Solving the continuous folding problems of polyhedral figures under rigidity conditions

2023 Fiscal Year Final Research Report

• Project/Area Number: 20K03726
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12030:Basic mathematics-related
• Research Institution: Meiji University
• Principal Investigator: Nara Chie 明治大学, 研究・知財戦略機構(中野), 研究推進員(客員研究員) (40147898)
• Co-Investigator(Kenkyū-buntansha): 伊藤 仁一 椙山女学園大学, 教育学部, 教授 (20193493)
• Project Period (FY): 2020-04-01 – 2024-03-31
• Keywords: 多面体 / 折りたたみ / 移動折り目 / 連続的平坦化 / 剛性折り / 高次元正多面体 / 星形正多面体 / ひし形の翼折り

Outline of Final Research Achievements

① It is important to develop foldable (collapsible) products that can be opened and closed smoothly and for which manufacturing processes are simple. From these points of view, we have worked on the problem of continuously flattening the surfaces of polyhedra without cutting or stretching. We have found methods to do so for all polyhedra except for some special cases. Combining some methods, we have also solved the problem for the special case of regular star-polyhedra.
② We have given the best possible numbers of rigid faces (or edges) in the continuous flattening of some regular polyhedra. These results will be useful for designing foldable products and temporary house units at the time of disaster.
③ Extending the continuous flattening problem for 3-dimensional polyhedra, we proposed two approaches for higher dimensional polyhedra. Almost half of regular polyhedra have been solved in both cases.

Free Research Field

離散幾何学

Academic Significance and Societal Importance of the Research Achievements

多面体の剛性に関する問いに対して，コーシーによる凸多面体の剛性定理（1813年）はより一般の多面体の研究や平坦化の動きに関する研究，あるいは高次元化へと発展している。多面体を切り込みや伸縮なしで平坦化するためには，どれかの面の形を折り目によって変形し続ける必要がある。このような折り目の動きや占める領域を求めることが問われるようになった。3次元のみでなく高次元の多面体の連続的平坦化の問題にも取り組み，折り目の入らない（剛性）面や辺に注目して，具体的な折りたたみの方法とともに連続的平坦化のプロセスを提示した。折りたたみ式防災用帽子の過去の例と同様に，折りたたみ式製品開発への応用が期待できる。