Continuous Flattening and Shapeshifting of Parallelohedra
連続的平坦折り畳みと形状シフト －平行多面体の場合－
Chie Nara (奈良知惠)
Meiji University, MIMS(明治大学先端数理科学インスティチュート)
cnara@jeans@ocn.ne.jp
Can a polyhedron constructed from paper or similar flexible material be flattened without
stretching or cutting? This problem was proposed by E. Demaine et al. in 2001 ([1]). I. Sabitovproved that a polyhedron does not change its volume under flexing, if shapes of the faces are
fixed; so, we need an infinite number of line segments to move creases for changing shapes ofsome faces, leading to a flattened polyhedron. Three methods for continuous flattening of
convex (and some non-convex) polyhedra have been shown by the author et al. (see [2] forexample). We define a continuous flattening process of a polyhedron as a family ofpolyhedra each of which is intrinsically isometric to the original polyhedron and converges to aflat folded state where a polyhedron is permitted to touch itself without self-intersection. Fig. 1
shows a flat folded state of a rhombic dodecahedron.
Recently, Obervelde et al. presented interesting papers [3, 4] which were related to “Snapology” (see [5]).
For a convex polyhedron, remove all faces and attach an excluded prismatic tube to fit the boundary of eachface. The resulting figure may be flexible. If we choose, some faces remained as rigid faces instead ofattaching extruded tubes, and if the resulting figure is transformable to a shape such that the part of the original
polyhedron is flat folded, we say that the model has the shape-shifting property. They showed models withsuch property for 28 space-filling shapes. We investigated parallelohedra (whose typicalrepresentatives are the cube, the hexagonal prism, the truncated octahedron, the rhombic
dodecahedron, and the elongated rhombic dodecahedron) for the property. In this talk, we show
our results, and models in Fig. 2 are examples of shape-shifting ones of a rhombic
dodecahedron.

Fig. 1.

Fig. 2.

Acknowledgement. This research is supported by Grant-in-Aid for Scientiﬁc Research (C)(16K05258).
References[1] Demaine, E.D., O’Rourke, J, Geometric folding, Algorithms, Lincages, Origami, Polyhedra. Cambridge
University Press, 2007.
[2] Itoh, J-i., Nara, C., Vilcu, C. Continuous flattening of convex polyhedral. Comp. Geometry, EGC 2011,
LNCS 7579, Springer (2012) 85-97.
[3] Obervelde, J.T.B. et al. A three-dimensional actuated origami-inspired transformable metamaterial withmultipledegrees of freedom. Nat. Commun. 7, 10929 (2016).
[4] Obervelde, J.T.B., Weaver, J.C., Hoberman, C. Bestoldi, K.Rational design of reconfigurable prismaticarchitected materials. Nat. 541 (2017) 347-352.
[5] Webpage, http://www.boredpanda.com/shapeshifting-paper-puzzle-harvard/Three dimensional measuring of thin plate and membrane
Naoko Kishimoto
Department of Mechanical Engineering, Setsunan University
kishimoto@mec.setsunan.ac.jp
Keyword : surface shape measurement, image measuring method
As a satellite is launched into orbit by a rocket, large space structures such as solar power panels
and antennas must be deployable and modular assembled structures. According to increase inobservation frequency and sensitivity, not only larger but also more precision space structures are
required. Therefore, we have worked to develop new optical surface shape measuring methods
enable to grasp surface shape of large space structures with high precision and high speed on orbit
for future space antenna and telescope These methods are applicable to testing such structures onground. In these methods, we analyse phase values of projected or painted grating patterns on the
structures and perform calibration using a reference plane.
The advantages of our methods are as follows. 1) They realize high density measurement byanalysing the phase of the projected or imparted grating pattern. 2) They realize high speed ant
high precision measurement by calibration using reference plane. 3) They realize wide range
measurement without degrading measurement precision by integrated results of multiple
measurement systems.
In order to construct ultra-light weight large space structures, they must be composed of thin plate,membrane and mesh. In these flexible structures, shapes changed depending on application oftension, dynamics including vibration is complicated, and phenomenon such as wrinkling andbuckling are caused. Therefore simultaneous measurement over entire surface with high precisionand high speed is required. Furthermore it is important that measuring devices and systems mustbe simple and robust as possible considering measurement on orbit. Our measurement method
meets these requirements. In this presentation, we will show some measuring results for thin plateand membrane surface.

