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1
Superimposition of Rigid Origami Patterns
Xiang Liu1, Joseph M. Gattas2, Yan Chen 1,3*
1School of Mechanical Engineering, Tianjin University, Tianjin 300072, China
School of Civil Engineering, University of Queensland, St Lucia, QLD 4072, Australia
3
Key Laboratory of Mechanism Theory and Equipment Design of Ministry of Education,
Tianjin University, Tianjin 300072, China* Corresponding Author <yan_chen@tju.edu.cn>
2

1. Introduction
Origami-inspired engineering is used to apply origami science and technology to thedesign of engineering structures. The majority of existing self-folding structures either
use a bespoke crease pattern to form a single structure, or a universal crease patterncapable of forming numerous structures with multiple folding steps.
2. Superimposition of Patterns
We propose a fundamentally new method, called superimposition, for generation ofmultiple 1-DOF rigid-foldable configurations from pre-defined crease patterns, such thatkinematic independence and 1-DOF mobility of each individual pattern is preserved.
Sheets can thus be created that contain multiple states with different or complementaryfunctionality. The superimposed patterns can be classified into two families byconsidering the functionality, one is to fold the non-flat foldable patterns into a compact
flat folding (Fig. 1), while the other is to fold the patterns in two steps to achieve muchlarger deployable ratio (Fig. 2). The proposed method has substantial potential for
application in origami-inspired engineering design.

Figure 1 The thick-panel accordion shelter Figure 2 Two folding sequences of thebased on the superimposed Arc and Miura-ori superimposed pattern with different hill andpatterns, with rigid motion between the two valley crease assignment. 
discrete states.

1

2
A Consideration on Industrialization of Origami Structure
Ichiro HAGIWARA 1,
Meiji Institute for Advanced Study of Mathematical Sciences, Japan1

1. Introduction
Recently we have researched about pairing origami structures such as Nojima polyhedron and Tachi-Miura
polyhedron for crash energy absorbers. They are both easy to be manufactured but not so much absorbed
energy. On the other hand, reversed spiral cylindrical origami structure (RSC) has splendid ability for crashabsorbed energy but it is difficult to be manufactured. These structures are shown in Fig.1. The RSC structure
inspires us a new structure which has splendid crash absorbed energy and is easy to be manufactured. This is also
the Origami engineering.

2. Force-Deformation Characteristics of Origami Structure and Its Application
Fig.2 shows that the initial peak force sometimes too large for passengers which is a representative
Force-Deformation curve(F-D curve) for conventional hollow quadrilateral member. On the other hand, RSC
has the ideal F-D curve parallel to the Deformation axis. We used origami structures for the helmet of“amazing skills”contest in the NHK which must be smaller than 300mm×150mm×50mm box and even if theweight 5kg from a height of 1m over the top has fallen in the head, the impact force must be damaged less than 10%
without helmet. Our helmet consists of 3 origami structures shown in Fig.3. Fig.2 is the reason why RSC is
selected among 3 parts. This contest is concerned with Origami engineering and crush problem which areboth my measure. So we had complete victory. RSC anew inspired a splendid structure. And based on thishelmet, the helmet on Fig.4 is on the market.
      (a)       (b)        (c)Crash Box

Fig.1 (a)Nojima Polyhedron ,(b)Tachi-Miura Polyhedron   Fig.2 Crash Box(left) and Its load-displacement(c)Reversed Spiral Cylindrical Origami Structure      Characteristics(right)(a)Accordion Fold(b)RCS(c)Honey comb

Fig.3 Origami-helmet for “Sugowaza” in NHK Television

Fig.4 Helmet on the market

3
Crease Patterns and Mechanisms in Insect Wing Folding
Kazuya Saito1
1Institute of Industrial Sciences, The University of Tokyo

Foldable hindwings in insects are one of the ultimate deployable structures and have large potential forengineering applications, because they have sufficient strength and stiffness to tolerate 20–1000 beats per
second in the flight position, although they can be folded and unfolded instantly depending on the situation.
Among wing folding insects, beetles (Coleoptera) are known to use the most highly diverse wing folding
patterns and mechanisms [1]. Unlike six legs, the wings of insects are considered to be derived from thetergal exoskeleton. Their folding and unfolding mechanisms involve the elastic behaviors of structures,internal forces driven by muscle or blood pressure and external forces from other body parts. Understanding
these mechanisms requires not only analyzing the materials and structures of the wings but also carefulobservations of how they are actuated in living specimens. This study focuses on a group of beetles that haveelastic-deploying-type wings, and aims to reveal their wing-folding/unfolding mechanisms. These beetles use
wing elasticity for deployments and therefore can quickly take off. On the other hand, when it comes to wingstoring, they have to fold their wings against elastic forces. Usually, this process requires the supports ofother body parts: the abdomen and elytra. Note that these folding movements can enable the beetles toachieve two different objectives at the same time: quick wing-folding and efficient storage of elastic energy
for subsequent wing deployment. Therefore, detailed investigations of these movements are expected togenerate useful knowledge in the development of excellent self-deploying systems. This study uses
high-speed cameras to reveal the details of these motions. As a typical example of this type wings, this paper
focuses on ladybird beetles (Coleoptera: Coccinellidae)(Fig. 1). First, on the basis of the reports ofentomologists and our own observations, the factors relating to the actuation of insect wing folding andunfolding are described. Next the detailed motion involved during take-off and wing storing in ladybird
beetles are observed using high-speed cameras. The analysis of these movies reveals the detailed processesof wing-unfolding and folding in ladybird beetles. Finally, the characteristics of wing-folding mechanism aresummarized, and the potentials for engineering applications are discussed. The discussion also includes anoutline of the future works required on the engineering side to reveal the detailed systems of these excellent
deployable structures.

Fig. 1 Wing unfolding motion in a ladybird beetle

Reference[1] Kazuya Saito, Shuhei Yamamoto, Munetoshi Maruyama, Yoji Okabe (2014) Asymmetric hindwing
folding in rove beetles, Proc. Natl. Acad. Sci., 111(46), pp.16349-16352.

4
Characteristics of deformations on assembled structures by origami forming
Kousuke Terada1,
National Institute of Technology Fukushima College, Japan1

1. Introduction
Considering the earth environment and energy problem, the research as lightweight and high strength
structuresshould be important. Assembled structures which can be produced by origami forming, have big
merits on its application to improve the flexibility of structure designs. Characteristics of deformations ofassembled structures are 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 assembled structures, which consists of 10 cores, upper/lower
panel. This A-type has spot weld elements between cores and lower panel. On the other hand, B-type is
continuous structures as upper/lower panel and cores. Fig.2 shows a hydraulic system universal testingmachine to evaluate the bending stiffness by 3 points bending mechanism. Comparison between measured
data and FEM results for relationship with load and deformation shows FEM results are in good agreement
to measured data. Mises stress distribution of A-type (Fig.3) has stress concentration area at the contact area
between panels and cores much larger than that of B-type (Fig.4) and deformations of A-type are overall.

Fig.1 FEM model for assembled structures.
Shell element thickness is 1.5mm.

Fig.3 Mises stress distribution of A-type.

Fig.2 Assembled structures in the bending test.
Weight of aluminum structures is 6.5kg.

Fig.4 Mises stress distribution of B-type.

3. Conclusions
Assembled structures by origami forming that can be produced actually as high flexibility to design, areexpected to serve as internal structural reinforcement materials. Characteristics of deformations as them areshown clearly that the stress concentrations generate at many joint points and they deflect overall.

Reference[1] Terada, K., Tokura, S., Sato, H., Makita, A., and Hagiwara, I., Evaluation of the bending stiffness on assembled
lightweight and high strength panel,No.15-00039[DOI:10.1299/transjsme.15-00039],Vol.81,No.828,2015.

