Building with origami structures elainebonavia

BUILDING WITH ORIGAMI STRUCTURES

ELAINE BONAVIA

JUNE 2014

A dissertation completed in partial in partial fulfilment of the requirements for the Degree of Bachelor of Engineering and Architecture (B.E.& A. (Hons)) at the University of Malta.

Building with Origami Structures

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ABSTRACT Transformable structures are capable of existing and working with forces in both

static

and

dynamic

forms

often

with

the

ability

to

undergo

large

geometrical and volumetric transformations, enabling efficient compaction.

The ancient art of origami comes as an inspiration for the design and study of such structures. Rigid origami can be produced on a human scale due to its ability to

enable

models

not

are

folding without the

only

deformation of its

scale-independent

but

applications

facets. These within

various

engineering disciplines are widespread. This dissertation is concerned with using active rigid origami tessellations for architectural and building purposes.

When

building

with

origami

structures,

issues

concerning

deployment,

thickness, connection and boundary conditions will arise due to the scale of the model and material limitations. These practical limitations are studied and investigated through two examples: a canopy tessellated with the Miura-ori pattern and a rigid foldable cylinder based on the Tachi-Miura Polyhedron. A discussion that analyses these constraints which directly influence buildability then

ensues,

proposing

the

most

practical

methods

from

a

geometric

perspective.

A final discussion regarding the usability of these structures in the building industry and the potential that they may or may not have concludes this study.

KEYWORDS: Rigid Origami, transformable, Miura-Ori, Rigid-foldable cylinder, origami structures, origami tessellations.

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Authenticity form

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Dedicated to my parents

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ACKNOWLEDGEMENTS I would like to thank Professor S. Buhagiar B.E.&A. (Hons), MSc (Lond.)

D.I.C., Ph.D. (Lond), M.I.Struct.E, C.Eng, Perit for his continuous support throughout this study and for his patience, interest and belief in the topic.

I would also like to thank Professor A. Torpiano B.E&A (Hons), M.Sc. (Lond), Ph.D. (Bath), D.I. C., M.I.Struct.E, C.Eng., Eur.Ing, A&CE, Perit for being the

foremost example in the development of my architectural education

throughout this course of studies. My thanks also go to the academic staff at the Faculty of the Built Environment for sharing their knowledge with utmost dedication.

I would also like to extend my gratitude to my parents for their understanding and patience throughout the past five years, to my siblings for bearing the late nights and the clutter, and to Chris, who has unconditionally been caring and supportive at every stage.

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TABLE OF CONTENTS BUILDING WITH ORIGAMI STRUCTURES...........................................i ABSTRACT...............................................................................ii ACKNOWLEDGEMENTS................................................................v LIST OF FIGURES......................................................................ix CHAPTER 1.

INTRODUCTION.....................................................1

1.1 Building with Origami Structures..........................................1 1.2 Scope..........................................................................1 1.3 Research Questions.........................................................3 1.4 Layout.........................................................................3 CHAPTER 2.

LITERATURE REVIEW...............................................5

2.1 What is Origami? ...........................................................5 2.2 Active Origami Tessellations...............................................7 2.3 Rigid Origami...............................................................12 2.4 Origami and Mathematics................................................16 2.5 Rigid Origami Mathematics..............................................19 2.5.1 Mathematical models of rigid origami.........................20 2.5.2 Kinematic Models of rigid origami.............................22 2.6 Rigid Folding in Nature....................................................23 2.7 Origami in Engineering....................................................24 2.7.1 Aerospace Engineering..........................................25 2.7.2 Medicine............................................................27 2.7.3 Material science...................................................28 2.7.4 Nanotechnology and Self folding Origami....................29 vi

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2.7.5 Other Applications................................................31 2.8 Origami in the Building Industry: Architecture and Engineering....31 2.8.1 Static interpretation of Origami in Architecture..............31 2.8.2 Non –static interpretation of Origami in Architecture and Structural Engineering....................................................43 2.9 Conclusion..................................................................52 CHAPTER 3.