Deformation mechanism on gripper bending process of origami forming
Kousuke Terada1,
1

National Institute of Technology Fukushima College, Japan

1. Introduction
Origami forming has big merits on its application to improve the flexibility of structure designs. In order to
get the high quality of products by origami forming, deformation mechanism on bending process using
gripper is investigated in this paper based on measurements and FEM calculations [1].

2. Measurements and FEM calculations
Fig.1 represents FEM model as an example of grooved steel sheets, which consists of 6 types (=2×3,
grooving in bending inner/outer side, long/standard/short grooving length size). The depth of grooving is halfof sheet thickness. The standard grooving length is the same size as thickness. Mises stress distribution in
Fig.3, which is big spring back case, has stress concentration area at all bending arc due to long grooving
length. On the other hand, that of Fig.4, which is small spring back case, has stress concentration area atlimited bending arc. Because grooving length in Fig. 4 is much shorter than that of Fig.3. FEM results show
clearly deformation mechanism on gripper bending.

Fig.1 FEM model for gripper bending.

Fig.3 Mises stress distribution of big spring back case.

Fig.2 An example of bending tests.

Fig.4 Mises stress distribution of small spring back case.

3. Conclusions
Deformation mechanism on gripper bending as many typical phenomena (big spring back, spring back less,cracks and wrinkles) can be explained clearly based on measured data and FEM results.

Reference[1] Terada, K., Sato, H., Tokura, S., Hagiwara, I. and Takahashi, S.,” The mechanism of metal sheets bending withgrooving” The Proceedings of the 66th Japanese Joint Conference for the Technology of Plasticity, No.553, 201.

Rep-cubes: Dissection of Cube to Nets∗
Ryuhei Uehara
School of Information Science, Japan Advanced Institute of Science and Technology,
Japan. uehara@jaist.ac.jp
A polyomino is a “simply connected” set of unit squares introduced by Solomon W. Golomb in 1954[7]. Since then, polyominoes have been playing an important role in recreational mathematics (see, e.g.,[5]). In 1962, Golomb also proposed an interesting notion called “rep-tile”: a polygon is a rep-tile oforder k if it can be divided into k replicas congruent to one another and similar to the original (see [6,Chap 19]).
From these notions, Abel et al. proposed a new notion [1]; a polyomino
is said to be a rep-cube of order k if it is a net of a cube (or, it can fold toa cube), and it can be divided into k polyominoes such that each of themcan fold to a cube. If all k polyominoes have the same size, we call theoriginal polyomino a regular rep-cube of order k. We note that crease lines
are not necessarily along the edges of the polyomino. For example, a regular
rep-cube of order 2 folds to a cube by folding along the diagonals of unit
squares. In Figure 1, each√T shape√ can√ fold to a cube, and this shape itself
can fold to a cube of size 2 × 2 × 2 by folding along the dotted lines.
In [1], Abel et al. propose regular rep-cubes of order k for each k =22, 4, 5, 8, 9, 36, 50, 64, and also k = 36gk 0 for any positive integer k 0 and
an integer g in {2, 4, 5, 8, 9, 36, 50, 64}. In other words, there are inﬁnitely
Figure 1: A regular repmany k that allow regular rep-cube of order k. On the other hand, they
cube of order 2 [1].
left an open problem that asks if there is a rep-cube of order 3. We solved
this question negatively. There are no regular rep-cube of order 3. From this result, we imply a weakdichotomy of positive integers k that may allow or not to have regular rep-cubes of order k.
We enumerated all possible regular rep-cubes of order k for small k. We mention that the following
problem is not so easy to solve eﬃciently; for a given polygon P , determine if P can fold to a cube ornot. Recently, Horiyama and Mizunashi developed an eﬃcient algorithm that solves this problem for agiven orthogonal polygon, which runs in O((n + m) log n) time, where n is the number of vertices in P ,and m is the maximum number of line segments that appears on a crease line [8]. We remark that theparameter m is hidden and can be huge comparing to n. In our case, P is a polyomino, and this hidden
parameter is linear to the number of unit squares in P , and hence our algorithm is simpler.
Finally, we investigated non-regular rep-cube. In [1], Abel et al. also asked if there exists a rep-cubeof area 150 that is a net of a cube of size 5 × 5 × 5 and it can be divided into two nets of cubes of size
3 × 3 × 3 and 4 × 4 × 4. This idea comes from a pythagorean triple (3, 4, 5) with 32 + 42 = 52 . We
gave a partial answer to this question by dividing into more pieces than 2. Precisely, we proposed a
general method for any pythagorean triple (a, b, c) with a < b < c to obtain a ﬁve piece solution. Thatis, for any given pythagorean triple (a, b, c) with a < b < c, we construct a polyomino that is a net ofa cube of c × c × c, and it can be divided into 5 pieces such that one of 5 pieces can fold to a cube ofa × a × a, and gluing the remaining 4 pieces, we can obtain a net of a cube of b × b × b. An example forthe pythagorean triple (3, 4, 5) is given in Figure 2, and another one for the pythagorean triple (5, 12, 13)
is given in Figure 3.