5
The Waterbomb Origami Tube
Zhong YOU1
1

Department of Engineering Science, University of Oxford, U. K.

1. Introduction
In origami, the waterbomb tube refers to an origami structure made from a paper patternobtained by tessellation of the so-called waterbomb bases. One of the most distinctive
characteristics of the waterbomb tube is that it has a negative Poisson’s ratio: when compressed,
both its length and diameter get smaller. This has led to a number of notable practicalapplications including an expandable medical stent graft, a transformable worm robot, as well asa deformable robot wheel. Despite these applications, its precise motion behaviour when being
compressed and the resultant mechanical property have remained ambiguous, especially whether
the shape change is rigid origami, in which the facets remain flat and un-deformed whilst
rotating about bounding creases, or whether the facets themselves have to deform. To know thisis critically important for it to be used to create a metamaterial and for other current and futureapplications. In this talk, I shall uncover the true motion behaviour of the waterbomb tube
through a detailed kinematic analysis and structural simulation thereafter.
2. Characteristics of the Watermomb tube
The behaviour of a waterbomb tube depends on its geometric parameters. It turns out thatsome tubes are capable of transferring between a pure rigid origami to structural deformation
under external loadings, and there exists a mechanism-structure-mechanism transition in its
motion, Fig. 1. More interestingly, the tube has an inherited ability of being switched among thevarious rigid motion paths that corresponding to different mechanical properties. The talk will
provide detailed information on when this is happening and how it can be tuned. I anticipate that
this research will provide a solid foundation for full exploitation of this ancient origami object tocreate novel metamaterials, shape-changing structures and soft robots.

Fig. 1. A waterbomb tube folds from the expanded configuration (i) to the fully contracted
configuration (vi). (i) to (iii), and (iv) to (vi) are rigid folding whereas (iii) to (iv) is not.

6
ICMMA 2016 (Nov. 2-9)

Continuous Flattening of Boxes with Thickness
CHIE NARA
Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University, Japan

1. Introduction
When we want to flatten an empty box, how to manage it? There are many ways to do so; some people
may cut and open it, or others may push it by force. Here, we treat such problem a little precisely by using
mathematical definitions which depend on materials. First, we consider boxes made of paper withzero-thickness, which means papers can be folded by creases. Next, we discuss on the case that the box ismade of rigid faces with suitable hinges (that is, panel-hinges). Finally, we study on the case that the box ismade of rigid faces with thickness. For the sake of industrial applications, we prefer methods which aresimple and elegant.

(K. Matsubara-C.N.)

2. Continuous motions
The problem of continuous flattening polyhedra made of paper, without cutting and stretching, wasoriginally proposed by E. Demaine, M. Demaine and A. Lebiw in 2001, and .published in 2007 in [2]. If the
shape is a convex polyhedron, there were at least three methods given by author et al. In this talk we use thekite method shown in [3, 4] among them, and introduce several ways of flattening a zig-zag belt. We discuss
the difference between the ways for crease patterns. The idea may help us to find an efficient way, as we
worked on the cupboard foldable helmet shown in the following figures.

If a box is made of thick material, we need to manage the continuous motion for flattening without
self-intersection more carefully. There are known results for folding thick panels [1] which we use forflattening boxes with thickness.

3. Future works
There are many polyhedral figures with thickness which are expected to be foldable. Therefore, we would
like to continue finding suitable moving crease patterns and places of hinges.
Reference[1] Yan Chen, Rui Peng, Zhong You. Origami of thick panels. Science, 349 (2015), 396-400.
[2] E. D. Demaine and J. O'Rourke. Geometric folding algorithms, Linkages, Origami, Polyhedra.
Cambridge University Press, 2007.
[3] J. Itoh and C. Nara. Continuous flattening of Platonic polyhedral. Computational geometry, Graphs andApplications, 7033, LNCS, Springer -Verlag (2011), 108-121.
[4] C. Nara. Continuous flattening of some pyramids. Elem. Math. 69 (2014), 45-56.

7
A survey on computational complexity of finding good folded state with few
crease width
Ryuhei Uehara1,
School of Information Science, Japan Advanced Institute of Science and Technology, Japan1

1. Introduction
For a given crease pattern, there exists an infinite number of folded states which are consistent of the pattern.
More precisely, for a given random string of mountain and valley of length n, there are f(n)=Θ(1.65n) folded
states on average for a paper strip of length n+1 having crease pattern at regular intervals. This function f(n)has upper bound O(2n) and lower bound Ω(1.53n) from mathematical point of view (see [1] for the details).
From the practical point of view, among this exponentially many folded states, it is natural to aim to find agood folded state that has better property than the others. From this viewpoint, we introduced a notion ofcrease width that is the number of paper layers between two paper segments joined at the crease. From theviewpoint of theoretical computer science, we have investigated computational complexity of theminimization problems for crease width. We showed hardness for some problems, and gave some efficientalgorithms for the other. A survey of this research will be given.

2. Stamp folding and crease width
We focus on a 1D strip paper of length n+1. The input of the problem is a string over “mountain” and“valley” of length n. Then there are exponentially many folded state of unite length of this paper with respect
to this string. A special case is that the case of the string “MVMV…” which yields to a pleat folding. The
number of folded state is one if and only if the string gives pleat folding. The other case, we have two or
more folded state for a given string. It is still open estimation of the number of folded state for a given crease
pattern except this pleat case. Especially, we do not know a string of length n that gives most folded state.
When a crease pattern is given, it is natural to find a good folded state from the viewpoint of accurate folding
and stress of paper. From this viewpoint, we like to minimize the number of paper layers at each crease. We
show that this problem is intractable in general in [2]. That is, most cases are NP-complete, which means that
we cannot make any program that solves this problem efficiently. We also prove that some cases can besolved efficiently. Precisely, we can find a minimum total crease width when this total value is reasonably
small. In the term of theoretical computer science, we show that this problem is fixed parameter tractable inthis situation. Recently, we extend the origami model from regular intervals to general case in [3]. That is,
we extend the results for regular intervals to general non-regular intervals. We again show that mostproblems are intractable in this new model.

3. Conclusions

From the mathematical point of view, we have some unsolved problem in stamp folding. Especially, what themost complex string that gives most folded states? From the viewpoint of theoretical computer science,developing efficient algorithms seem to be interesting problem. Extension to 2D is another issue.

Reference[1] Ryuhei Uehara, Stamp foldings with a given mountain-valley assignment in ORIGAMI5, pp. 585-597, CRC
Press, 2011.

[2] Takuya Umesato, Toshiki Saitoh, Ryuhei Uehara, Hiro Ito, and Yoshio Okamoto, Complexity of the stamp
folding problem, Theoretical Computer Science, Vol. 497, pp. 13-19, 2012.
[3] Erik D. Demaine, David Eppstein, Adam Hesterberg, Hiro Ito, Anna Lubiw, Ryuhei Uehara and Yushi Uno,
Folding a Paper Strip to Minimize Thickness, The 9th Workshop on Algorithms and Computation (WALCOM
2015), Lecture Notes in Computer Science Vol. 8973, pp. 113-124, 2015.

8
Principle and experimental evaluation of origami-inspired vibration isolators
Sachiko Ishida1
Joint work with Kohki Suzuki1 and Haruo Shimosaka1
1

Department of Mechanical Engineering, Meiji University, Japan

1. Introduction
In this talk, the author shows a principle to isolate vibration using a foldable cylinder with a torsional
buckling pattern and experimental results of a prototyped vibration isolator designed based on the principle.

