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By Shizuo KAJI

This work was supported by Japan Science and Technology Agency (JST) PRESTO Grant Number JPMJPR16E3, Japan.

• Kaleidocycle.mw is the main Maple script that computes the shape and motion of Kaleidocycles satisfying various conditions. It has some visualisation features including creating animation and plans of paper models.
• Kaleidocycle.nb is a similar script in Mathematica but comes with less functionality.
• 3d_model directory contains 3D printable models. They are modelled by Kamagata Design Studio
• paper_model directory contains paper models foldable by hand.
• hexagon directory contains a paper model of the hexaflexagon, a two-dimensional cousin of Kaleidocycle.

The shape is patented, but please feel free to use any of the material here for personal use.

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

For details, please look at the following materials (click the title to get the document)

• (presentation slides) Geometry of Kaleidocycles presented at the Kyushu-Illinois Strategic Partnership Colloquia Series #2 Mathematics Without Borders - Applied and Applicable, 11 Mar. 2021
• (preprint) Shizuo Kaji, Kenji Kajiwara, Shota Shigetomi, An explicit construction of Kaleidocycles
• (paper) Shizuo Kaji, Kenji Kajiwara, Hyeongki Park, Linkage Mechanisms Governed by Integrable Deformations of Discrete Space Curves, in Nonlinear Systems and Their Remarkable Mathematical Structures, Volume 2, pp 356--381, CRC Press, 2019
• patent publication
• (older conference abstract) Shizuo Kaji, A closed linkage mechanism having the shape of a discrete Mobius strip, the Symposium of the 2018 Spring meeting of the Japan Society for Precision Engineering, Tokyo, 17 Mar. 2018.
• (older conference presentation) Shizuo Kaji, Eliot Fried, Michael Grunwald, Johannes Schoenke, Geometry of closed kinematic chain, IMI Workshop Mathematics in Interface, Dislocation and Structure of Crystals, Nishijin plaza, Fukuoka, 29 Aug. 2017
• (in Japanese) 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.
• (in Japanese) 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, 2019.
• (in Japanese) 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, 2021.
• (presentation slides in Japanese) Kaleidocycle, 13 Mar. 2021

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.