0 This paper is a survey of recent results in [1, 11].

1

Figure 2: The set S(3, 4, 5) of ﬁve polyominoes
that folds to (a,b) two cubes of size 3 × 3 × 3
and 4 × 4 × 4, and (c) one cube of size 5 × 5 × 5.

Figure 3: The set S(5, 12, 13) of ﬁve polyominoes
that folds to (a,b) two cubes of size 5 × 5 × 5 and12 × 12 × 12, and (c) one cube of size 13 × 13 × 13.

References[1] Z. Abel, B. Ballinger, E. D. Demaine, M. L. Demaine, J. Erickson, A. Hesterberg, H. Ito, I. Kostitsyna,
J. Lynch, and R. Uehara. Unfolding and Dissection of Multiple Cubes. In JCDCG3 , 2016.
[2] J. Akiyama. Tile-Makers and Semi-Tile-Makers. American Mathematical Monthly, Vol. 114, pp. 602–609,
2007.
[3] Y. Araki, T. Horiyama, and R. Uehara. Common Unfolding of Regular Tetrahedron and Johnson-Zalgaller
Solid. J. of Graph Algorithms and Applications, Vol. 20, No. 1, pp. 101–114, 2016.
[4] E. D. Demaine and J. O’Rourke. Geometric Folding Algorithms: Linkages, Origami, Polyhedra. Cambridge,
2007.
[5] M. Gardner. Hexaflexagons, Probalitity Paradoxes, and the Tower of Hanoi. Cambridge, 2008.
[6] M. Gardner. Knots and Borromean Rings, Rep-Tiles, and Eight Queens. Cambridge, 2014.
[7] S. W. Golomb. Polyominoes: Puzzles, Patterns, Problems, and Packings. Princeton Univ., 1996.
[8] T. Horiyama and K. Mizunashi. Folding Orthogonal Polygons into Rectangular Boxes. 19th Korea-Japan
Joint Workshop on Algorithms and Computation, 2016.
[9] J. Mitani and R. Uehara. Polygons Folding to Plural Incongruent Orthogonal Boxes, CCCG 2008, pp. 39-42,
2008.
[10] D. Xu, T. Horiyama, T. Shirakawa, and R. Uehara. Common Developments of Three Incongruent Boxes ofArea 30. COMPUTATIONAL GEOMETRY: Theory and Applications, Vol. 64, pp. 1–17, 2017.
[11] D. Xu, T. Horiyama, and R. Uehara. Rep-cubes: Unfolding and Dissection of Cubes. 29th Canadian
Conference on Computational Geometry, 2017.