2. Principle to isolate vibration using foldable structures
It has been known that a nonlinear spring with high-static-low-dynamic stiffness is theoretically able to work
as a vibration isolator around a static equilibrium position that is in a low-stiffness range [1]. By numericalanalysis, it is identified that an origami-based foldable cylinder with a torsional buckling pattern has ahigh-static-low-dynamic-stiffness-spring characteristics that implies the possibility to apply it to isolator [2].

3. Experimental results
Figure 1 shows a prototyped vibration isolator using versatile mechanical components based on theabove-mentioned foldable cylinder. To see the possibility to practical usage, the vibration responses againstz-dimensional signals of seismic waves were examined. As a result, it was confirmed that the isolator
enabled to mitigate vibration at high frequency range (Figure 2). On the current configuration, the applicablefrequency range is over 6 Hz, which is to be improved for the future.

Figure 1 Prototyped vibration isolator. The
figure was extracted from [3].

Figure 2 Seismic waves generated on the Tohoku-Pacific Ocean
Earthquake and response waves by the prototyped isolator. The
figure was extracted from [3].

Reference[1] Carrella, A., Passive Vibration Isolators with High-Static-Low-Dynamic-Stiffness, VDM Publishing, 2010.
[2] Ishida, S., Uchida, H., Shimosaka, H., and Hagiwara, I., Design Concepts and Prototypes of Vibration
Isolators Using Bi-stable Foldable Structures, Proceedings of ASME 2015 International Design Engineering
Technical Conferences & Computers and Information in Engineering Conference, DETC2015-46409.
[3] Ishida, S., Suzuki, K., and Shimosaka, H., Design and Experimental Analysis of Origami-inspired Vibration
Isolators with Quasi-zero-stiffness Characteristic, Proceedings of ASME 2016 International Design
Engineering Technical Conferences & Computers and Information in Engineering Conference,
IDETC2016-59699.

9
ICMMA 2016

Folding Paper: Visual Art Meets Mathematics
Erik Demaine
Massachusetts Institute of Technology, USA

My father Martin Demaine and I like to blur the lines between art and mathematics,
by freely moving from designing sculpture to proving theorems and back again. Paper
folding is a great setting for this approach, as it mixes a rich geometric structure with abeautiful art form. Mathematically, we are continually developing algorithms to foldpaper into any shape you desire.
Sculpturally, we have been exploring curved creases, which remain poorly understoodmathematically, but have potential applications in robotics, deployable structures,
manufacturing, and self-assembly. By integrating science and art, we constantly find
new inspirations, problems, and ideas: proving that sculptures do or don't exist, or
illustrating mathematical beauty through physical beauty. Collaboration, particularly
as a father-son team, has been a powerful way for us to bridge these fields. Lately weare exploring how folding changes with other materials, such as hot glass, opening a new
approach to glass blowing, and finding new ways for paper and glass to interact.

10
Four-dimensional origami model of the alveolar structure in the human lung
+LURNR.LWDRND1,
1

Division of Engineering Technology, JSOL Corporation, Japan

1. Introduction
The alveolar system in the mammalian lung is one of the most complicated structures in the living organ.
The air pathway is space-filling in spite of a part of a rooted tree originated from the trachea, and its
configuration changes during respiratory cycle. I previously proposed a computational model of the 4D
alveolar structure [1], but very few people understood it because of geometric complexity. Then, I haveconstructed an origami model which is almost equivalent to the computational model [2]. One can make a realsolid model by oneself, and understand how it would be moved during respiratory cycle.

2. Single origami alveolus
The left part in Fig.1 indicates a sheet for a single alveolar origami model. When neighboring edges of pingparts are connected by cellophane tape, the sheet becomes a 3D object, and a pink ring, corresponding to thealveolar mouth, is generated. As the pink ring is folded at black lines, its opening becomes smaller, and finally isclosed (right part in Fig.1).

The whole volume becomes the minimum when the ring is closed. This behavior

mimics the real alveolar defamation during respiratory cycle.
3. Origami alveolar duct
The alveolar duct is an air duct whose wall is completely replaced by several alveoli. Therefore, the origami
duct is generated by connecting single origami alveoli in 3D space (left in Fig 2). Branching and space-filling ductsystem is generated by connecting multiple duct units (right in Fig 2).

Fig.1

Single alveolar model.

Fig.2

Alveolar duct model

3. Conclusions
My 4D alveolar model in computer had been neglected for several years, because it was inconsistent withconventional respiratory physiology and pathophysiology. Owing to the origami modelling, novel
understandings for alveolar structure and function are spreading in medical people. Seeing is believing. I
dare say “Making is convincing”.
Reference[1] Kitaoka H, Nieman GF, Fujino Y, Carney D, DiRocco J, Kawase I. A 4-dimensional model of the alveolar
structure. J Physiol. Sci. 57: 175-185, 2007.

[2] Kitaoka, H. A 4D model generator of the human lung. Forma

26: 19-24, 2011.

11
Formation of 3D co-culture microstructuresusing MEMS and Origami folding techniques
Kaori Kuribayashi-Shigetomi and He Qian
1

Institute for Advanced Study of Mathematical Science, Meiji University, Japan

1. Introduction

Interactions between different kinds of cells play an extremely important role for cell functions and proliferation [1, 2].
Recently, many methods to produce 3D co-culture structures have been developed [3, 4], however, they are complicated,
and the shapes of the 3D structures are restricted. We have developed a method to produce 3D cell-laden microstructure
which formatted by cell traction force as the motive power [5]. This technique solved the above problems. It is easy andrapid to format and the shape can be changed simply. Our previous results observed by confocal microscope showed ahollow inside of dodecahedron 3D microstructure after folding. Therefore, it is possible to culture other kinds of cells in
the hollow and create 3D co-culture microstructures.

2. Method of forming 3D co-culture microstructures

Cells were cultured the micro-sized plates that was fabricated on a glass substrate by a standard lithography techniqueof micro electro mechanical systems (MEMS). In order to make sure that the cells will only stay on the microplates,
2-methacryloyloxyethyl phosphorylcholine was (MPC) coted on the substrate (Fig. 1). In this research, we use 3T3 and
HepG2 cells to form the 3D co-culture microstructures. This method for cells culture is the same as conversional 2D
culture without any other complicate processes. After these processes, alginate lyase was applied to remove alginate that
was a sacrificial layer (Fig. 1), and the microplates were folded to format the 3D co-culture microstructures because of
the traction forces of 3T3 cells. Finally, the location of 3T3 and HepG2 cells was determined.

3. Results

We successfully formed the 3D microstructure and co-culture of 3T3 and HepG2 cells. HepG2 cells located in thecentral of microstructure were surrounded by 3T3 cells which distributed in the periphery (Fig 2). The folding time

can be controlled by using alginate comparing with using gelatin in previous research [5], and a number of 3D
microstructures can be folded at one time during 1 to 2 minutes.
The 3D co-culture HepG2 and 3T3 cells system can be easily acquired by applying microplates formatted 3D
cell-laden microstructures. And the specific hepatic genes and the drug metabolism will be detected for investigation offunction and response of co-culture microstructure.

Acknowledgments

The authors are grateful to Takahiro Okajima for his useful discussions and Kazuhiko Ishihara for porviding MPC
polymers. This work was supported by JSPS KAKENHI Grant-in Aid for Scientific Research (B), Grant Number
26286030.

