On the rigidity of regular bicycle (n,k)-gons
Balázs Csikós · Contributions to Discrete Mathematics · 2007
A bicycle (n, k)-gon is an equilateral n-gon whose k-diagonals are equal. In this paper, the order of infinitesimal flexibility of the regular n-gon within the family of bicycle (n, k)-gons is studied. An equation characterizing first order flexible regular bicycle (n, k)-gons were computed by S. Tabachnikov in [7]. This equation was solved by R. Connelly and the author in [4]. S. Tabachnikov has also constructed nontrivial deformations of the regular bicycle (n, k)-gon for certain pairs (n, k). The main result of the paper is that if the regular bicycle (n, k)-gon is first order flexible, but is not among Tabachnikov’s examples of deformable regular bicycle (n, k)-gons, then this bicycle polygon is second order flexible as well, however, it is third order rigid.