Design note: the shadow of a higher-dimensional grid

The geometry on this site is inspired by the 2026 disproof of Erdős's unit distance conjecture. A construction in a higher-dimensional space becomes an unexpected arrangement of points in the plane. The figure on our home page is a small, interactive illustration of that change of perspective—not a rendering of the proof.

The problem

Place $n$ points in the plane. How many pairs can be at distance exactly $1$? In 1946 Erdős showed that a carefully scaled square grid achieves

$$u(n) \ge n^{1 + c/\log\log n}$$

and conjectured that this is essentially optimal: $u(n) \le n^{1+o(1)}$. That stood for eighty years.

The disproof

On May 20, 2026, OpenAI announced a proof generated by an internal model and checked by mathematicians. The primary manuscript constructs a high-dimensional arithmetic lattice and projects it into the plane. A human-authored companion paper explains and generalizes the argument. Will Sawin's explicit lower bound, submitted the same day, shows that arbitrarily large point sets can have more than

$$n^{1.014}$$

unit distances, using algebraic number fields of large degree and small discriminant.

Why it fits a research lab

Changing the representation can make hidden structure visible. Our illustration keeps the underlying grid fixed while changing its projection. It is a reminder to distinguish the thing being studied from the way it is represented.

The catalogue

OpenAI's openai/math collection separately inspired the publication index: stable numbered families, visible materials, and citations alongside results. It contains work at different verification stages; inclusion is not a blanket certification. Our source labels likewise describe what a reader can inspect, not independent verification. Changes are recorded in the release history. The original unit-distance disproof is linked above, not attributed to this repository.

What the home page draws

The animation is a toy version of the idea, not the actual construction. Take the grid $\{0,\dots,N-1\}^d$ and choose unit vectors $u_1,\dots,u_d$ in the plane. Send each grid point $a$ to

$$\pi(a) = \sum_{j=1}^{d} a_j\, u_j .$$

Grid neighbours differ by exactly one $u_j$, so every drawn edge has length exactly $1$ in mathematical units. The whole figure uses a common screen scale. The red segment marks one such edge. You can pause the motion or change the projection with the slider; the edge lengths remain fixed.

With $d = 4$ and $N = 3$, the grid has $81$ vertices and $4 \cdot 3^3 \cdot 2 = 216$ edges. Projected vertices can coincide at special angles, so these are counts of the source grid, not a claim about distinct planar points at every slider position. This toy construction does not achieve the theorem's superlinear bound. That requires the special arithmetic lattices described in the papers.

@misc{tbc_unit_distance,
  title = {Design note: the shadow of a higher-dimensional grid},
  author = {{TBC Research}},
  year = {2026},
  note = {F-010, catalogue version 2},
  howpublished = {\url{https://tbcresearch.org/research/unit-distance/}}
}