A crescent configuration is a set of points in Euclidean space with exactly distinct interpoint distances whose unordered-pair multiplicities, in some ordering, are 1, 2, ..., . In dimension , the general-position convention requires no points in a hyperplane and no points on a hypersphere. The distances need not be numerically ordered by their multiplicities (Burt et al. 2016).

For planar configurations, the conditions are no three collinear points and no four concyclic points. Their existence for arbitrary is problem 217 in the Erdős problems collection.

Chen (2026) constructed three-dimensional crescent configurations for every using

where . Pairs with index difference have squared distance

and multiplicity . These distances increase with , and the configuration has no four coplanar or five cospherical points. Integer-coordinate constructions are also supplied. Chen (2026) reports GPT-5.5 Pro assistance in searching for the constructions.

Wang (2026) ruled out a planar crescent configuration with nine points by an exact computation over possible distance patterns. The existence of larger planar crescent configurations remains open.

See also

Erdős Problems, General Position

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References

Burt, D.; Goldstein, E.; Manski, S.; Miller, S. J.; Palsson, E. A.; and Suh, H. "Crescent Configurations." Integers 16, A38, 2016. https://doi.org/10.5281/zenodo.10606286.Chen, Y. "Crescent Configurations in Three Dimensions." 28 Oct 2026. https://arxiv.org/abs/2610.38368.Wang, Y. "The Nonexistence of Nine-Point Crescent Configurations." 27 Oct 2026. https://arxiv.org/abs/2611.03688v1.

Cite this as:

Weisstein, Eric W. "Crescent Configuration." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CrescentConfiguration.html

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