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PreprintCC BY-SA 4.0arXiv

Crescent configurations in three dimensions

  • Yuda Chen
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Abstract

A crescent configuration of n points in ℝd is a set in general position whose distinct pairwise distances have multiplicities 1,…,n−1. We construct such a configuration in ℝ3 for every n≥3, giving a uniform bound of three for the minimum dimension. The proof uses equally spaced parameters on a short circular helix and Rolle's theorem. We also give explicit integer-coordinate configurations with O(nlog n) bits per coordinate, and prove that every finite initial segment of the sequence (cos j,sin j,2j)j≥0 of length at least three is a crescent configuration.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026crescent,
  title = {{Crescent configurations in three dimensions}},
  author = {Yuda Chen},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.38368},
  eprint = {2610.38368},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.38368},
  note = {CC BY-SA 4.0},
}
RIS
TY  - UNPB
TI  - Crescent configurations in three dimensions
AU  - Yuda Chen
PY  - 2026
DA  - 2026/10/28
PB  - arXiv
DO  - 10.48550/arxiv.2610.38368
UR  - https://arxiv.org/abs/2610.38368
AB  - A crescent configuration of n points in ℝ^d is a set in general position whose distinct pairwise distances have multiplicities 1,…,n−1. We construct such a configuration in ℝ^3 for every n≥3, giving a uniform bound of three for the minimum dimension. The proof uses equally spaced parameters on a short circular helix and Rolle's theorem. We also give explicit integer-coordinate configurations with O(nlog n) bits per coordinate, and prove that every finite initial segment of the sequence (cos j,sin j,2j)_j≥0 of length at least three is a crescent configuration.
ER  -