Classification of rational angles in plane lattices

Dvornicich, Roberto; Veneziano, Francesco; Zannier, Umberto

doi:10.1090/bull/1723

Classification of rational angles in plane lattices
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by Roberto Dvornicich, Francesco Veneziano and Umberto Zannier;

Bull. Amer. Math. Soc. 59 (2022), 191-226

DOI: https://doi.org/10.1090/bull/1723

Published electronically: August 31, 2021

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Abstract:

This paper is concerned with configurations of points in a plane lattice which determine angles that are rational multiples of $\pi$. We shall study how many such angles may appear in a given lattice and in which positions, allowing the lattice to vary arbitrarily.

This classification turns out to be much less simple than could be expected, leading even to parametrizations involving rational points on certain algebraic curves of positive genus.

References

Jack S. Calcut, Gaussian integers and arctangent identities for $\pi$, Amer. Math. Monthly 116 (2009), no. 6, 515–530. MR 2519490, DOI 10.4169/193009709X470416
J. W. S. Cassels, An introduction to the geometry of numbers, Die Grundlehren der mathematischen Wissenschaften, Band 99, Springer-Verlag, Berlin-New York, 1971. Second printing, corrected. MR 306130
J. H. Conway and A. J. Jones, Trigonometric Diophantine equations (On vanishing sums of roots of unity), Acta Arith. 30 (1976), no. 3, 229–240. MR 422149, DOI 10.4064/aa-30-3-229-240
Roberto Dvornicich and Umberto Zannier, On sums of roots of unity, Monatsh. Math. 129 (2000), no. 2, 97–108. MR 1742911, DOI 10.1007/s006050050009
Roberto Dvornicich and Umberto Zannier, Sums of roots of unity vanishing modulo a prime, Arch. Math. (Basel) 79 (2002), no. 2, 104–108. MR 1925376, DOI 10.1007/s00013-002-8291-4
P. Gordan, Ueber endliche Gruppen linearer Transformationen einer Veränderlichen, Math. Ann. 12 (1877), no. 1, 23–46 (German). MR 1509926, DOI 10.1007/BF01442466
E. Lucas, Théorème sur la géométrie des quinconces, Bull. Soc. Math. France 6 (1878), 9–10 (French). MR 1503766, DOI 10.24033/bsmf.124
Henry B. Mann, On linear relations between roots of unity, Mathematika 12 (1965), 107–117. MR 191892, DOI 10.1112/S0025579300005210
Gerald Myerson, Rational products of sines of rational angles, Aequationes Math. 45 (1993), no. 1, 70–82. MR 1201398, DOI 10.1007/BF01844426
Bjorn Poonen and Michael Rubinstein, The number of intersection points made by the diagonals of a regular polygon, SIAM J. Discrete Math. 11 (1998), no. 1, 135–156. MR 1612877, DOI 10.1137/S0895480195281246
J. F. Rigby, Adventitious quadrangles: a geometrical approach, Math. Gaz. 62 (1978), no. 421, 183–191. MR 513855, DOI 10.2307/3616687
W Scherrer, Die Einlagerung eines regulären Vielecks in ein Gitter, Elemente der Mathematik 1 (1946), 97–98.