2026-05-19 京都大学
左からそれぞれ6個、9個、8個のパーツで構成されたカライドサイクル。カライドサイクルの本質的な形状は隣り合う蝶番の位置関係のみで決まるので、構成する要素は四面体でなくてもよい。また、紙だけでなくプラスチックや金属でも作ることができる。
＜関連情報＞
- https://www.kyoto-u.ac.jp/ja/research-news/2026-05-19-3
- https://onlinelibrary.wiley.com/doi/10.1111/sapm.70224
楕円シータ関数によるカレイドサイクルの明示的な構成 An Explicit Construction of Kaleidocycles by Elliptic Theta Functions
Shizuo Kaji, Kenji Kajiwara, Shota Shigetomi
Studies in Applied Mathematics Published: 13 May 2026
DOI:https://doi.org/10.1111/sapm.70224
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 k≥6. 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.
