Configurations with Rational Angles and Trigonometric Diophantine Equations

Laczkovich, M.

doi:10.1007/978-3-642-55566-4_26

Configurations with Rational Angles and Trigonometric Diophantine Equations

M. Laczkovich^5,6

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Part of the book series: Algorithms and Combinatorics ((AC,volume 25))

Abstract

A subset E of the plane is said to be a configuration with rational angles (CRA) if the angle determined by any three points of E is rational when measured in degrees. We prove that there is a constant C such that whenever a CRA has more than C points, then it can be covered either by a circle and its center or by a pair of points and their bisecting line. The proof is based on the description of all rational solutions of the equation

$$ \sin \pi p_1 \cdot \sin \pi p_2 \cdot \sin \pi p_3 \cdot = \sin \pi q_1 \cdot \sin \pi q_2 \cdot \sin \pi q_3 . $$

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References

J. Beebee, Some trigonometric identities related to exact covers, Proc. Amer.Math. Soc. 112 (1991), 329–338.
Article MathSciNet MATH Google Scholar
J. H. Conway and A. J. Jones, Trigonometric diophantine equations, Acta Arithmetica 30 (1976), 229–240.
MathSciNet MATH Google Scholar
W. J. R. Crosby, Solution to problem 4136, Amer. Math. Monthly 53 (1946), 103–107.
MathSciNet Google Scholar
H. B. Mann, On linear relations between roots of unity, Mathematika 12 (1965), 107–117.
Article MathSciNet MATH Google Scholar
G. Myerson, Rational products of sines of rational angles, Aequationes Math. 45 (1993), 70–82.
Article MathSciNet MATH Google Scholar
M. Newman, A diophantine equation, J. London Math. Soc. 43 (1968), 105–107.
Article MathSciNet MATH Google Scholar
M. Newman, Some results on roots of unity, with an application to a Diophantine problem, Aequationes Math. 2 (1969), 163–166.
Article MathSciNet MATH Google Scholar

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Authors and Affiliations

Department of Analysis, Eötvös Loránd University, Pázmány Péter sétány 1/C, Budapest, Hungary, 1117
M. Laczkovich
Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, England
M. Laczkovich

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M. Laczkovich
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Editors and Affiliations

Department of Computer and Information Science, Polytechnic University, Six Metro Tech Center, Brooklyn, NY, 11201, USA
Boris Aronov
School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332-0160, USA
Saugata Basu
City College and Courant Institute, 251 Mercer St., New York, NY, 10012, USA
János Pach
School of Computer Science, Tel Aviv University, Tel Aviv, 69978, Israel
Micha Sharir

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Laczkovich, M. (2003). Configurations with Rational Angles and Trigonometric Diophantine Equations. In: Aronov, B., Basu, S., Pach, J., Sharir, M. (eds) Discrete and Computational Geometry. Algorithms and Combinatorics, vol 25. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55566-4_26

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DOI: https://doi.org/10.1007/978-3-642-55566-4_26
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-62442-1
Online ISBN: 978-3-642-55566-4
eBook Packages: Springer Book Archive

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