsistencies” between their various official documents.

The inflexibility of our systems, in circumstances where any human with a brain can understand perfectly why there is an innocent reason for these kinds of disparities, is just maddening.

2002: I buy a prepaid SIM card, giving my name as “Greg Egan” ...

2026-08-12T11:27:18Z

2002: I buy a prepaid SIM card, giving my name as “Greg Egan” on the account application. Asked for ID, I show my passport, which gives my first name as “Gregory” and includes my middle name.

Human checking this (who, being a human, understands how humans use their informal and full legal names): Everything is fine, enjoy your new mobile phone!

2026: Telco’s automated message to me: We must verify your identity. Please supply the details of a suitable ID document.

Me: Supplies the details on my passport.

Verification algorithm: Details do not match your account.

So I am now looking forward to hours of time wasted on getting through to a human in the telco who knows exactly the right solution to this “problem”.

And I know that people with non-Anglophone names face this kind of needless idiocy a hundred times more often.

Automating paperwork and bureaucracy is great, but 3 decades into the process, why are we still so crap at handling even the simplest, most obvious pitfalls?

[LLMs won’t fix this, they’ll just change the failure modes.]

These days, I can imagine a significant proportion of the people ...

2026-08-10T21:13:41Z

In reply to nevent1q…7r6n
_________________________

These days, I can imagine a significant proportion of the people who wield political or cultural power watching the whole of the movie and still not understanding that “going up to eleven” is not measuring something real.

Breaking: Mark Zuckerberg reveals that Meta plans to create ...

2026-08-10T11:49:34Z

Breaking: Mark Zuckerberg reveals that Meta plans to create Artificial Super Intelligence by redrawing the scale on the Intelligence knob so it goes up to 11.

A conference in honor of Jean-Pierre Serre on the occasion of his ...

2026-08-09T21:14:18Z

A conference in honor of Jean-Pierre Serre on the occasion of his 100th birthday will be held in Paris on September 15 and 16, 2026.

Speakers: Pierre Deligne, Ramon van Handel, Peter Sarnak, Maryna Viazovska, Don Zagier and possibly Jean-Pierre Serre.

Venue: Institut Henri Poincaré, 11 Rue Pierre et Marie Curie, 75005 Paris.

https://www.reddit.com/r/math/comments/1qbgrm4/serre_100_a_conference_in_honor_of_jeanpierre/

No laser? No problem? “Here, we demonstrate, for the first ...

2026-08-07T06:50:57Z

No laser? No problem?

“Here, we demonstrate, for the first time, that sunlight … can produce entangled-photon pairs via [spontaneous parametric down-conversion] despite its lack of optical coherence.”

https://www.scientificamerican.com/article/physicists-harness-sunlight-for-quantum-entanglement/

Open-access paper: https://opg.optica.org/optica/fulltext.cfm?uri=optica-13-8-1508

The form says “Main occupation during working life”, so not ...

2026-08-07T01:00:30Z

In reply to nevent1q…mwhw
_________________________

The form says “Main occupation during working life”, so not “retired”, but “main” is open to some ambiguity (most hours in total? best paid? most fulfilling?)

Not sure why I need to provide my parents’ occupations on the ...

2026-08-06T12:27:47Z

Not sure why I need to provide my parents’ occupations on the form for donating my body to the local medical school. It’s not as if we’re living in a medieval village where John, son of the blacksmith John, needs to be carefully distinguished from John, son of the innkeeper John.

If we define the radius of a circle on a sphere as being the ...

2026-08-06T10:59:29Z

In reply to nevent1q…z4df
_________________________

If we define the radius of a circle on a sphere as being the minimum of the two possible choices (i.e. we always choose the radius that is at most π/2) then the average will be less than π/2, i.e. 7/6.

We *could* pose the question in a different way that made a great circle the average, but if we actually pick 3 random points and always choose the smaller radius, we get 7/6.

If the 3 points are close to lying on a great circle, they will ...

2026-08-06T10:47:11Z

In reply to nevent1q…k2tu
_________________________

If the 3 points are close to lying on a great circle, they will be close to lying on a great circle, which has radius 1 measured in the ambient 3-dimensional space, rather than π/2 as measured on the surface.

But I’m not sure what point you are trying to make.

There was a viral joke recently about the 3 largest cities in ...

2026-08-06T10:33:46Z

There was a viral joke recently about the 3 largest cities in Germany all lying on a circle … but whether that made you laugh or groan, it gives me an excuse to mention 3 non-trivial things about the circle through 3 points on 3 kinds of surface.

[1] If you pick 3 points at random on a unit sphere, what is the average radius (measured along the surface) of the circle they define?

If we integrate from r=0 to π/2:

Mean(r) = ∫ r sin³r dr / ∫ sin³r dr
= 7/6

[2] What happens on a flat surface? Here, there is no upper bound on the radius, and it turns out that it grows so high for nearly-colinear points that the average becomes infinite.

[3] For a region in the hyperbolic plane, there is a further twist: sometimes 3 points *don't* lie on any circle, but a different curve known as a hypercycle.

This is easiest to see if we model the hyperbolic plane as one surface of the hyperboloid −𝑡²+𝑥²+𝑦²=−1. The image shows 2 cases, where the plane through 3 points intersects the hyperboloid either in a closed curve or an open one, with the latter corresponding to a hypercycle.

