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Kaleidocycle: Geometry of Closed Linkages

Kaleidocycles are closed linkage mechanisms shaped like discrete Möbius bands. Each unit consists of tetrahedral links that flip inside-out by rotating around hinged joints, producing everting motions reminiscent of bubble rings. This repository gathers asset needed to model, simulate, and fabricate these mechanisms: Python tooling, Maple/Mathematica notebooks, paper templates, and 3D-printable parts.

Repository Tour

• python/ — Python scripts (package sources under src/kaleidocycle/, notebooks, examples)
• Kaleidocycle.mw / Kaleidocycle.nb — Maple and Mathematica scripts for symbolic construction, visualization, and template export
• 3d_model/ — STL for 3D printing (models by Kamagata Design Studio)
• paper_model/ — printable templates for hand-folded models
• hexagon/ — hexaflexagon cousin plus a pictorial folding guide for kaleidocycles
• LICENSE — MIT license for the code (see the patent note below)

Quick Start

Python toolkit

1. cd python
2. uv venv .venv && source .venv/bin/activate
3. pip install -e .[dev]
4. pytest (or pytest -k geometry) to run the suite
5. jupyter lab notebooks/FindKaleidocycles.ipynb for interactive exploration

Maple / Mathematica

Open Kaleidocycle.mw (Maple) or Kaleidocycle.nb (Mathematica) to reproduce symbolic computations. Both scripts include visualization helpers and export functions for paper templates and motion studies.

Physical model assets

• Print STL files from 3d_model/ to fabricate kaleidocycles
• Use paper_model/ templates for hand-folded prototypes; the hexagon/ folder also contains instructions

References & Further Reading

Technical publications

• Shizuo Kaji, Kenji Kajiwara, Shota Shigetomi, An explicit construction of Kaleidocycles
• Shizuo Kaji, Kenji Kajiwara, Hyeongki Park, Linkage Mechanisms Governed by Integrable Deformations of Discrete Space Curves, in Nonlinear Systems and Their Remarkable Mathematical Structures, Vol. 2, CRC Press, 2019
• Shizuo Kaji, A closed linkage mechanism having the shape of a discrete Möbius strip, JSPE Spring Meeting Symposium, 17 Mar 2018

Japanese-language articles

• プレスリリース: 現代数学と折紙から生まれた新しい機構「メビウス・カライドサイクル」
• 鍛冶静雄, 動く折り紙「カライドサイクル」をめぐる機構学と幾何学の出会い, 京都大学数学教室同窓会誌, Vol 9, 2025
• 鍛冶静雄, ユークリッド空間への図形の配置と設計への応用
• 鍛冶静雄, 曲線の幾何学から生まれた閉リンク機構, 2018年度精密工学会春季大会 シンポジウム資料集, pp. 62–65
• 鍛冶静雄, 数理のクロスロード／かたちと動きの数理基盤／(1) リンク万華鏡, 数学セミナー 2019年6月号, 日本評論社
• 鍛冶静雄, かたちを算する／おもちゃのかたち, 数学セミナー 2021年1月号, 日本評論社

Patent

The specific shape of the Möbius kaleidocycle is covered by the patent

• Shizuo Kaji, Johannes Schoenke, Eliot Fried, Michael Grunwald, Moebius Kaleidocycle, JP2018-033395 (Japan), 2019JP007314 (PCT), WO 2019167941, filed 27 Feb 2018.

However, feel free to use the material here for personal and educational projects. For commercial use, contact the Japan Science and Technology Agency.

Some of the results described above are also published in

• Johannes Schoenke and Eliot Fried, Single degree of freedom everting ring linkages with nonorientable topology, PNAS 116 (1), 90–95, 2019

Gallery

License

All code and assets in this repository are released under the MIT License (see LICENSE). Cite the references above when using or extending the material in academic work.