BUILDING

WITH

RIGID

FOLDABLE

ORIGAMI

STRUCTURES..........................................................................54 3.1 Introduction.................................................................54 3.2 Basic forces in active rigid origami tessellations.....................54 3.3 Existing problems..........................................................56 3.3.1 Thickness...........................................................56 3.3.2 Connections........................................................62 3.3.3 Deployment........................................................65 3.3.4 Boundary...........................................................68 3.4 Discussion...................................................................69 3.5 Conclusion..................................................................73 CHAPTER 4.

CASE STUDIES....................................................74

4.1 Introduction.................................................................74 4.2 Deployable Roof Structure...............................................74 4.2.1 Pattern Selection..................................................74 4.2.2 Usability of Structure.............................................76 4.2.3 Connection and Thickness......................................78 4.2.4 Boundary and Deployment......................................83 4.3 Foldable Cylinder Structure...............................................87 vii

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4.3.1 Pattern Selection..................................................88 4.3.2 Size and Proportion...............................................91 4.3.1 Building components.............................................92 4.3.2 Thickness, Connection and Water-tightness.................93 4.3.3 Boundary and Deployment......................................93 4.4 Conclusion..................................................................94 CHAPTER 5.

FINAL DISCUSSION...............................................95

5.1 Research Questions.......................................................95 5.2 Why foldable origami structures in the building industry?..........95 5.3 The Concept of Transformable Architecture..........................98 5.4 Usability and Performance...............................................99 5.5

Research

trends

in

the

field..............................................100 CHAPTER 6.

CONCLUSION.....................................................102

6.1 Conclusions...............................................................102 6.2 Recommendations for Future Work...................................102 BIBLIOGRAPHY......................................................................104 APPENDIX I – Software Description..............................................116 APPENDIX II – Crease patterns....................................................118 APPENDIX III – Grasshopper Definitions.........................................120 APPENDIX IV – Documentation of physical models...........................122

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LIST OF FIGURES Figure 2-1: Scheme of a Rabbit Origami Model taken from Akira Yoshizawa’s book ‘Creative Origami’. ................................................................. 5 Figure 2-2:

The four basic steps to create an origami form. Source: (Lang,

2008) ........................................................................................ 7 Figure 2-3 Origami Tessellations as described by (Garibi, 2011) ............... 8 Figure 2-4: Action origami models. Source: (Wikipedia, 2012-2014) ......... 9 Figure

2-5

:

Various

examples

of

Active

Origami

Tessellations.

Source:

(Piker, 2009) ..............................................................................10 Figure 2-6: Regular Ron Resch pattern (top) and the process of obtaining a freeform origami tessellation (bottom). ..............................................11 Figure 2-7: Origamizer and Resch’s pattern showing tucks. Source: (Tachi, 2013) .......................................................................................11 Figure 2-8: Rigid Origami square twist fold model simulation. Source: (Hull, 2003) .......................................................................................12 Figure 2-9: Triangular mesh model for rigid origami. Source: author. ........14 Figure

2-10:

Diagrams

explaining

how

to

model

a

single

rigid

vertex

mathematically using the matrix model. Source: (Hull, 2004) ..................20 Figure 2-11: Diagrams explaining the basic concept of a Gauss mapping in the context of folding. Source: (Hull, 2004) ........................................21 Figure 2-12 : A classic square twist fold, an example of non-rigid origami .22 Figure 2-13: Folded plate structure in Chamaerops humilis. Source: (left) (Trautz & Herkrath, 2009); (right) Author. ...........................................23 Figure

2-14:

(Left)

The

Eyeglass

Telescope

by

Robert

Lang.

Source:

Laurence Rivermore National Library. (Right) The Miura-ori solar panel array shown in space. Source: (Miura, 1994) .............................................26 Figure 2-15: An origami stent. Source: (You & Kuribayashi, 2006)............27

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Figure 2-16: The Origami Paper Analytical Device (oPAD). Source: (Liu & Crooks, 2011) ............................................................................28 Figure 2-17: Cellular meta-material made from stacking a number of folded sheets. .....................................................................................29 Figure 2-18: Programmed Origami Folding. Source: (Hawkesa, et al., 2010) ...............................................................................................30 Figure 2-19: Image showing load distribution in folded structures. Source: (Trautz & Herkrath, 2009) ..............................................................33 Figure

2-20:

Image

showing

different

types

of

folded

plate

structures.