If we pick 2 points at random in a disk in the hyperbolic plane with radius ρ around our 3rd point, what is the probability that those 3 points *will* lie on a circle?

This turns out to be:

\[ \frac{\text{csch}^4\left(\frac{\rho }{2}\right) \left(-30 \sinh \left(\frac{\rho }{2}\right)+2 \sinh
\left(\frac{3 \rho }{2}\right)+36 \tan ^{-1}\left(\tanh \left(\frac{\rho }{4}\right)\right)+3 \pi
\right)}{12 \pi }-\frac{\left(\text{csch}^4\left(\frac{\rho }{2}\right)-4\right) \cos ^{-1}\left(1-2
\text{sech}^2\left(\frac{\rho }{2}\right)\right)}{4 \pi } \]

which starts at 1 for ρ=0, falls to about 3/4 at ρ=1, and approaches 0 as ρ→∞.

Data centres powered by gas or coal should be 100% steampunk, ...

2026-08-06T05:36:50Z

Data centres powered by gas or coal should be 100% steampunk, full of hissing, clanking Babbage engines, tended by grimy-faced urchins who risk being crushed between the gears as they clear out all the sparrow's-nest malware with chimney-sweep brushes.

https://www.abc.net.au/news/2026-08-06/ai-data-centres-gas-northern-territory/107002860

The fatal Hendra virus was an infamous spillover from bats to ...

2026-08-03T10:19:41Z

The fatal Hendra virus was an infamous spillover from bats to horses to humans in Australia in the 1990s. Since then, ecologists, bat experts and virologists have learned a lot about these events, perhaps including mitigating strategies.

“The findings suggested that planting eucalyptus trees that feed bats in winter, when food elsewhere is scarce, could help prevent spillover by drawing bats from all over the country. When there are rare pulses of flowering in remnant winter habitat, the bats leave the cities and farms, and go back to native habitat. Every time this happened in the 25-year study period, there were no spillover events. A pulse of winter flowers rich enough to draw more than 100,000 flying foxes to an individual roost would substantially reduce the risk of spillover, the researchers concluded. And it is possible that a sufficiently large tract of eucalypts could protect the entire country from the virus.”

https://www.theguardian.com/science/2026/aug/03/hendra-virus-australia-sentinels-book-extract-animal-to-human-disease-spread

“All previous identifications of asteroids with a neck have ...

2026-08-03T07:04:14Z

“All previous identifications of asteroids with a neck have been interpreted as ‘bilobates’, having just two lobes, but (44) Nysa has a unique ‘trilobate’ structure.”

https://www.eso.org/public/videos/potw2631a/

I quite like “Sugar” but honestly they could have just made ...

2026-08-01T12:01:33Z

I quite like “Sugar” but honestly they could have just made him a quirky cinephile detective with some other secret than [redacted as might still be a spoiler for some people], which is completely ridiculous and has no effect on 90% of what happens.

I also loved “White Noise”! But then I read “Ratner’s ...

2026-07-30T10:19:31Z

In reply to nevent1q…mvrc
_________________________

I also loved “White Noise”!

But then I read “Ratner’s Star”, which seemed like an attempt to satirise the practice of science by someone without the slightest idea what scientists did, and I hated it so much I didn’t read any more of his books ... until now, decades later.

I didn’t like “Zero K” as much as “White Noise”; it’s nowhere near as sharp and funny. But it’s readable, and not too long, and has a kind of tone and vision that I could enjoy and appreciate.

When setting a default name for the PDFs that you're ...

2026-07-30T07:22:01Z

When setting a default name for the PDFs that you're encouraging your customers to download for their records, try to aim for the Goldilocks zone that lies somewhere between Mercury ("getPDF.pdf" for every file) and the Oort Cloud ([64 hex digits guaranteed unique across the universe, and utterly meaningless to humans].pdf)

This is a great account of the perils of trying to reach ...

2026-07-29T23:37:34Z

This is a great account of the perils of trying to reach statistical conclusions from scans of old astronomical photographic plates, when the noise and artifacts in the data turns out to be very far from uniformly distributed.

https://www.iheart.com/podcast/105-daniel-and-kellys-extraord-29862087/episode/did-aliens-put-satellites-into-earths-338322285/

I liked the coinage “veer” in Don DeLillo’s Zero K for an ...

2026-07-29T23:23:54Z

I liked the coinage “veer” in Don DeLillo’s *Zero K* for an elevator that moves sideways.

After recent observations with the VLT: “While there was ...

2026-07-28T12:44:45Z

After recent observations with the VLT: “While there was evidence for the existence of this companion star, including a possible direct detection with the Gemini North Telescope in Hawaiʻi, USA, this is the strongest evidence for, and clearest image yet of, Betelgeuse B.”

https://www.eso.org/public/news/eso2611/

The formal definition of the Burau representation is a bit ...

2026-07-27T08:27:57Z

In reply to nevent1q…am56
_________________________

The formal definition of the Burau representation is a bit daunting, but Vaughan Jones gives a vivid intuitive description. If you think of a braid as a bowling alley with n lanes that can cross over and under each other, and at every crossing there is a probability t of a ball in the upper lane falling down to the lower lane, the (i,j) entry in the Burau matrix is just the probability that a ball starting in lane i exits in lane j.

[This metaphor only works for braids that can be built by stringing together any sequence of the “standard generators” where strand i crosses over strand i+1. To get the whole group you need to throw in the inverses of those generators as well.]