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PreprintCC BY 4.0Zenodo

Two Congruence Classes of Extremal Twenty-Point Sets

  • Alper Ferudun1
  1. 1Mercury Software GmbH
Open at sourcePDFDOICite

Abstract

Alexeev, Mixon and Parshall proved that twenty points in the plane have at most 54 unit-distance pairs and that every maximizing set has the same abstract unit-distance graph. We determine its Euclidean realization classes: there are exactly two, even when reflections are allowed. Both are explicit Minkowski sums of the unit diamond and a five-point triangular fan. Cartesian-product squares force every injective realization to have this form, and one additional edge leaves two relative rotations. A fourth distance moment separates the resulting sets. The proof uses the established abstract-graph enumeration and otherwise consists of elementary geometry and exact algebra. This is a finite congruence classification; it does not resolve the asymptotic question in Erdős Problem 668.

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Cite

BibTeXRIS
BibTeX
@misc{ferudun2026congruence,
  title = {{Two Congruence Classes of Extremal Twenty-Point Sets}},
  author = {Alper Ferudun},
  howpublished = {Zenodo},
  year = {2026},
  month = oct,
  doi = {10.5281/zenodo.23440649},
  url = {https://zenodo.org/records/23440649},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Two Congruence Classes of Extremal Twenty-Point Sets
AU  - Alper Ferudun
PY  - 2026
DA  - 2026/10/17
PB  - Zenodo
DO  - 10.5281/zenodo.23440649
UR  - https://zenodo.org/records/23440649
AB  - Alexeev, Mixon and Parshall proved that twenty points in the plane have at most 54 unit-distance pairs and that every maximizing set has the same abstract unit-distance graph. We determine its Euclidean realization classes: there are exactly two, even when reflections are allowed. Both are explicit Minkowski sums of the unit diamond and a five-point triangular fan. Cartesian-product squares force every injective realization to have this form, and one additional edge leaves two relative rotations. A fourth distance moment separates the resulting sets. The proof uses the established abstract-graph enumeration and otherwise consists of elementary geometry and exact algebra. This is a finite congruence classification; it does not resolve the asymptotic question in Erdős Problem 668.
ER  -