Reference

[1]
[2]
[3]
[4]
[5]

Shimaoka S, Nakamura T and Ichihara A, Exp Cell Res, 172(1) (1987), 228-242.
Tsuda Y, Kikuchi A, Yamato M and et al, Biochem Biophys Res Commun, 348(3) (2006), 937-944.
Matsunaga YT, Morimoto Y and Takeuchi S, Adv Mater, 23(12) (2011), H90-94
Yamada M, Utoh R and Ohashi K, Biomaterials, 33(33) (2012), 8304-8315
Kuribayashi-Shigetomi K, Onoe H and Takeuchi S, PLoS One. 7(12) (2012), e51085

12
Development of a Paper-based Rapid Prototyping System
for Orthopaedic Surgical Planning
Luis DIAGO 1,
1

Meiji Institute for Advanced Study of Mathematical Sciences, Japan

1. Introduction
3D printing or rapid prototyping (RP) has emerged as a revolutionary technique that overcomes the limitations
produced by the use of flat screens and 2D drawings for the visualization of three-dimensional (3D) imaging databy producing graspable 3D objects which can be applied for surgical planning, training, prosthetics and related
applications. 3D printing may be costly but in complicated surgical cases, additional costs of RP may becompensated by reduced operating times and higher success rate of the surgical procedure. The time needed for
producing a 3D object limits its use in surgery to elective cases and makes it unsuitable for emergency cases. In
order to overcome above limitations we have been working on the development of a new RP system thatincorporates a new origami-pattern generation algorithm to produce paper-based copies of physical objects [1].
Figure 1 shows the general flowchart of our current research that deals with two problems: 1) the generation of 3D
digital models from X-ray images [2] and 2) the development of a paper-folding robot by direct teaching fromexperiments [3] in order to speedup the creation of paper-models by proposed folding machines [4].

Figure 1: General flowchart of the proposed research

2. Bone Shape Modeling and Paper-based Construction by “Norigami” Folding Machines.
Figure 2 (left) shows a comparison of a model of a Tibia constructed from X-ray images using [2], printed with Cube
3D printer and proposed method [1]. Three folding trajectories in Fig.2 (middle) are used to create a round object byproposed “norigami” folding machines (right). Feed-forward control is needed to reduce the error of the machines.
Two neural networks (i.e. ADALINE and proposed HNN) are compared to follow the trajectories. HNN is faster andaverage error in the folding position by HNN is 2 times smaller than ADALINE’s error (~2.6mm vs ~5.6mm).

Figure 2: Models of a right tibia and “Norigami” folding machines with folding trajectories for a round object
References[1] B. Yu, M. Savchenko, J. Shinoda, L. Diago, I. Hagiwara and V. Savchenko “Producing Physical Copies of the Digital Models via Generating 2D Patterns for “Origami
3D Printer” system”, Journal of Advanced Simulation in Science and Engineering, Vol. 3 (2016) No. 1 p. 58-77.
[2] V. Karade and B. Ravi, “3D femur model reconstruction from biplane X-ray images: a novel method based on Laplacian surface deformation” Int J CARS Vol.10(2015) No. 4 p. 473-485.
[3] Y. Kihara and Y. Yokokohji, Skill Transfer from Human to Robot by Direct Teaching and Task Sharing: A Case of Study with Origami Folding Task” In 11th IFAC
Symposium on Analysis, Design, and Evaluation of Human-Machine Systems Vol. 11 (2010) p. 454-459[4] J. Romero, L. Diago, J.Shinoda, C. Nara and I. Hagiwara. “Norigami Folding Machines for Complex 3D Shapes” In Proc. ASME IDETC/CIE (2016) (in Press)

13
Maekawa Jun1
1

Origami Artist

1. Introduction
Dividing a cube in various ways to see diverse sections is an effective exercise for artistic design. It has beenpracticed by some educational programs including Bauhaus in German and Kuwasawa Design School inJapan, and has resulted in some art works including "Wine Cube" by Hiroshi Tomura[1]. It is also a usefuleducational material for Mathematics. The following geometric fact is well-know:
When a cube is divided with a plane, the section can be a square, an equilateral triangle, a rhombus, arectangle, a regular hexagon, and so on, but cannot be a regular pentagon.

2. Origami works
Some origami works have also designed on cube division, including:Figure 2-1.

Figure 2-2.

Figure 2-3.

Cube with a regular hexagon [2].

Cube with a hyperbolic paraboloid[3].

Cube trisection (modular origami)[4].

3. An application
While studying cube division origami models, the author came up with an idea that does not have directconnection with origami techniques.
Basic measuring spoons for cooking have volumes 15cc, 10cc, and 5cc, which are 1/2, 1/3, and 1/6 of 30cc,
respectively. Meanwhile, a cube can be divided in 1/3 or 1/6 easily because, as we have seen in 2-3, it has
rotational symmetry of order 3. Noticed these facts, the author devised a spoon that can measure threedifferent volumes by rotating three axes.

Figure 3-1. Large, medium, and small integrated measuring spoon

References[1]Tomura H. “Kihon Keitai no Kozo”, Bijutsu Shuppan Sha, 1974
[2]Hull T., “Origami3 Third International Meeting of Origami Science, Mathematics, and Education”, A K Peters,
2002[3]Maekawa J. , “Oru Kikagaku(Folding Geometry)”, Nippoon Hyoron Sha, 2016[4]Maekawa J. , “Otte Tanoshimu Origami Seminar”, Sugaku Seminar, Vo;. 55 No. 5(2016), pp. 88-89

14
Tilt-up concrete dome skeleton construction by
Rotational Erection System (RES)1

Yoshinobu Miyamoto1,
Aichi Institute of Technology, Japan

1. Introduction

We propose a novel construction method for dome skeleton using Tilt-up concrete and Rotational Erection
System (RES)[1]. The construction procedures are 1) setting the edge form and rebar on the level floor pad,2) casting structural concrete, 3) hoisting up the hub on the temporary support, 4) tilting-up the arms andfastening them to the hub. We studied the two types of rigid frame configuration with a) intersecting arms
and b) dual layer arms and one type of cable-stay configuration (Fig. 2) [2,3]. We investigated a modification
to RES for rigid folding that would be applicable to the alternative jack-up erection method with fully linked
elements.

2. Three types

We apply RES geometry to
determine the efficient layout of theconcrete forms on the floor slab (Fig.
1). The interactive parametric design
tool written in GeoGebra was used
to find the arm profile appropriate
for the concrete structure.

Intersecting arms

3. Structural analysis

FEM Structural analysis showed
30m span domes in the three
configurations are feasible with theconventional concrete materials. The
seismic load C0=0.5 was applied as
the critical condition (Fig. 3).

4. Rigid foldable RES

Dual layer arms

Cable stayed arms

Fig. 1 Patterns.

Fig. 2 Dome types.

Fig. 3 Displacements.

types.
Adding two hinges for each arm at the hub make
RES rigid foldable (Fig. 4).

Fig. 4 Rigid folding sequence (RES n=3, α= 60°, β= 90°).

5. Conclusions
Taking advantage of existing know-how of tilt-up concrete method, RES could achieve the visual effect andspecial impression that was never done with tilt-up concrete. Further parametric optimization and refinementin the construction procedure would find a competitive position among the existing construction methods.

Reference[1] Miyamoto Y., Rotational Erection System (RES): Origami Extended with Cuts, in Origami6 II:
Technology Art, Education, Miura K (ed.), AMS, 2015, 537-544.
[2] [3] Miyamoto Y., Itoh T., Harada H., Tilt-up concrete dome skeleton construction by Rotational Erection
System (RES), part 1 & 2, Proceedings of the IASS Annual Symposium 2016.

