"For decades, mathematicians believed there was a strict upper limit to how fast those equal-distance pairs could grow.
But OpenAI’s internal model found something unexpected:
There may be no universal limit at all.
Using high-dimensional geometry and abstract number structures far beyond human visual intuition, the AI generated hundreds of pages of rigorous reasoning showing you can create vastly more unit-distance relationships than experts ever thought possible.
Why this matters:
It challenges one of the longest-standing open problems in discrete geometry.
It shows AI isn’t just solving known problems faster it’s discovering entirely new mathematical territory.
It raises the real possibility that machines can now explore abstract realities humans never even thought to search for..."
Well according to the AI, "the confusion comes from mixing up two different questions. The problem is not asking for the maximum number of points where every pair is the same distance apart (that is indeed 3 in the plane, an equilateral triangle). Four points with all pairs equidistant requires 3D space. The actual Erdős unit distance problem asks:
Given n points in the plane, what is the maximum number of pairs that can be exactly distance 1 apart?
The recent AI breakthrough found new constructions showing you can achieve significantly more unit-distance pairs than previously thought possible, a polynomial improvement (roughly n^{1+δ} for some δ > 0). This is considered a genuine advance in discrete geometry."
AI found genuinely new mathematical structures in an area where human intuition (rooted in low-dimensional geometry and simple lattices) had been stuck for 80 years.
"It’s saying you can arrange large numbers of points to create far more unit-distance connections than experts thought possible. "
I still do not get it. Put arbitrarily 10000 of unit distance pairs. We have created 10000 unit distance connections. So where is the problem. I guess AI is not very good at stating the problem precisely. Or maybe my glasses are not strong enough to read what is written?
You have a vast night sky, and you can place as many stars as you want (n stars).
The question is not: “How many stars can I place so that every single pair of stars is exactly the same distance apart?”
(That’s only possible with 3 stars in a plane, an equilateral triangle.)
The actual question is:
“How many pairs of stars can be exactly the same distance apart?”
With a small number of stars, it’s easy.
With 100 stars, you might create a few hundred such pairs.
With 1,000 stars, maybe a few thousand pairs.
For 80 years, mathematicians believed there was a relatively strict upper limit on how fast that number could grow as you add more and more stars (millions, billions, etc.).
The AI breakthrough found new patterns for arranging enormous numbers of stars so that the number of exact same-distance pairs grows significantly faster than those old upper limits allowed.
It discovered new mathematical structures that let you create way more “same-distance” connections in the sky than experts thought was possible.
On X: 'A geometry problem posed by legendary mathematician Paul Erdős in 1946 may have just been overturned… by an AI.
The problem sounds simple on the surface:
How many pairs of dots can exist on a flat plane while all being exactly the same distance apart?' https://x.com/CharlesMullins2/status/2058013449213649117
By intuition the answer is 3 - equilateral triangle, And what did AI say?
Strangely enough...
"For decades, mathematicians believed there was a strict upper limit to how fast those equal-distance pairs could grow.
But OpenAI’s internal model found something unexpected:
There may be no universal limit at all.
Using high-dimensional geometry and abstract number structures far beyond human visual intuition, the AI generated hundreds of pages of rigorous reasoning showing you can create vastly more unit-distance relationships than experts ever thought possible.
Why this matters:
It challenges one of the longest-standing open problems in discrete geometry.
It shows AI isn’t just solving known problems faster it’s discovering entirely new mathematical territory.
It raises the real possibility that machines can now explore abstract realities humans never even thought to search for..."
Still I do not see how you can have, say 4 equidistant points on the plane. I am not getting it.
Well according to the AI, "the confusion comes from mixing up two different questions. The problem is not asking for the maximum number of points where every pair is the same distance apart (that is indeed 3 in the plane, an equilateral triangle). Four points with all pairs equidistant requires 3D space. The actual Erdős unit distance problem asks:
Given n points in the plane, what is the maximum number of pairs that can be exactly distance 1 apart?
The recent AI breakthrough found new constructions showing you can achieve significantly more unit-distance pairs than previously thought possible, a polynomial improvement (roughly n^{1+δ} for some δ > 0). This is considered a genuine advance in discrete geometry."
AI found genuinely new mathematical structures in an area where human intuition (rooted in low-dimensional geometry and simple lattices) had been stuck for 80 years.
"It’s saying you can arrange large numbers of points to create far more unit-distance connections than experts thought possible. "
I still do not get it. Put arbitrarily 10000 of unit distance pairs. We have created 10000 unit distance connections. So where is the problem. I guess AI is not very good at stating the problem precisely. Or maybe my glasses are not strong enough to read what is written?
LoL!
Think of it like this:
You have a vast night sky, and you can place as many stars as you want (n stars).
The question is not: “How many stars can I place so that every single pair of stars is exactly the same distance apart?”
(That’s only possible with 3 stars in a plane, an equilateral triangle.)
The actual question is:
“How many pairs of stars can be exactly the same distance apart?”
With a small number of stars, it’s easy.
With 100 stars, you might create a few hundred such pairs.
With 1,000 stars, maybe a few thousand pairs.
For 80 years, mathematicians believed there was a relatively strict upper limit on how fast that number could grow as you add more and more stars (millions, billions, etc.).
The AI breakthrough found new patterns for arranging enormous numbers of stars so that the number of exact same-distance pairs grows significantly faster than those old upper limits allowed.
It discovered new mathematical structures that let you create way more “same-distance” connections in the sky than experts thought was possible.
That’s the advance.
Beautiful Feline!