2

An Algebraic Representation of Flat Origamis
平坦に折り畳まれる折り紙の代数表現
Kosuke Nosaka
野坂 康祐
Department of Mathematics, Kyoto University of Education
京都教育大学 大学院数学教育専修

1

Introduction

Imaginary flat origamis, which are suggested in this paper, are algebraic representations of flat origamis which represent
the relation of folding steps configured a pair of operations and partial orders. Flat origamis are not always represented
by composite mappings because there exists methods of folding such that developing some parts of a paper (see Figure 1
right). However, operations can represent developing a folded paper. Moreover we can verify that the Maekawa’s theorem
and Kawasaki’s theorem hold as properties of imaginary flat origamis. Therefore, we can prove them as algebraic theorems.

2

Imaginary flat origamis

An imaginary flat origami is a partially ordered set defined as follows:((fm1 · · · fmnm ) · · · (f11 · · · f1n1 )A, ⪯m )(1)

A is a compact set in R2 and f11 , . . . , f1n1 are folding lines passed the interior of A. They separate some parts A1 , A2 , . . . , Ak1
from the set A. We define an operator (f11 · · · f1n1 ) as a reflection by one of f11 , . . . , f1n1 (or the composite function or theidentity function) in each A1 , A2 , . . . , Ak1 , that is, the image (f11 · · · f1n1 )A represents the state of A folded by f11 , . . . , f1n1 .
Moreover, We define Ai ≺1 Aj or Ai ≻1 Aj for any parts Ai , Aj such that the intersection of the interior of (f11 · · · f1n1 )Aiand (f11 · · · f1n1 )Aj is empty. Then the partially ordered set ((f11 · · · f1n1 )A, ⪯1 ) is a imaginary flat origami.
Next, we consider that the image (f11 · · · f1n1 )A is folded by the lines f21 , . . . , f2n2 . Then f11 , . . . , f1n1 and the developed fonding lines ((f11 · · · f1n1 ))−1 f21 , . . . , ((f11 · · · f1n1 ))−1 f2n2 separate smaller parts B1 , B2 , . . . , Bk2 from the set A thanA1 , A2 , . . . , Ak1 . We define an operator (f21 · · · f2n2 )(f11 · · · f1n1 ) as a reflection by one of f21 , . . . , f2n2 (or the composite
function) in each (f11 · · · f1n1 )B1 , (f11 · · · f1n1 )B2 , . . . , (f11 · · · f1n1 )Bk2 . Moreover, We define Bi ≺2 Bj or Bi ≻2 Bj forany parts Bi , Bj such that the intersection of the interior of (f21 · · · f2n2 )(f11 · · · f1n1 )Bi and (f21 · · · f2n2 )(f11 · · · f1n1 )Bj is
empty. Then the partially ordered set ((f21 · · · f2n2 )(f11 · · · f1n1 )A, ⪯2 ) is a imaginary flat origami.
By induction, we can define (1) as a generalized imaginary flat origami. It can make a paper crane and its 10 folding steps
for example.

Figure 1: Some folding steps of a paper crane3

Properties of Imaginary Flat Origamis

1-vertex foldable flat origamis are imaginary flat origamis which are not broken and have only one point where creases
meet in the interior of the set A. The following statements hold.
Theorem 1 (Maekawa [1]). the diﬀerence in the numbers of mountain creases and valley creases is 2
Theorem 2 (Kawasaki [1]). the sum of the alternate angles of each adjacent creases is π

References[1] T. Hull, On the Mathematics of Flat Origamis, Congressus Numerantium, Vol.100 (1994)

Special Talk

The Central Role of Combinatorics in Origami

Dr. Thomas C. Hull
(Western New England University)
The art of origami is inherently geometrical, and a lot of themathematical research done on origami centers aroundgeometry. However, combinatorics (the mathematics of counting
things) plays a very important role in understanding how paper,or any material, can fold. Maekawa's Theorem is a well-known
example. This talk will describe some of the mathematical results
known (and not known) for counting the number of ways anorigami crease pattern can fold flat, that is, counting the numberof valid ways to assign mountain and valley creases to a crease
pattern. We will highlight the importance of such enumeration
research in recent applications of origami in physics andengineering.