Source: (Trautz & Herkrath, 2009) ....................................................33 Figure 2-21: (Left) Eugene Freyssinet’s Orly Hangar in pre-stressed concrete.

Source: (Anon., 1916) ..................................................................35 Figure 2-22: (Top Left) Chapel of St Loup, Switzerland. Source: Milo Keller.

(Top

Right)

Timber

pavilion,

Osaka.

Source:

Beijing

International

Design

Triennial. (Bottom Left & Right) Prototype testing of folded plate structure made from cross-laminated timber panels. Source: (Buri & Weinand, 2008) ...............................................................................................38 Figure 2-23: Detail of the Hinzert Museum exterior. Source: Robert Miguletz 39 Figure 2-24: US Air Force Academy Chapel during construction. Source: The Denver Post Library Archive ............................................................40 Figure 2-25: Plate House Model by Joe Gattas. Source: www.joegattas.com ...............................................................................................41 Figure 2-26: Mezzanine ceiling called "Kielsteg", made from timber. Source: (Ĺ ekularac, et al., 2012)................................................................42 Figure 2-27: Design of an irregular folded plate structure. Source: (Trautz & Herkrath, 2009) ...........................................................................43 Figure 2-28: Temporary structures built using thin profile materials ...........44 Figure 2-29: Arch model of a potential temporary structure modelled using thick aluminium sheets. Source: (Gioia, et al., 2011) ............................45

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Figure 2-30: (Top) Images showing the foldable aluminium structure and the existing supporting structure. (Bottom) Details of the ‘mountain’ and ‘valley’ connections. ..............................................................................47 Figure 2-31: Design of a foldable passage between two buildings (left) and its construction details (right). Source: (Tachi, 2010) ............................48 Figure 2-32: Collapsible beam prototypes built using aluminium sheeting.

Source: www.joegattas.com ...........................................................49 Figure 2-33: Foldable bar structure based on the geometry of foldable plate structures. Source: (Temmerman, 2007) ............................................51 Figure 3-1: Diagrams showing the behaviour of forces inside a thin paper model (left) and a thick material model (right). Image source: Author. .......55 Figure

3-2:

Diagrams

explaining

the

basic

problem

of

thickness.

Image

source: Author. ...........................................................................57 Figure 3-3: (Left) Diagrams showing Hoberman’s proposal for thickness.

Image from (Hoberman, 1988) (Right) An example of sliding hinges model proposed by Trautz and Kunstler. Here the sliding value is accumulated along the hinges on the right. Image from (Tachi, 2010) ................................58 Figure 3-4: (Top) Two approaches for enabling thick panel origami. (a) Axisshift (b) The proposed method based on trimming by bisecting planes. Red path

represents

the

ideal

origami

without

thickness.

(Bottom)

origami with modified panels using the Tapered Hinges Model.

Freeform

Both images

from (Tachi, 2010) .......................................................................60 Figure 3-5: Image showing a model using the Constant Thickness Method.

Source: (Tachi, 2010) ...................................................................61 Figure 3-6: Doubly expandable shell with reinforcement. Image from (Resch & Christiansen, 1970) ...................................................................62 Figure 3-7: Vacuum hinges as proposed by (Tachi, et al., 2012) .............63 Figure 3-8: (Left) Typical example of a living hinge made from PETG. Image from http://www.sdplastics.com; (Right) Diagram explaining the concept of

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internal

and

external

trusses

used

for

stiffening

connection

zones

and

facilitating deployment. Image from (Gentry, et al., 2013) ......................64 Figure 3-9: Images showing the principle of ‘connectionless’ joints at the vertices. Left image from (Tachi, 2010) and right image from (Gattas, 2012) ...............................................................................................65 Figure 3-10: Diagram showing the construction process of a deployable structure powered by vacuumatics. Image from (Tachi, et al., 2012) .........66 Figure

