An AI just knocked over a famous geometry problem!

This is a major step forward in AI intelligence.

(First published on my Substack where you can get #NerdNews, marvellous maths and general geekery.)

Oh this has got Spence genuinely excited! 

An OpenAI system has just helped crack one of the most famous geometry questions of the century.

And some of the best in the business are certainly impressed.

“There is no doubt that the solution to the unit-distance problem is a milestone in AI mathematics … No previous AI-generated proof has come close to that.” – Tim Gowers, Fields Medallist.

There’s a tiny bit of maths here, but don’t freak out. NerdNews will walk you through this one step at a time. It’s genuinely worth it. Let’s go.

What was the problem, and what did the AI actually do?

Back in 1946, the brilliantly and wonderfully eccentric maths demon Paul Erdős asked a question so simple you could explain it to a ten-year-old.

Draw some dots on a piece of paper. How can you arrange the dots so that the largest possible number of pairs of dots are exactly the same distance apart?

Check these diagrams to see what the big E was asking.

You can arrange 4 dots in a square with 4 equal length sides.

In fact you could squash the square into a rhombus with the shorter diagonal equal to the sides and hey presto you’ve got 5 equal lengths.

Same with 6 dots in a regular hexagon: 6 sides, 6 unit segments, that’s as good as you can do with 6 dots.

That’s as good as you can get with 4 points.

Clearly 6 dots can be arranged in a regular hexagon: 6 sides, 6 unit segments. But there are arrangements of 6 dots out there with 9 equal sides. Check these bad boys out.

One of my favourite examples involves 7 points. Even though we can do better than 11 equal sides, I just love myself a good Moser Spindle!

Basically Erdos wanted to understand how big and complicated these patterns got and the number of equal length connections we could cram into them. Especially as the number of dots on the page got massive.

Erdos did not think the number of possible equal connections grew very fast.

And a lot of people agreed with him … for the last 80 years, until last week.

Ok that’s enough maths, Adam – why is this impressive?

As I said the popular belief was that Erdős was probably right.

As the number of dots on the page got larger even the best configuration of points would only give you “almost” a linear number of unit length pairs.

The new proof shows that’s wrong.

It shows that as the number of points on the page grows, the number of equal lengths in the best diagram curves upwards, a lot faster than a straight line.

But the really cool bit is where the proof comes from.

The AI didn’t just draw better grids on graph paper.

It dived into deep algebraic number theory. This is the mathematics of structures built to understand prime numbers and symmetries. It used these tools to them build point patterns in the plane with far more unit distances than anyone expected.

The companion paper stresses that what’s new is this bridge. Using tools from very abstract number theory to control a simple sounding geometry puzzle about dots and line segments.

“I really couldn’t believe it. I had trouble sleeping. I was just blown away.” – Mark Sellke, OpenAI researcher.

So that’s us done?

So that’s it? Humans and maths are done?

Yeah … nah.

As groundbreaking as this result is, and it is truly a landmark in human Ai collaboration, the proof still needed human expertise at every stage.

We chose the problem. We checked the argument. We simplified it and made sure it fit into the existing landscape of number theory and geometry.

Gowers points out that this is a result that would have deserved a top tier journal slot and mainstream coverage even if a human had done it alone.

And the way it went about getting to the solution, seems almost human to at least one world class Aussie maths whiz.

“Looking at the chain of thought you can see a point where the AI almost realises “wow I might be onto something here”. And then it has to just keep pushing. Exactly the sensation I have had countless times.” — Geordie Williamson, University of Sydney.

At the same time, the tools the AI used weren’t magically new. Instead it found a clever way to deploy existing deep number theory on a problem most people thought they understood.

“The model did not invent something fundamentally new that nobody saw coming. It just executed like an amazing mathematician.” — Sébastien Bubeck, OpenAI in Scientific American.

Harvard’s Melanie Matchett Wood pushes back on the idea that this shows humans are suddenly outclassed.

She argues that if the same group of experts had decided in advance to hunt for a counterexample with the same intensity they later spent checking the AI’s answer, they probably could have found one themselves.

She also points out that current systems are terrible at the social side of mathematics. AI platforms will reuse ideas without credit and present every trick as if they invented it. If a human did that, we’d call it malpractice.

As Wood puts it, the community urgently needs to decide how to use AI in ways that keep proofs rigorous, properly credited and understandable.

Summing it all up.

Yes, this feels like a turning point.

An AI system has helped crack a classic geometry problem in a way that impresses some of the best mathematicians on the planet.

“I think it will be looked back on as quite an important moment in the history of mathematics.” – Tim Gowers.

But it also reminds us that, at least for now, AI not a superintelligent wizard operating on a level no human can understand.

It’s more like a frighteningly persistent colleague who can roam through deep technical machinery, maintain a 20 page argument, and occasionally stumble on a construction that thousands of very smart people have walked past.

And with a human looking over its shoulder, it can achieve something magic!

Further reading;

• OpenAI announcement – “An OpenAI model has disproved a central conjecture in discrete geometry” (includes links to the proof and an abridged chain of thought).

• “Planar Point Sets with Many Unit Distances” – the main 18 page proof from the OpenAI team.

• “Remarks on the disproof of the unit distance conjecture” – explanatory companion by Noga Alon, Thomas Bloom, Tim Gowers, Arul Shankar, Jacob Tsimerman and others.

• Scientific American: “AI Just Solved an 80-Year-Old Erdős Problem, and Mathematicians Are Amazed”