厚みのある剛体折紙
舘 知宏

Thick Rigid Origami
Tomohiro Tachi

剛体折紙はパネルとヒンジによる折紙の機構モデルであり、繰り返し折りたためる展開構造物
や機構のデザインに⽤いられる。通常折紙は厚み 0 のモデルで考えるが、物理的な実装の際には、パネルの厚みを考慮しその⼲渉を解消する必要がある。パネル厚を解決するため、多くの⼿法が提案されてきている。(a)折り線を厚み⽅向にオフセットする⼿法[1,2]では、折り線が⼀点で交わる頂点を、対称性を⽤いた⼀点で交わらない過拘束メカニズムに置き換える。そのため180°の完全な折り畳み状態を実現できるが、対象となるパターンは特定の⼀⾃由度パターンに限られる。またパネルの厚みがパターンに依存すること、ヒンジの位置が同⼀平⾯上にないこと
から製造は多くのパーツの組み⽴てによらざるを得ない。 (b)折り線の位置を移動せずにパネルにテーパーやオフセットを施す⽅法[3,4]は、厚み 0 の折紙のメカニズムを保持して、⼲渉しない部分にのみ厚みを加える仕組みである。最も汎⽤性が⾼く、またヒンジ層を挟み込んだパネル
層を平⾯状態で加⼯するだけで良いため、製造⾏程において折紙の可展性のメリットが⽣かせる。ただし、折り畳み⾓と厚みとの間にトレードオフがあり、特に 180 度に折り畳むことはできない。(c)(b)の改良として、折り線パターン⾃体を、あらかじめ変更し、折り線を⼆重線とする
ことで、折り畳み⾓の制限から逃れる⽅法[5,6]が提案されている。
著者らが提案した[6]の⼿法では適⽤可能なパターンはまだ限られるものの、⾯に⽳を作らずに折り線を⼆重線とできるため、家具・建築などの厚みのある折紙構造の実現⽅法として有⼒であると考える。
[1] C. Hoberman. Reversibly expandable three-dimensional structure. United States Patent No.
4,780,344, 1988
[2] Y. Chen, R. Peng, and Z. You. Origami of thick panels. Science, 349(6246): 396‒400, 2015.
[3] T. Tachi. Rigid-foldable thick origami. In Origami5, pages 253‒263., 2011.
[4] B. J. Edmonson, R. Lang, M. R. Morgan, S. P. Magleby, L. L. Howell, Thick rigidly foldable
structures realized by an offset panel technique, In Origami6, Vol. 1., pp. 149-161, 2015.
[5] J. S. Ku and E. D. Demaine. Folding flat crease patterns with thick materials. Journal of
Mechanisms and Robotics, 8(3): 031003‒031009, 2016.
[6] T. C. Hull and T. Tachi. Double-line rigid origami, 11th Asian Forum on Graphic Science,
Tokyo, August 6-10, 2017.

Application of pairing origami structure to aluminum cans―Comparison of TMP and NP from the viewpoint of rigid folding and crushing force

２枚貼り折りによるアルミ缶適用に関する検討―TMP と NP の圧潰力と剛体折りの観点からの比較

阿部 綾 Aya ABE (Meiji University)
Abstract. Polyhedrons by Nojima and by Tachi-Miura, which both are two symmetrical origami structures,
can be folded in the axial or radial direction, and it is convenient if they can be applied to aluminum cans. We
studied the crushing characteristics of both structures from the viewpoint of rigid folding, and explore their
possibility.