15
An iterative simulation-based design of the origamiperforming robot
Maria Savchenko
Meiji Institute for Advanced Study of Mathematical Sciences

Abstract
The robot design based on experiment only is difficult, expensive, and time-consuming. Then thesimulation approach based on software becomes the main option in the real-world related with arobotic activity for creating basis for hardware design. We present a study: the iterative simulationbased design of the origami-performing robot’s hardware. The basic premise underlying the study isthat each folding operation of the crease patterns of origami is considered as a function of themechanical systems such as a robot. The aim of the research is to manipulate the foldable objects suchas a sheet of paper in the simulation environment in the relation with understanding the robot design
instead of creating the real robotic prototypes. In this case, dynamic and kinematic behaviour of therobot arms in the formation of origami models can be simulated by using LS-DYNA software. In
simulating, we consider two types of crease patterns: i) in ordinary paper folding; ii) in the thickened
paper folding. In the both cases the goal of the simulation is to form the 3D origami models. Fig.1
illustrates the design process from the schematic stage to the final design. Fig.2 shows forming abending crease and the 3D shapes by the robot end-effectors in the finite element models of crease
patterns.

The 3D origami simulation byusing the FEM in LS-DYNA

Fig.1 Design process: schematic stage- simulating - the final design stage

Grippers
Holder

Fig.2 Results of the simulation: ordinary paper and carton models

16Some observation of regular polygons in origami paper
Haruo Hosoya
Department of Information Sciences, Ochanomizu University (Emeritus), Japan

1. Introduction
The standard size of origami paper in Japan is 15cm×15cm, from which I have found several interesting
ways for cutting out regular polygons. Although metric ruler and scissors are necessary for these operations,
I believe that these findings are helpful not only for mathematics education to junior high school students but
also for popularization of origami to the public. In this talk the secret of “Nana-kin-san Silhouette Puzzle(Golden Heptet Triangles)” will also be disclosed, which is motivated from the pair of golden triangles andwas recently designed by me.
.

2. Patterns
2 cm

44

4 cm

2 × 44 2 = 62.23(4.019)

4 cm

15 cm

152 + 4 2 = 241 = 15.524 cm

132 + 7.52 = 15.008cm

Fig. 1.

112 × 2 =242 = 15.556 cm

Largest triangles in origami.

Fig. 2.

Largest octagons in origami.

97.5

48.5

60

35.5

4

44

62

60

158

114.5

1
γ

35.5

60

48.5

97.5
146

Fig. 3.

Relation among golden triangles,

τ
1

τ
τ

τ2 δ

τ
α
xδ

τ3

2

γ

τ3

β

τ

α τ

α

β

α

τ2

τ2
Fig. 4.

Golden Heptet Triangles.

regular pentagon and square.

References[1] H. Hosoya, Seven Secrets of Triangles (in Japanese), Blue Backs, B-1823, Kodansha, Tokyo, 2013.
[2] Golden Heptet Triangles, Manufactured by Image Missions Inc., Shizuoka, 2015.

τ2
τ

17
Paper Models of Projection of 4-dimensional Regular Polytopes
Koji Miyazaki

Professor Emeritus, Kyoto University, Doctor of Engineering

Abstract
In this paper, the real shape of each face of orthogonal
projectioninto

3-spaceof

4-dimensional

regularpolytopes are shown to make their Origami-likehandmade models. The result will serve to flourish theworld of paper models which is active in 4-space.

Unique point of this paper
There are 6-kinds of regular 4-polytopes: the 5-, 8-,16-, 24-, 120-, and 600-cell (Fig.1, from top to bottomrows). They are composed of 3-dimensional regularpolyhedra and can be orthogonally projected into 3-spacein typical 4 types: vertex-, edge-, face- and cell-centeredprojections (Fig.1, from left to right column).

Fig.1

In spite of such variety, merely some of simple
vertex- or cell-centered projections are usually adopted
as projections of regular 4-polytopes because of
usefulness for easy understanding and manufacturing.
To make innovations such present conditions, in thispaper, the real shape of each face which composes all
projections shown in Fig.1 are represented.

Conclusion
The real shape of each face of projections shown in
Fig.1 coincides with any of the numbered plans orelevationsofrotational

3-dimensional

regularpolyhedra as is shown in Fig.2. Each polyhedra

Fig.2

composes any of regular 4-polytopes.
Fig.3 shows Origami-models representing orthogonal
projections of regular 4-polytopes constructed by the realshapes shown in Fig.2. Each is composed of small
flattened regular polyhedra shown below.

Fig.3

18
Computing Area, Circumradius, and their Integrated Formulae
for Cyclic Polygons
Shuichi Moritsugu1,
Faculty of Library, Information and Media Science, University of Tsukuba, Japan1

1. Introduction
In this study, we consider a classic problem in Euclidean geometry for cyclic polygons; that is, 𝑛𝑛-gons
inscribed in a circle, given by the length of sides 𝑎𝑎1 , … , 𝑎𝑎𝑛𝑛 . In particular, we focus on the relation between
the area 𝑆𝑆 and the circumradius 𝑅𝑅 of cyclic pentagons and hexagons.
The classic results derived by Heron in the 1st century gives the formulae for triangles. For cyclic
quadrilaterals, Brahmagupta’s formula was obtained in the 7th century. Nowadays, we can rewrite theformulae for 𝑛𝑛 = 3,4 into unified and simplified forms as follows, using elementary symmetric functions
𝑠𝑠1 = 𝑎𝑎12 + 𝑎𝑎22 + 𝑎𝑎32 + 𝑎𝑎42 , … , 𝑠𝑠4 = 𝑎𝑎12 𝑎𝑎22 𝑎𝑎32 𝑎𝑎42:
 
𝑥𝑥 − (−𝑠𝑠12 + 4𝑠𝑠2 + 𝜀𝜀 ∙ 8√𝑠𝑠4 ) = 0
Area (𝑥𝑥 = (4𝑆𝑆)2 ):(−𝑠𝑠12 + 4𝑠𝑠2 + 𝜀𝜀 ∙ 8√𝑠𝑠4 )𝑦𝑦 − (𝑠𝑠3 + 𝜀𝜀 ∙ 𝑠𝑠1 √𝑠𝑠4 ) = 0
Circumradius (𝑦𝑦 = 𝑅𝑅 2):
2
𝑍𝑍 − (𝑠𝑠3 + 𝜀𝜀 ∙ 𝑠𝑠1 √𝑠𝑠4 ) = 0
Integrated (𝑍𝑍 = (4𝑆𝑆𝑆𝑆) ):In these expressions, ε is called crossing parity with 0 for a triangle, 1 for a convex quadrilateral, and−1 for a non-convex quadrilateral. Our goal is the computation of these types’ formulae for polygons 𝑛𝑛 ≥ 5,
which remained unsolvable for 1,300 years since Brahmagupta.
n5Area formula

Circumradius formula

Integrated formula

Robbins (1994)

Takebe (1683)

Moritsugu (2014)

Pech (2006)

Izeki (1690)

Moritsugu (2015)

Moritsugu (2015)

Robbins (1994)
Pech (2006)

6

Robbins (1994)

Moritsugu (2011, 2016)

Moritsugu (2015)

Moritsugu (2015)
7

Maley et al. (2005)

Moritsugu (2011, 2016)(unsolved)

8

Maley et al. (2005)(unsolved)(unsolved)

Fig.1 A cyclic pentagon

Table 1 Contributions to each formula

2. Algorithms and results
We applied elimination algorithms by resultant using computer algebra systems. Table 1 shows the recent
results. We should note that old Japanese mathematicians in the 17th century had already solved circumradius
problem for pentagons based on the theory of resultants. Meanwhile, the area formula discovered by Robbins
in 1994 was a great breakthrough. In contrast, the author’s main contributions are the following computation:(1) The area formulae (𝑛𝑛 = 5,6) by only simple elimination method(2) The circumradius formulae (𝑛𝑛 = 6,7) and their compact expressions by elementary symmetric functions(3) The integrated formulae (𝑛𝑛 = 5,6) which have been “missing” since Brahmagupta
To our best knowledge, we believe that these results are quite original.

