@roryreckons.bsky.social writes:

"The question "What do 2 and 7 have in common?" on autism tests is extremely interesting in the responses, and relates to differences in reasoning. I will discuss these in this thread, but think about it for a second.

1) The most common neurotypical response is that they are both numbers, which relies on broad categorisation.

2) Autistic people usually discuss either shape, their mathematical significance (primes, factors of 14, roman numerals), rotation, angles etc instead of the most broad category.

When asked this question, it never occurred to me that they would be asking about whether they are numbers. It's obvious they are numbers to me, that isn't where I would start to look for similarities.

Autistic people tend to use bottom-up processing, and neurotypicals top-down. So, the neurotypical expectation if someone asked you this question would be "they are numbers"

If you replied with something else, they would likely indicate that this showed a deficit on your part for not picking the most "obvious" answer, or imply you are being pedantic. But I feel like it's generally insulting to my intelligence to assume that I didn't already know they are numbers, and that pointing that out might be insulting the intelligence of the person I was talking to also."

As for me, "they're both numbers" was way too obvious to dream of answering that way. Like:

Q: What do a duck and platypus have in common?
A: They're both animals.

Duh!

Since I'm a mathematician, it's also rather dull to say 2 and 7 are both prime. It's acceptable - but are too many primes for it to be actually interesting.

So I set out looking for a better answer.

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Oh-oh.

"The crux of our work is to demonstrate that advances in AI enable a standalone, self-replicating malware that is able to propagate across heterogeneous computer networks with no prior assumptions about host vulnerabilities, network topology, or system configurations. Existing works on cyber-offence predominantly use frontier models that excel at both logical reasoning and decision-making. The compute footprint of these models however precludes them from being integrated into a standalone malware, whose operation is independent from the availability of a model provider. We thus turn to smaller, open-weight models."

More geometry in music: the 'cube dance'. This shows all the chords called major, minor and augmented triads. Two of these chords are connected by an edge if they differ by just one note. We get 4 cubes connected at their corners!

Augmented triads are choke points: if you want to get from one cube to another, you have to go through an augmented triad.

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The rhombicosidodecahedron is an Archimedean solid with 12 pentagons, 20 equilateral triangles, and 30 squares as faces. You can draw it with all its vertices having coordinates of the form a + b√5 with a,b rational. If you replace √5 by -√5, you get something kind of wonderful! The squares stay square. The triangles stay equilateral triangles. But the pentagons turn into pentagrams!

Here I've only drawn the pentagrams, because it gets a bit messy otherwise.

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The new CEO of Wikipedia worked at J.P. Morgan and Lehman Brothers. The Wikimedia Foundation has now fired the a longtime lead developer and disbanded the team whose job was to listen to volunteers. Most of the people they fired were union organizers. Wikipedia’s editors are now threatening to strike in solidarity. To stand in solidarity with them, sign the petition:

https://en.wikipedia.org/wiki/Wikipedia:Wiki_Workers_United_solidarity

For more, read on!

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"Every city in the Midwest has a half-finished canal in it."

I love canals, so this makes me want to do a tour of the midwest US and its half-finished canals - and a boat tour of the *finished* canals. But I'm in Scotland so... let me do it online.

First up: Indianapolis!

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[image from https://mathstodon.xyz/@[email protected]/116635972190246192]

Bad at lab work? Don't give up hope.

The man who discovered the electron through some of the most delicate vacuum-tube experiments ever performed was a complete disaster with his hands.

My friend's son was a math prodigy, and now he has one-upped OpenAI, which recently wowed the world by disproving a conjecture due to Erdos. OpenAI found a way to put n points on the plane with more pairs at distance 1 from each other than anyone thought possible. Will Sawin quickly found a way to get even more pairs at distance 1.

Math is *not* mainly about solving problems that were already posed. Machines may beat humans at that. I'm glad Will is keeping humans in the competition. But math is about exploration and understanding... as Will knows.

https://arxiv.org/abs/2605.20579

Wow: you can knock out a plant with anesthetics - the same anesthetics that work on people!

It's easist to see for plants that move, like a Venus fly trap. But experiments have shown it's true for others too.

We're still struggling to figure out what this means. We don't really know how anesthetics work, but here's a clue: you don't need to have neurons to get anesthetized!

Marie Curie died of aplastic anemia in 1934. Her daughter Irène, who also studied radiation, died of leukemia in 1956. Irène's husband Frédéric followed two years later, also of leukemia, in 1958. Three Nobel laureates, all dead of radiation-induced blood cancers.

Marie Curie's notebooks and papers are so radioactive that they may only be opened by people wearing protective suits - and they're stored in lead-lined boxes at the Bibliothèque Nationale.

Yet again, Mastodon helps me crack my math problems.

I never knew what was so great about the "great icosahedron". Now I do.

Take a regular icosahedron whose vertices have coordinates in ℚ[√5]. Apply the nontrivial element of the Galois group: that is, replace √5 by -√5. You get the great icosahedron, shown below!

More precisely, take the regular icosahedron whose 12 vertices are

(±1,±Φ,0), (0,±1,±Φ), (±Φ,0,±1)

where Φ = (1 + √5)/2 is the golden ratio. Take

points (vertices of the icosahedron),
lines (containing edges of the icosahedron),
and
planes (containing faces of the icosahedron)

in ℚ[√5]³.

Now replace √5 by -√5 in all your formulas. You get the equations for new

points (vertices of a great icosahedron),
lines (containing edges of this great icosahedron),
and
planes (containing faces of this great icosahedron).

So a Galois transformation makes the icosahedron "great"!

If you apply the Galois transformation again you get back the icosahedron, and if you apply it yet again you make the icosahedron great again.

I thank @nprofile1q... and @nprofile1q... for their help on this.