概要．野島ポリへドロンと舘-三浦ポリへドロンは両方とも2枚貼りの対称な折紙構造であり,軸方向にも半径方向にも折り畳むことができる．それらをアルミニウム缶に適用することができれば便利である.我々は,両構造の圧潰特性および剛体折りの観点からの比較を通して,その可能性を探る．

Optimal NP
節点数

20359

Optimal TMP
18261

高さ 200mm，底面完全固定，上面から構造体を圧縮する条件で有限要素法によるシミュレーション結果から圧潰力を小さくするための最適化で得られた 2 枚貼

TMP（上図）は剛体折りのため折り畳むと平坦化するがり折り解析モデルNP（下図）は剛体折りでないため平坦に折り畳めない━ Optimal TMP━ Optimal NP

TMP はヒンジ部分の折り線の多さや剛体折りであるために底面固定部分で圧潰に無理が生じる現象が顕著となることから, 剛体折りでない NP よりも圧潰荷重値が大きく,また底つきが早いと考察される
参考文献

阿部綾,楊陽,陳詩暁,戸倉直,奈良知惠,安達悠子,萩原一郎,“２枚貼り折りによるペットボトル適用に関する検討”,日本応用数理学会論文集に投稿

折紙工学カンファレンス 2017

Origami in Architecture: Concept | Configuration | Construction
折紙的建築術：構想・構法・構成

Yoshinobu Miyamoto, Aichi Institute of Technology
愛知工業大学宮本好信

We will show how origami technique or folding has been used for architecture both in design concepts
and their realization. Folding is a method for manipulation of form, form finding and composition of
spaces. It is useful for adjusting functional spaces to the programming requirements and enhancing
visual effects for facades and interior spaces. Folding is useful also in tectonic aspects in architecture
and building construction. Corrugated metal sheet, folded metal roof, folded plate structure and foldable
building elements are ubiquitous. We look into the future possibility of strategic integration of folding
from the concept to the realization.
建築における折り紙の応用は、構想/企画/意匠の分野と構造/構法/材料の両分野で行われている。
前者では「折り」は形態操作、形態生成、空間構成の手法として、建築企画と機能空間構成の整合、外観や内部空間の視覚的演出のために利用される。後者では機能性・安全性・経済性の具現化手法
として利用され、金属波板、金属折板屋根、折板鉄筋コンクリート構造、展開構造などが普及している。構想から建設まで折り紙を総合的に活用する可能性を展望する。

Endless House (1924,1950,1960)
Frederick Kiesler (1890–1965)
The un-built project demonstrated the continuous
connectivity of spaces or folded spaces.
link

Villa VPRO model (1993–1997)
MVRDV (1993–)
The public broadcasting center’s folded floor was
realization of the integration of the program
requirements and the functional space.

link

3D-Modeling for the Developments of Polyhedra
Takashi Horiyama
Graduate School of Science and Engineering, Saitama University, JapanA development of a polyhedron is a simple polygon obtained by cutting edges or faces of the polyhedron andunfolding it into a plane. While we can realize a development by a paper (i.e., we can draw it on a paper), we
may have troubles when we fold it into a polyhedron. Since it has no thickness, the folded polyhedron isfragile. More precisely, the hinge is flexible enough to be bend with at most 180 degrees, and thus we cannot
fix the dihedral angles between adjacent faces.
To avoid such trouble, we use the technique of rigid-foldable thick origami [1]. By this technique,zero-thickness ideal facets (denoted by red lines) are realized by thick panels: First offset the ideal facets byconstant distance in two directions, and then trim facets by the bisecting planes of dihedral angles between
adjacent facets.
If we trim the facets in the same side, all facets are folded in that side. If two polyhedra have a common
development of the same shape (see e.g., [2], [3]), we can realize it so that it can be folded into the twopolyhedra: We prepare hinges on the place where at least one polyhedron has a folding line. One side of thehinges is trimmed if they correspond to a polyhedron, and the other side is trimmed if they correspond toanother polyhedron.