References[1] Moritsugu, S., Computing Explicit Formulae for the Radius of Cyclic Hexagons and Heptagons, Bulletin of
JSSAC, Vol.18, No.1, pp.3-9, 2011.

[2] Moritsugu, S., Integrated Circumradius and Area Formulae for Cyclic Pentagons and Heptagons, ADG 2014,
Lecture Notes in Artificial Intelligence, Vol.9201, pp.94-107, 2015.

19
On the Enumeration and Counting of Developments of Polyhedra
Takashi Horiyama1
1*UDGXDWH6FKRRORI6FLHQFHDQG(QJLQHHULQJ6DLWDPD8QLYHUVLWy, Japan

1. Introduction
A development of a polyhedron is a simple polygon obtained by cutting along edges of the polyhedron andunfolding it into a plane. The cut edges of a development of a polyhedron form a spanning tree of the
1-skeleton (i.e., the graph formed by the vertices and the edges) of the polyhedron, and vice versa. In thistalk, we discuss the techniques for enumerating developments of polyhedra and counting the number ofdevelopments.

2. Enumeration of developments
We use ZDDs (zero-suppressed binary decision diagrams) for enumerating developments of polyhedra (i.e.,
spanning trees of the 1-skeletons) [1, 2]. A ZDD is a directed acyclic graph representing a family of sets. Fig.
1 is an example of a ZDD representing {{e1, e3, e5}, {e1, e4}, {e2, e3, e4}, {e2, e5}}. Each path from theroot node to the 1-node (the sink node labeled “1”) corresponds to a set. The paths e1-e3-e5-1 and e1-e3-e4-1
in Fig. 1 respectively correspond to {e1, e3, e5} and {e1, e4}. Note that the labels of the nodes with dottededges are ignored. Figs. 2 and 3 illustrate partial lists of the developments of a dodecahedron and an
icosahedron, respectively.

Fig. 2 Partial list of the developments of a dodecahedron.

Fig. 1 ZDD.

Fig. 3 Partial list of the developments of an icosahedron.

3. Counting the number of developments
The easiest way for counting the number of developments of a polyhedron is to enumerate those and countthe number of obtained ones. By utilizing the group theory, however, we can count the number ofdevelopments without enumerating them [3]. In this talk, we will give the number of developments ofArchimedian solids.

Reference[1] T. Horiyama, and W. Shoji. Edge-Developments of Platonic Solids Never Overlap, In Proc. of the 23rd
Canadian Conference on Computational Geometry (CCCG 2011), pp. 65-70, 2011.
[2] J. Kawahara, T. Inoue, H. Iwashita, and S. Minato. Frontier-based Search for Enumerating All Constrained
Subgraphs with Compressed Representation, Technical Report TCS-TR-A-14-76, Division of Computer Science,
Hokkaido University, 2014.
[3] T. Horiyama, and W. Shoji. The Number of Different Unfoldings of Polyhedra, In Proc. of the 24th
International Symposium on Algorithms and Computation (ISAAC 2013), Lecture Notes in Computer Science,
vol. 8283, pp. 623-633, 2013.

20
Modeling by the infinite foldTomoko Fuse
Origami Artist, Independent
1. Introduction
Folding a sheet of paper sometimes results in unintentional discovery of an interesting shape,
sometimes in finding a rhythm of folds. Modeling by folding paper is more constrained thanother genres of art. That is the reason why the found shapes are usually geometrical.
As examples of such findings, the author reports some works with the infinite fold, wherefolding patterns can be repeated endlessly: two-dimensional tessellation pieces andthree-dimensional accordion-like works.
2. Two-dimensional works
Most of origami pieces involve sequences of bisection. The most common technique is dividing
into smaller parts (figure 1). In tessellation, the basic fold is the square twist (figure 2).

Figure 1.[1]Figure 2.

3. Three-dimensional works
Bisection is also the major technique in three-dimensional models. The author presents five
types of pieces with their variations that can be closed flat (figure 3) (figure 4).

Figure 3.

Figure 4.

4. Conclusion
The infinite fold has various types, both two- and three-dimensional. Its pieces have shapes thatemerge from the rules of origami, not that represent known shapes or nature. The mostimportant thing is that a complete world can be perceived in the actual folding process.
Reference [1] Shuzo Fujimoto, Ajisai-ori, Seibundo Shinkosha, December 2010.

21
Folding of Deployable Membrane Space Structures
Hiroshi Furuya1,
Department of Mechanical Engineering, Tokyo Institute of Technology, Japan1

1. Introduction
The folding/deployment techniques in space engineering are significant to realize satellites, space antennas,
and space stations, because the limited volume and weight of capacity for launching the equipment by rocket
as well as the increasing size of it due to the complex space missions. As the results, to fold structures on the
ground and to deploy easily in space environment are requested. Recently, several deployable membrane
space structures have been proposed and constructed, for de-orbiting satellites within 25 years after thecompletion of their primary missions. Furthermore, membrane structures will enable innovative missions
including solar sails, solar arrays, phased array antennas, and sun shields. This paper introduces
boom-membrane integrated space structures for small satellites.

2. Boom-membrane integrated space structures
For small satellite, the retraction size and the easy deployment with small deployment mechanisms areimportant to design the deployable membrane. We have proposed the rotationally skew fold (Fig.1). This
folding pattern enables the membrane to be stowed in a desired height of fold pitch without increasing the numberof tangential fold lines. The deployment of the membrane is performed by the self-deployable boom in Fig.2.
Finally the boom-membrane integrated structures are configured as shown in Fig.3. The deployment procedures
are indicated in Fig.4.

Fig.1 Rotationally skew fold.[1]Fig.2 Triaxially woven CFRP

Fig.3 Deployed configuration

Fig.3 Deployment procedures with cable suspension system.

Reference[1]Furuya, H. and Masuoka, T., Concept of Rotationally Skew Fold Membrane For Spinning Solar Sail,
CD-ROM Proc. 55th International Astronautical Congress, IAC-04-I.1.05, Vancouver, (2004), pp.1-5.
[2]Furuya, H, Mori, O., Sawada, H., Okuizum, N., Shirasawa, Y. Natori, M., Miyazaki, Y., Matunaga,S.,"Manufacturing and Folding of Solar Sail "IKAROS"," 12th AIAAGossamer Systems Forum, Denver,
AIAA-2011-1967, (2011), pp.1-4.

22
Enumeration of formal crease patterns in the square/diagonal grid andverification of their flat-foldability
Jun Mitani1, Yoshihisa MATSUKAWA1, Yohei YAMAMOTO2
University of Tsukuba, 2GIKEN, LTD.

1

1. Introduction
The square/diagonal grid (Fig. 1(a)), which consists of arrayed squares and their diagonals, is commonlyused not only in basic origami design but also in the basic structure of recent self-folding origami robots[1].
However, the number of shapes made by folding lines of the subset of the grid pattern is not known as far as
we have surveyed.

2. Locally flat-foldable crease patterns and their folded shapes
We enumerated the number of locally flat-foldable crease patterns[2] and their folded shapes without
considering mountain/valley parity. The numbers of crease patterns and the folded shapes in 2x2, 3x3, and4x4 gird patterns we enumerated are: {116/27}, {58,530/ 366}, and {259,650,300/13,452} respectively.
We verified that all the shapes are realizable, i.e. actually flat-foldable with a physical sheet without
self-intersections. At the same time, we found some locally flat-foldable crease patterns that are impossible
to be folded.

