On the Flat Folding of Origami (Developments of Language, Logic, Algebraic system and Computer Science)

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(1)30. 数理解析研究所講究録第2051巻 2017年 30-31. On the Flat. Tomoko. Adachi,. Folding. Kaisei. of. Origami. Ishibashi, Kohei. of Information. Kume. Department Sciences, University 2‐2‐1 Miyama, Funabashi, Chiba, 274‐8510, Japan E‐mail:. Toho. adachi@is.sci.toho‐u.ac.jp. 1. Introduction. Japan, Origami is one of the most popular toys with a paper for children. Hence, we have been very familiar to Origami. On the other hand, many studies of Origami have been mathematically re‐ searched. In this paper, we introduce some results on the flat folding of Origami. In particular, we make mention of single vertex fold and multiple vertex folds ([1], [2]). Moreover, we touch upon new study about mathematical model of flat origami ([3]) In. Single Vertex Fold. 2.. We start with the. single one. simplest case. The simplest case for flat origami folds is a single vertex fold to be a creases pattern with only. vertex fold. We define. vertex in the interior of the paper and all. There. two famous theorems for. are. (Maekawa. Theorem 2.1 creases. vertex. and Justin. and V be ihe number. fold. Then. a. single. (1987)). of valley. crease. lines incident to it.. vertex fold. follows.. as. Let M be the number. cTeases. adjacent. to. a. of mountain vertex in a single. it holds M-V=\pm 2.. (Kawasaki(1989), Justin(1989)). Theorem 2.2 a. single. creases.. vertex. Then. fold v. is. and let $\alpha$_{1}, $\alpha$_{2}, a. flat. vertex. \cdots,. Let. v. be. a. of degree 2n in angles between the. vertex. $\alpha$_{2n} be the consecutive. fold if and only if. $\alpha$_{1}-$\alpha$_{2}+$\alpha$_{3}-$\alpha$_{4}+\cdots -$\alpha$_{2n}=0.. 3.. Multiple. Vertex Folds. Next, we describe some results of multiple vertex folds. workshop, we mention the proof of Theorem 3.1.. In. our. talk at this.

(2) 31. Theorem 3.1. (Hull (1g94) Ỉ21). (resp. (resp. valley) creases, U (resp. D ) denote the number of up (resp. down) vertices, and M_{i} (resp. V_{i} ) denote the number of interior mountain (resp. valley) creases. Then it holds. V). ,. denote the number. of. Given. a. multiple. vertex. flat‐fold, let. M. mountain. M-V=2U-2D-M_{i}+V_{i}.. (Kawasaki (1997)_{f} Justin(1997)). Theorem 3.2. Let. us. denote. R(m_{i}). to be the. reflection in the plain, along a line m_{i} Given a multiple vertex fotd, let $\gamma$ be any closed, vertex‐avoiding curve drawn on the crease pattern which crosses crease lines m_{1}, m_{2} m_{n} in order Then, if the crease pattern can fold flat, we have .. ,. ,. ,. R(m_{1})\times R(m_{2})\times\cdots\times R(m_{n})=I Where I denotes the. identity transformation.. 4. Mathematical Model of Flat. Finally,. we. touch upon hot. Origami. study about mathematical. model of flat. origami. ([3]). A is. a. compact. having. set. the. interior, and f. is. a. fold line. A is divided into. the parts A_{1}, A_{2} by f Let us denote R_{f} to be the reflection in the a line f Then, all points x\in A is mapped as follows: .. plain, along. .. (f)x=\left\{ begin{ar ay}{l} R_{f}(x)&(x\inA_{1})\ x&(x\inA_{2}) \end{ar ay}\right.. A_{1}, A_{2} we define A_{2}<A_{1} using a partial order <. Using such a partial order, Nosaka try to describe mathematically several flat foldings of Origami, and give strictly mathematical proofs of above theorems. For. ,. References. [1]. Thomas C. Hull. (2002);. The Combinatorics of Flat Folds: A Survey Origami3: Meeting of Origami Science, Mathematics, and Educa‐ Hull, Editor, A K Peters, Natick, Massachusetts, pp. 29‐38.. Third International. tion, Thomas. [2]. Thomas C. Hull. (1994);. Numerantium, 100,. [3]. Kosuke Nosaka. On the mathematical of flat. origamis Congressus. pp. 215‐224.. (2016);. Mathematical model of flat. Session, Conference 2016, Origami‐Based Modeling and 12, 2016. International. origamis arts, Poster Modeling and Application Analysis, Meiji University, November, 9‐. on. Mathematical.

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