An Explicit Construction of Kaleidocycles by Elliptic Theta Functions

Abstract

We study a configuration space consisting of ordered points on the two-dimensional sphere satisfying a system of quadratic constraints. We construct explicit periodic orbits in the configuration space using elliptic theta functions. The constructed orbits simultaneously satisfy semi-discrete analogues of the modified KdV and sine-Gordon equations. This configuration space arises as the state space of a linkage mechanism called a Kaleidocycle, and the constructed orbits describe the characteristic motion of the Kaleidocycle. A key consequence of our construction is the proof that Kaleidocycles exist for any number of tetrahedra (Formula presented.). Our approach is founded on the relationship between the deformation of spatial curves and integrable systems. The motion of the mechanism is interpreted as a deformation of a closed discrete spatial curve with constant torsion angle. This provides an explicit example in which an integrable system is solved to generate periodic orbits in a real solution set of polynomial equations arising from geometric constraints.