Abstract
Let \({\mathcal {S}}\) be a finite set of integer points in \({\mathbb {R}}^d\), which we assume has many symmetries, and let \(P\in {\mathbb {R}}^d\) be a fixed point. We calculate the distances from P to the points in \({\mathcal {S}}\) and compare the results. In some of the most common cases, we find that they lead to unexpected conclusions if the dimension is sufficiently large. For example, if \({\mathcal {S}}\) is the set of vertices of a hypercube in \({\mathbb {R}}^d\) and P is any point inside, then almost all triangles PAB with \(A,B\in {\mathcal {S}}\) are almost equilateral. Or, if P is close to the center of the cube, then almost all triangles PAB with \(A\in {\mathcal {S}}\) and B anywhere in the hypercube are almost right triangles.
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Anderson, J., Cobeli, C. & Zaharescu, A. Counterintuitive Patterns on Angles and Distances Between Lattice Points in High Dimensional Hypercubes. Results Math 79, 94 (2024). https://doi.org/10.1007/s00025-024-02126-2
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DOI: https://doi.org/10.1007/s00025-024-02126-2