3-11:

Morphing

Section 2.8.1.6.

panels

using

the

Foldcore

concept

described

in

Images by (Gattas, 2012) ........................................67

Figure 4-1: Detail of a Miura-Ori module, Source: Author .......................76 Figure 4-2: Deployment stages of a Miura-ori tessellated canopy. Source: Author ......................................................................................77 Figure 4-3: Model of a single Miura-ori module modelled with thickness (top left) and with tapered panels (top right) .Source: Author.........................79 Figure 4-4: Plan view and detail of a Miura-ori module modelled using the tapered hinges model showing the zones where hinging can occur. Source: Author. .....................................................................................80 Figure 4-5: Figures showing a series of Miura-ori modules as modelled using the constant thickness panel method. Source: Author ...........................81 Figure 4-6: Physical model using fabric sticks and thread to assimilate and test out the concept of a fabric pocket. Source: Author, 2013 .................83 Figure 4-7: All deployment stages of the Miura-ori roof showing the position of the fixed perimeter edges. Source: Author ......................................85 Figure 4-8:

All deployment stages of the Miura-ori roof showing compaction

towards the centre. Source: Author ...................................................86 Figure

4-9:

A

series

of

non-rigid

deployment states. (From Left)

foldable

cylinders

and

their

different

Cylindrical bellows cylinder, classic bellows

cylinder and deployable cylinder based on the Yoshimura pattern. Source: Author. .....................................................................................88

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Figure 4-10: Deployment stages of a Zig Zag cylinder from (Tachi & Miura, 2013). Image source: Author ..........................................................89 Figure 4-11: (Top) Images showing the TMP rigid foldable cylinder crease pattern

and

deployment

stages

as

proposed

in

(Tachi,

2010).

(Bottom)

Diagram showing a typical TMP cylinder highlighting Miura-ori fold within.

Source: Author ............................................................................91 Figure 4-12: Varying widths are possible by simply modifying the crease pattern dimensions. .....................................................................92 Figure 4-13: Schematic diagram depicting the insertion of clear glass panels.

Source: Author ............................................................................92

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CHAPTER 1. INTRODUCTION

1.1 Building with Origami Structures This dissertation deals with the prospect of applying origami as the basis of a new,

transformable

building

typology.

It

is

also

coming

from

a

continued

fascination with the way origami can provide transformation in such a neat and efficient manner.

1.2 Scope Our built environment is increasingly becoming more susceptible to physical change such that the changing needs and wants of society are not only more frequent but also more widespread. As a result, it could be said that the usability of traditional buildings could be seen as decreasing due to their inability to transform spatially to meet new trends. Research in this topic has shown that the current and future nature of the working and living spaces will undergo such a drastic change that the built environment may have to slowly make way for completely new building structure typologies.

While traditional dwellings are designed for a life of 50 years, today we are faced with

buildings

that

require

alterations

within

at

least

5

years

due

to

users’

changing spatial desires. Buildings that cannot respond to these changes are usually either completely or partly demolished leading to considerable amount of physical, financial and time waste. Thus, it may be argued that architects should begin taking

into consideration the

building’s

life

cycle

and design for

this

transformability from an early stage.

Of course, it is unnecessary to think that these new structural typologies need to be so drastic so as to transform the shape of the building itself. The solution 1

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may be more of a question of detailing internal and external components.

However, in this context, taking a revolutionary approach using origami models as the basis for transformation may provide an interesting solution which could perhaps

be

the

only

solution

in

remote

locations

such

as

space

or

harsh

environments.

Yet, a building is and has always been a static structure intended to provide inhabitants with shelter and usable space and it is a strange thought to aspire for a

building

that

is

designed

to

incorporate

both

the

static

and

dynamic.

Essentially, repetitive movement would mean that the building becomes a sort of machine. Luckily, various origami models showing different levels of movement already

exist

and

can

provide

a

base

transformation

path.

