An explicit construction of Kaleidocycles by elliptic theta functions

We consider the configuration space of points on the two-dimensional sphere that satisfy a specific system of quadratic equations. We construct periodic orbits in this configuration space using elliptic theta functions and show that they satisfy semi-discrete analogues of mKdV and sine-Gordon equations. The configuration space we investigate corresponds to the state space of a linkage mechanism known as the Kaleidocycle, and the constructed orbits describe the characteristic motion of the Kaleidocycle. Our approach is founded on the relationship between the deformation of spatial curves and integrable systems, offering an intriguing example where an integrable system generates an orbit in the space of real solutions to polynomial equations defined by geometric constraints.