# Distances and packing

> 5 AI-made results on distances and packing, each explained in plain English. How points can be spread out, how often a distance can repeat and how…

Canonical: https://closedform.nishachay.tech/topic/distances-and-packing/
Last updated: 2026-10-14

How points can be spread out, how often a distance can repeat and how tightly shapes can be packed.

5 AI-made results on distances and packing, from 6 papers in the OpenAI math collection, each explained in plain English. 4 have a formal Lean proof listed; 1 is claimed in a paper and not yet machine-checked. They span 3 fields, led by combinatorics (3), differential geometry (1) and theoretical computer science (1). The oldest question here dates back to 1946: [The higher-dimensional Erdős distinct-distances conjecture](https://closedform.nishachay.tech/e/openai-math-2026-166/). Rated Landmark: [Planar distinct distances and unit-distance bounds](https://closedform.nishachay.tech/e/openai-math-2026-167/).

## Results (5)

- [Bin packing and unbounded configuration-LP gaps](https://closedform.nishachay.tech/e/openai-math-2026-118/) - Entry 118 · Rounding up the fractional bin count can be wrong by any fixed number of bins. (Formal proof listed)
- [The higher-dimensional Erdős distinct-distances conjecture](https://closedform.nishachay.tech/e/openai-math-2026-166/) - Entry 166 · Any n points in d ≥ 3 dimensions span at least a constant times n^(2/d) distinct distances, the paper claims. (Claimed, not checked)
- [Planar distinct distances and unit-distance bounds](https://closedform.nishachay.tech/e/openai-math-2026-167/) - Entry 167 · Proves almost every point in a large planar set sees nearly n distances, and lowers the unit-distance exponent 4/3. (Formal proof listed)
- [Counterexamples to infinite matroid intersection and packing/covering](https://closedform.nishachay.tech/e/openai-math-2026-185/) - Entry 185 · Two self-dual infinite matroids break the infinite matroid packing/covering and intersection conjectures. (Formal proof listed)
- [Symplectic ball packing in higher dimensions](https://closedform.nishachay.tech/e/openai-math-2026-343/) - Entry 343 · Proves Siegel and Yao's conjecture: from dimension six, ball packings are limited only by volume and the two-ball test. (Formal proof listed)

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