Paper

folding

techniques can be efficient due to their ability to transform from a flat sheet of paper

to any

shape. Action origami

may be seen as one

possible

means

through which designers can start to cater for the need of transformability in architecture. Rigid origami, a sub-topic of origami has certain properties that can (theoretically) enable the realisation of human scale system using rigid materials.

Therefore, there is scope in identifying the main challenges associated with using origami structures as a potential building tool in the future. In hindsight it may be easy to associate origami models with architectural uses, but in truth a number of limitations may be met in the process of realisation.

Today, origami has infiltrated a number of research topics, especially in various engineering

fields.

This

study

will

also

question

where

today’s

research

is

heading and whether there is potentiality for such structures to make an impact or co-exist within the current building industry.

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1.3 Research Questions Given the technological difficulties that exist, is the adoption of foldable origami tessellations realistic in today’s building industry? How and why is this so?

What can we say about the direction being taken from the way research has dealt with this topic thus far? Is it leading anywhere and are the applications justifiable?

1.4 Layout This document is divided into 6 chapters, each of which is described below:

Chapter 2 is a compilation of literature which initially introduces the topic of origami and then proceeds to describe the principles and properties associated with rigid origami tessellations. Following this, the usefulness of rigid origami is demonstrated though a series of studies on origami in various engineering fields.

Work related to the building industry is treated separately and divided by virtue of its mechanical nature. Origami work is classified as static and non-static, citing examples in both cases. The chapter concludes with a few thoughts on the nonstatic work that has been mentioned.

Chapter 3 focuses on the tessellations are

scaled up

problems that

emerge

when active rigid

to human sizes. Problems related to

origami

thickness,

connections, deployment and boundary conditions are presented in more detail though studies and observations. A discussion that points out the most viable methods and discusses how these critical issues can be realistically dealt with ensues.

Chapter 4 presents two case studies – a transformable roof based on the Miuraori tessellation and a bi-directionally deployable rigid foldable cylinder based on

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the Tachi Miura Polyhedron. The issues brought forward in Chapter 3 are applied to the case studies in a realistic scenario and various methods are discussed.

Chapter 5 discusses the research questions in light of the conclusions derived from the case studies.

Chapter 6 concludes this dissertation and proposes suggestions for future work.

Appendix I, II and III respectively include information about the software used in this dissertation, the Grasshopper definition for the Miura-ori tessellation and crease

patterns

for

both

the

Miura-ori

tessellation

and

the

Tachi

Miura

polyhedron Cylinder.

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CHAPTER 2. LITERATURE REVIEW

2.1 What is Origami?

‘The art of paper folding’ - origami, is known to many across the world as a Japanese tradition. Its origins are widely disputed as evidence is unclear, although records showing the Chinese paper folding (Zhe Zhi) tradition predate those of origami. Most likely, these two crafts were developed contemporarily and initially for ceremonial purposes. (Wikipedia, 2004-2014) Of course, in the case of origami, this was possible only after the importation of paper from China in the 6th century.

Origami art is made from a sheet of paper that is subject to a series of folds that eventually transform it into a figure. This figure often represents an animal or an inanimate object but ultimately, practically any shape is possible. If

Figure 2-1: Scheme of a Rabbit Origami Model taken from Akira Yoshizawa’s book ‘Creative Origami’.

unfolded, the lines or more accurately, the ‘creases’ used to make the final shape are revealed on the paper, and the result is known as a crease pattern.

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Throughout the years, studying folds and crease patterns led to the development of different techniques for generating the final origami shape, in many cases enabling more physical detail to be achieved and in doing so prompting further research into the subject. Ultimately, such investigations revealed that the craft could be developed much further.

