About this figure
The two views
Each of the 625 dots is built from four basic directions in the plane:
the horizontal and vertical of a square grid, plus two more axes
tilted into it. Combine these four directions in integer steps from
−2 to +2 (that's 54 = 625 combinations) and you get the
cloud of points you see.
Two ways of connecting the dots:
- Unit distance. A line is drawn between every pair
of dots that sits exactly the same distance apart — pick
any pair joined by a line and the gap between them is identical.
Lines are coloured by how far that pair lies from the centre.
- All pairs ≤ √3. Also include pairs that sit
slightly farther — up to about 1.73 times the unit. This reveals
the next "shells" of distances underneath the first view. Lines
are coloured by length.
The breathing animation sweeps a soft window through the colour
spectrum. Edges of similar length light up together — short ones
first, then progressively longer — so each ring of distances reads
as its own hidden sub-pattern.
The 80-year-old question
In 1946 the Hungarian mathematician Paul Erdős asked something
deceptively simple: scatter n points on a sheet of paper —
how many pairs can be exactly the same distance apart?
Random arrangements give very few. Clever ones — like a square
grid — give many more. Erdős guessed there was a hard ceiling:
roughly n times a barely-growing factor, and no more. For
eight decades that was the consensus, and everyone working on the
problem assumed Erdős was right.
What the paper proves (May 2026)
A new result,
Planar Point Sets with Many Unit Distances
(OpenAI), shows Erdős was wrong. They construct point arrangements
whose count of same-distance pairs grows strictly faster than
Erdős's ceiling — by a fixed margin that doesn't fade away as
n gets large.
The trick: don't stay flat. Build a very high-dimensional symmetric
arrangement first, using deep machinery from number theory, then
project it down onto the plane in a way that preserves many of the
original length-1 relationships. The toy figure here uses just four
directions in two dimensions; the paper uses arbitrarily many
directions in arbitrarily many dimensions. Even this small
four-direction version already produces many more unit pairs than
a plain square grid does.
The paper notes that the proof was first produced in a fully
automated way by an internal AI model, then verified, simplified,
and rewritten by human mathematicians.