Cassification of first-order flexible regular bicycle polygons

Robert Connelly; Balázs Csikós

doi:10.1556/sscmath.2008.1074

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Studia Scientiarum Mathematicarum Hungarica

Volume/Issue:: Volume 46: Issue 1

Cassification of first-order flexible regular bicycle polygons

Authors:

Robert Connelly

Robert Connelly Cornell University Ithaca Department of Mathematics Malott Hall, Room 433 NY 14853 USA

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Balázs Csikós

Balázs Csikós Eötvös University Institute of Mathematics Pázmány Péter sétány 1/C H-1117 Budapest Hungary

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Pages:: 37–46

Online Publication Date:: 17 Oct 2008

Publication Date:: 01 Mar 2009

Article Category:: Journal Article

DOI:: https://doi.org/10.1556/sscmath.2008.1074

Keywords:: Primary 11D61; 11R18; 52C25; Trigonometric diophantine equations; bicycle polygon; floating body problem; rigidity

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A bicycle ( n , k )-gon is an equilateral n -gon whose k -diagonals are equal. S. Tabach-nikov proved that a regular n -gon is first-order flexible as a bicycle ( n , k )-gon if and only if there is an integer 2 ≦ r ≦ n -2 such that tan (π/ n ) tan ( kr π/ n ) = tan ( k π/ n ) tan ( r π/ n ). In the present paper, we solve this trigonometric diophantine equation. In particular, we describe the family of first order flexible regular bicycle polygons.

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Studia Scientiarum Mathematicarum Hungarica

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Frédéric MEUNIER (École des Ponts ParisTech)
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János PACH (Rényi Institute of Mathematics)
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Andrew SUK (University of California, San Diego)
Zoltán SZABÓ (Princeton University)
Martin TANCER (Charles University, Prague)
Gábor TARDOS (Rényi Institute of Mathematics)
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Studia Scientiarum Mathematicarum Hungarica
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