3. Conclusions
We developed an application to search for a crease pattern that is folded into a shape specified by the user.
By using this application, we found the shapes of the alphabet and numbers as shown in Fig. 1(b). In addition,
we concluded that the two crease patterns shown in Fig. 1(c) are the smallest and simplest non-flat-foldable
ones.

(a)

(b)

(c)

Fig.1 (a) 2×2, 3×3, and 4×4 square/diagonal grid patterns. (b) Examples of folded shapes and their crease patterns
in the 4x4 grid pattern. (c) The simplest and smallest impossible-to-flat-fold crease patterns.

Reference[1] E. Hawkes, B. An, N. M. Benbernou, H. Tanaka, S. Kim, E. D. Demaine, D. Rus, R. J. Wood,“Programmable matter by fold-ing”, Proceedings of the National Acade-my of Sciences, 2010.

[2] Thomas C. Hull, “The Combinatorics of Flat Folds: A Survey”, Origami3: The Third International
Meeting of Origami Science, Mathematics, and Education, 2002.

23
ICMMA 2016

Folding Patterns and Deployment Processes inMorphological Changes of Insects
Naoko Kishimoto
Setsunan University, Japan

Insects need several ecdysis processes, since they have external skeleton.
Especially through eclosion processes, they deploy thin membrane structures fromhighly packed situation.
Influx of body fluid into branching veins induces deployment of membrane wings.
We show eclosion processes of some insects and folding patterns of membranes insidetheir sheaths.




24
Origami Based Responsive System Proposal for a Building Surface
Prof.Dr.Arzu Gönenç Sorguç, Özlem Çavuş, Fatih Küçüksubaşı, Serkan Ülgen, FıratÖzgenel, Müge Kruşa
METU Department of Architecture
Introduction
Architecture is not just a response to the needs but is always a manifestation of theavailable technology of its age. Today both these needs and enabling technologies
are increasing symmetrically and become more and more complex. In addition tothese advents, increasing number of environmental problems and their severity
force architecture to investigate new and innovative design approaches, newchallenging forms and materials as well as new construction /fabrication/manufacturing technologies.
Responsiveness in architecture either to soft (social, economic, cultural etc.) or to
hard (light, energy, earthquake natural disasters etc.) conditions is always a design
goal. In present, computational design approaches related parametric/generativemodels together with developments in electronics, mechatronics, control systems
allow architects to re-explore responsiveness with the use of many interesting
kinetic structures.
The Design of the Component
Responsive system design in architecture is a culmination of computational design
knowledge and knowledge on dynamics and related enabling technologies. Thus
teaching and learning responsive architecture in the present context necessitates
modifying curriculum as well.
In this study, design and development of a kinetic surface based on origami ispresented. Origami is always a source of inspiration for architects. It is possible tosee many building forms derived from origami folding.
However it is still novel to integrate real kinetic building structures based on origami
due to material, design and/or kinetic constraints and their implementations. In thepresent study a kinetic component but also the space itself, which can betransformed and provide different ambient conditions is shown.
In the design process, it is required developing a system to adopt different naturallight conditions, various natural ventilation levels and possibility of having different
volumetric relations.

1/3

The parametric/generative computational model is developed by using Grasshopper
and the fabricated prototype is run by Ardunio responding to light and air movement.
The basic design idea is shown in Figure 1. The initial form is obtained by truncating
the a simple nurb surface, than each surface is folded in two directions.

Figure 1. Initial Design of the Component
The design of the component is achieved in parametric environment as shown in
Figure 2

Figure 2. Parametric/Generative Model and the Resulting Component

Figure 3. Final Component and the Details
There are two different modes of deployment in each surface responding the needand the ambient conditions. Surface materials and textures are also determinedaccordingly.
The prototype is shown in Figure 4.

2/3

Figure 4. Working Prototype
Conclusion
The parametric modelling toolls and computational design approaches re-value
origami in design. Particularly the design of responsive systems and their kinetic
mechanisms can easily be deciphered by using origami which provides an invaluable
medium of design analysis for complex deployable responsive structures.

3/3

25
Optimizing Unfoldings of Convex Polyhedra
Anna Lubiw(The University of Waterloo, Canada)
It is an open problem whether every convex polyhedron has an edge unfolding, where wecut along its edges and unfold without overlap. However, if we allow cuts through faces
there are a number of known unfoldings, in particular, the star and source unfoldings andseveral generalizations of them.
In this talk, I will survey the known unfolding methods and explore them from theperspective of optimizing various criteria:- minimize the perimeter of the unfolding- minimize the diameter or the bounding box of the unfolding- maximize the minimum angle of the unfolding- minimize the number of leaves of the cut tree of the unfolding (a zipper unfolding has
two leaves)
The talk will include many open problems.

26
Applications of cut locus and Intuitive geometry--continuous flattening of polyhedra-Jin-ichi Itoh1,
Faculty of Education, Kumamoto University, Kumamoto, 860-8555, Japan1

j-itoh@kumamoto-u.ac.jp
Cut locus is defined by Poincare in the beginning of 20 century, and studied long time in global differential
geometry. In this talk, first we will review its history and recent results, a generalization of Jacobi’s laststatements and fractal cut locus, and discuss several applications (for example, shortest path problem,
unfolding of polyhedron, continuous flattening of polyhedron).
Theorem (J.I., C.Nara & C.Vilcu). Every convex polyhedron has infinitely many continuous flat folding
processes.
It is proved by using cut locus and Alexandrov'e glueing theorem.
Next we will show several problems of Intuitive Geometry. ”Intuitive geometry” is the title of book written
by Hilbert and Cohn-Vossen in the first half of 20 century. We will enjoy such kind of interestiong (amusing)geometric models made by 3D printer.

Continuous flattening of tetrahedron and cube made by thick panels

Surfaces of constant width

Rollable surface

Reference[1] Jin-ichi Itoh, Chie Nara & Costin Vilcu, Continuous flattening of convex polyhedral,
In: Comuputational Geometry, LNCS 7579, 85-97, Springer, 2012.

27
ICMMA 2016

Rigid Foldability and Self Foldability
Tomohiro Tachi
University of Tokyo, Japan

Rigidly foldable origami structures are parallel mechanisms made of rigid plates andhinges, which is useful for the designs of deployable structures and self-folding systems
in different scales. To design such transformable systems, we need to know if a given
pattern forms a finite mechanism (rigid-foldability) and if the mechanism can becontrolled to self-fold into desired states (self-foldability). Rigid-foldability and selffoldability have an inherent hardness, especially when dealing with the singularity andthe degeneracy. However, lots of interesting behaviors of origami patterns come fromthese wicked cases; for example, bifurcation at the singular state of origami can lead toreprogrammable folding patterns, and the degenerate constraints yield overconstrained
mechanisms with high stiffness and flexibility. The speaker talks about design methods
to exploit such unusual behavior of origami patterns and to control the self-folding
behavior of rigid origami.