Reference[1] T. Tachi, Rigid-Foldable Thick Origami, Origami 5 (Fifth International Meeting of Origami Science,
Mathematics, and Education), pp. 253-263, 2011.
[2] Y. Araki, T. Horiyama, R. Uehara, Common Unfolding of Regular Tetrahedron and JZ Solid, Journal of Graph
Algorithms and Applications, vol. 20, no. 1, pp. 101-114, 2016.
[3] T. Biedl, T. Chan, E. Demaine, M. Demaine, A. Lubiw, J. I. Munro, and J. Shallit, Notes from the University ofWaterloo Algorithmic Problem Session, September 8, 1999.

3.

4.

(60-15)(15+15+15+15)(60-15)(15+15+15)(15+15+15)

1.

PIECESASSEMBLAGE(15)ASSEMBLAGE(60)2.

ASSEMBLAGE(3)CONNECT / ASAO TOKOLOASSEMBLAGE(9)ASSEMBLAGE(25)(81)

スリット付き正多面体の裏返し(Reversing of regular polyhedra with slits)
Jin-ichi Itoh (Kumamoto University)(Joint work with Naofumi Horio)
Recently H. Maehara defined an origami-deformation of a polyhedral surface withboundary in Euclidean space, and he showed that every rectangular tube can be
subdivided so that it becomes reversible (it is called s-reversible).
In this talk we discuss on the reversibility of regular polyhedra with several slits aroundvertices. In the case of cube and icosahedron, if we cut them along all edges around
antipodal vertices, we get tubes, then they become s-reversible.
Theorem. Octahedron with slits as Figure 1 is s-reversible. Tetrahedron with slits as
Figure 2 is s-reversible. Cube with slits as Figure 3 is s-reversible.

Figure 1

Figure 2

Figure 3[1] H.Maehara: Reversing a polyhedral surface by origami-deformation, European Journalof Combinatorics 31 (2010), 1171-1180

Development of Origami structure superior to present energy-absorbing
vehicle structure by ultra-cheap forming method
自動車の現行エネルギー吸収材特性を凌ぐ折紙構造体及びその超安価な製造法の開発

Ichiro Hagiwara 萩原一郎 (Meiji University)
Abstract. Current vehicle energy absorbers have two defects during collision, only 70 % collapsed in its
length and high initial peak load. We have found so called Reversed Spiral Origami Structure (RSO) cansolve these defects. However, the manufacturing cost is too high to be applied in real vehicle structure. To
address the problems, a new structure, named Reversed Torsion Origami Structure (RTO), has been developed,
which can be manufactured at a low cost by using simple torsion of Origami engineering. This structure ispossible to replace conventional energy absorbers and expected to be widely used such as in building
structures.

概要．現行の自動車のエネルギー吸収材[1]には、１）初期ピーク荷重が高い、２）自長の70％しか潰れない、の二つの欠点がある。折紙構造でこれらの欠点を解消できることは先に見出していた[2]が製造費が高いという課題があった。ここに、廉価な製造で同様に上の特性を有す構造が得られた[3]。

Fig.1

Load-displacement characteristics
of RSC and conventional one.

Fig.2

Schematic drawing of the three steps in the torsion
forming process: (1) fix the first segment and twist
the third segment through a certain angle, thus
deforming the second segment; (2) move the dies
one segment along the axial direction; (3) repeat step(1), twisting in the opposite direction.

●自動車用衝撃吸収バンパー（クラッシュボックス）：衝突の際に求められる，高エネルギーを吸収し，かつ衝撃荷重を抑えることが可能．
●現行の中空ハット型断面構造より安価に製造可能
なお本研究は、埼玉工業大学の趙希禄教授および同教授研究室員との共同研究の成果である。
参考文献[1] 萩原一郎, 津田政明, 北川裕一, ビードの配置決定方法, 第 2727680 号(1991).
[2] 萩原一郎, 山本千尋, 陶金, 野島武敏, 反転らせん型モデルを用いた円筒形折り紙構造の圧潰変形特性の最適化検討, 日本機械学会論文集 A 編, Vol.70, No.689 (2004), pp.36-42.
[3] 萩原一郎、趙希禄、衝撃吸収体の製造装置および衝撃吸収体の製造方法、特願2017-089216（平成29年4月28日）