28
Quaternion expression of the condition for rigid origami
Naohiko Watanabe1,1National Institute of Technology, Gifu college, Japan

Abstract
The condition for rigid origami has been received significant interest from the standpoint of mathematics andengineering application. In the past research, the conditions that the combination of the motion velocities ofdihedral angles 𝜌𝜌𝑖𝑖̇ to be satisfied has been derived in consideration of the condition of a circuit around avertex with n crease lines, a series of facet angles 𝜃𝜃01 , 𝜃𝜃12 , ⋯ , 𝜃𝜃𝑛𝑛−1𝑛𝑛 and dihedral angles 𝜌𝜌1 , 𝜌𝜌2 , ⋯ , 𝜌𝜌𝑛𝑛

with using rotation matrix 𝝌𝝌𝒊𝒊 as described in eq.(1).㻌

𝝌𝝌𝒏𝒏 𝝌𝝌𝒏𝒏−𝟏𝟏 ⋯ 𝝌𝝌𝟏𝟏 = 𝑰𝑰
(1)
In this study, the condition for rigid origami is derived by using a quaternion which can make effective
expression of rotational operation. Use of the quaternion 𝑞𝑞̃𝑖𝑖 including the information of a series of unit
vectors of crease lines 𝒏𝒏𝟏𝟏 , 𝒏𝒏𝟐𝟐 , ⋯ 𝒏𝒏𝒏𝒏 and dihedral angles 𝜌𝜌1 , 𝜌𝜌2 , ⋯ 𝜌𝜌𝑛𝑛 around a vertex make a possible toexpress the constraint condition of a circuit around a vertex as follows.
𝑞𝑞̃𝑛𝑛 ⋯ 𝑞𝑞̃2 𝑞𝑞̃1 𝒓𝒓𝑞𝑞̃∗1 𝑞𝑞̃∗ 2 ⋯ 𝑞𝑞̃ ∗ 𝑛𝑛 = 𝒓𝒓㻌
(2)
𝜌𝜌

𝜌𝜌

𝜌𝜌

𝜌𝜌

(3)

Here, 𝑞𝑞̃𝑖𝑖 = (cos 2𝑖𝑖 , 𝒏𝒏𝒊𝒊 sin 2𝑖𝑖 ), 𝑞𝑞̃𝑖𝑖∗ = (cos 2𝑖𝑖 , − 𝒏𝒏𝒊𝒊 sin 2𝑖𝑖 )In consideration that every quaternions 𝑞𝑞̃𝑖𝑖 at the time t=t+dt also satisfy eq.(2), it can be expressed asfollows.

𝑑𝑑𝑞𝑞
̃(𝑞𝑞̃𝑛𝑛 (𝑡𝑡) +𝑑𝑑𝑞𝑞̃𝑛𝑛𝑑𝑑𝑞𝑞̃1𝑑𝑑𝑑𝑑) ⋯ (𝑞𝑞̃1 (𝑡𝑡) +
𝑑𝑑𝑑𝑑)
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑

= (𝑞𝑞̃𝑛𝑛 (𝑡𝑡) ⋯ 𝑞𝑞̃1 (𝑡𝑡)) + ( 𝑛𝑛 𝑞𝑞̃ 𝑛𝑛−1 (𝑡𝑡) ⋯ 𝑞𝑞̃ 1 (𝑡𝑡) + 𝑞𝑞̃𝑛𝑛 (𝑡𝑡)
𝑑𝑑𝑑𝑑

(A)

(B)

𝑑𝑑𝑞𝑞
̃𝑛𝑛−1
𝑑𝑑𝑑𝑑

⋯ 𝑞𝑞
̃ 1 (𝑡𝑡) + ⋯ ) 𝑑𝑑𝑑𝑑 + 𝑜𝑜(𝛿𝛿𝑞𝑞 2 ) = (1, 𝟎𝟎)(4)

Since the term (A) is equal to (1,0), the term (B) is to be equal to (0,0).
About the term (B), by using derivative of 𝑞𝑞̃𝑖𝑖 with respect to 𝜌𝜌𝑖𝑖 and 𝒏𝒏𝑖𝑖 as shown in eq.(5), it can be derived
the condition that the motion velocity of dihedral angles 𝜌𝜌𝑖𝑖̇ and direction of crease lines 𝒏𝒏̇𝒊𝒊 should satisfy.
The result of the differentiation with respect to the variable 𝜌𝜌𝑖𝑖 is described as shown eq.(6). This condition
corresponds to the geometric characteristic that “the sum of the weighted fold line vectors is equal to zero.”𝑑𝑑𝑞𝑞̃𝑖𝑖
𝜕𝜕𝑞𝑞̃𝑖𝑖
𝜕𝜕𝑞𝑞̃𝑖𝑖
𝜌𝜌𝑖𝑖
=(
𝜌𝜌̇ +
𝒏𝒏̇ ) = (𝒏𝒏𝒊𝒊 𝜌𝜌𝑖𝑖̇ 𝑞𝑞̃𝑖𝑖 + (0, 𝒏𝒏̇ 𝑖𝑖 sin ))𝑑𝑑𝑑𝑑𝜕𝜕𝜌𝜌𝑖𝑖 𝑖𝑖 𝜕𝜕𝒏𝒏𝑖𝑖 𝑖𝑖
2∑ 𝒏𝒏𝒊𝒊 𝜌𝜌𝑖𝑖̇ = 𝟎𝟎

(5)
(6)

Reference[1] W.Wu, Z.You, Modelling rigid origami with quaternions and dual quaternions, Proc. R. Doc. A466, (2010),
pp.2155-2174.
[2] N.Watanabe, K.Kawaguchi, The method for judging rigid foldability, Proc. 4th Int. Conf. on Origami in
Science, Mathematics, and Education (4OSME), AK Peters (2006), pp. 165-174.
[3] T.Tachi, Simulation of rigid origami, Proc. 4th Int. Conf. on Origami in Science, Mathematics, and Education(4OSME), AK Peters (2006), pp. 175-187.

29
Bifurcation diagram of disk packings on logarithmic spirals
Yoshikazu Yamagishi1 and Takamichi Sushida2
1

Department of Applied Mathematics and Informatics, Ryukoku University, Japan
2

Research Institute for Electronic Science, Hokkaido University, Japan

1. Introduction
Van Iterson’s bifurcation diagram for disk packings is a fundamental tool in the geometric study of spiral
phyllotaxis. He studied the cylindrical model with linear lattices, the centric model in the plane, and a cusp
model on the cone in 1907. The cylindrical model is popular recently, as it has PSL(2,Z) symmetry. It is
known that the bifurcation diagram of disk packings on linear lattices is a dual graph of the bifurcation
diagram of Voronoi tilings on the same linear lattices [1]. Here we show the duality between the bifurcation
diagrams of disk packings and Voronoi tilings in the centric model, on logarithmic spiral lattices [2,3].

2. Bounded distance with multiplicative symmetry.
j
A logarithmic spiral lattice   z jZ has a single generator z  C . Fig.1 is an example with disk
parastichy numbers 3, 5, and 8, where the dashed lines show the Voronoi tiling for  . In van Iterson’smodel, the radius of each disk R | z j | is proportional to | z j | . Our result is that the coefficient
R defines a bounded distance function in the plane, d ( z, w) | z  w | /(| z |  | w |) .

Fig.1 Disk packing and Voronoi tilingon a logarithmic spiral.

Fig.2 Bifurcation diagram of Voronoi tilings (dashed lines)
and van Iterson’s diagram

3. Disk parastichy number, Voronoi parastichy number, and continued fractions
In the phyllotaxis theory, the number of spirals is called a parastichy number. We show that the bifurcationof Voronoi tilings is written by an equation of the packing distance d . We show that, if the generator z
is fixed, a disk parastichy number is also a Voronoi parastichy number. The continued fraction expansion of
the divergence angle arg(z) / 2 plays an important role in the proof. This gives the duality of thebifurcation digrams, and a rigorous proof that van Iterson’s diagram in the centric model is connected andsimply connected, with the Farey tree structure.

Reference[1] Hellwig, H., and Neukirchner, T., Phyllotaxis, Die mathematische Beschreibung und Modellierung vonBlattstellungsmustern, Mathematische Semesterberichte 57 (2010), 17-56.
[2]Yamagishi, Y., Sushida, T., and Hizume A., Voronoi spiral tilings, Nonlinearity 28 (2015) 1077-1102.
[3] Yamagishi, Y., and Sushida, T., Spiral disk packings, submitted.