As a result, different categories of origami emerged, spinning people’s interests

Kirigami, which involves folding process; Wet folding, where the formation of curves; Pureland

in different directions. Today, these categories include cutting and gluing the paper throughout the the origamist uses damp paper to aid in

Origami

where a number of folds such as reverse folds and simultaneous folds

Origami Tessellations

are not allowed;

Action Origami Origami which

which involves the repetition of modules;

which is origami that moves in intelligent ways; and

Modular

involves stitching several pieces of separately folded objects

together. (Wikipedia, 2012-2014)

Centuries apart from discovery, a major step in the development of origami occurred in the 20th century due to Akira Yoshizawa: the first man to pursue origami as a career when it was only regarded as a children’s hobby. In doing so, Yoshizawa pioneered the wet folding technique amongst others and devised a system of representing and diagramming crease patterns. (Wikipedia, 20052014) (Fox, 2005). He also had the opportunity to exhibit his work at the Louvre in 1998, an event which undoubtedly put origami on the international artistic map. Several of today’s renowned origamists such as Tomoko Fuse, Peter Engel, Robert J. Lang, Samuel Randlett and John Montroll started to emerge at around this time.

Regarding the process of creating origami models, Robert J. Lang says that the principles behind any origami figure are based on the same simple four steps: (See Figure 2-2) First, starting out with the idea; Then reducing it to an abstract and simple form - a stick figure; Thirdly, getting a folded shape that has a part 6

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for every member of the abstract form; And lastly adding detail (i.e. more folds) to this folded shape so as to assimilate the original model more closely. (Lang, 2008)

Figure 2-2:

The four basic steps to create an origami form. Source: (Lang, 2008)

Today, origami is mainly an art form and many people continue to dedicate their lives to become origamists. Apart from this however, it has found its way into many disciplines: architecture, engineering, pedagogy, mathematics, fashion, computation, psychology, biology, medicine, to name a few. The fact that something 2D can be transformed into something 3D, and the process through which this occurs continues to be the main point of interest for many researchers.

2.2 Active Active Origami Tessellations It is not completely correct to use the word ‘origami’ to describe what this dissertation intends to study because the term is too general. More appropriately, we will be dealing with ‘origami tessellations’ which can be defined as a repetition of the same module or pattern of modules that follow a specific algorithm. As is usual in classic origami, these tessellations are formed from a single piece of paper meaning that no cutting and gluing is involved.

(Origami Resource Center, n.d.)

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Not surprisingly, there are different types of origami tessellations, although nobody has devised a complete classification system since they were popularised by Shuzo Fujimoto in the 1960s.

Ilan Garibi uses four main

descriptors to explain the different tessellation groups: 1) Classic, 2) Corrugation, 3) Recursive, 4) Organic (or ‘Back and Forth’).

Figure 2-3 Origami Tessellations as described by (Garibi, 2011)

The classic tessellations are based around hexagonal and square grids and their main property is that the module may be produced in any direction along the surface. The corrugation tessellation is perhaps the most common - its final appearance incorporates a module arranged in a series of ‘hills’ and ‘valleys’.

Recursive origami tessellations are similar to fractal theory and concerned with a pattern that is repeated, each time diminishing in size. According to Garibi, the rule for organic tessellations is the most basic: rows are folded forth. Repeat’.

‘one row folded back and two

Since Fujimoto, this branch of origami lives on

through artists such as Joel Cooper and Chris Palmer who have continued working with these tessellations, combining different genres and making up new patterns. (Garibi, 2011) While such tessellations are impressive within themselves, fusing this genre of origami with something like action origami provides interesting opportunities beyond the realm of paper. Most action origami models are used to make toys or impressive sculptures that can move occasionally although there are some 8

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which are based on tessellation patterns and can be compressed or expanded repeatedly. Examples include the Miura-Ori fold pattern and the Action Spring as shown in Figure 2-4: Action origami modelsFigure 2-4.

Figure 2-4: Action origami models. Source: (Wikipedia, 2012-2014)

This dissertation considers origami tessellations that are able to transform from a compact state to an incompact state. Such origami forms have not been officially named, thus, they will hereunder be referred to as Tessellations’

‘Active

Origami

. A number of examples are shown in Figure 2-5.

Various models of active origami tessellations have been created using a regular pattern. Perhaps the most renowned ones are the Ron Resch pattern and the Miura Ori pattern. In the 1960s and 70s Ron Resch discovered that certain folds in paper could produce interesting movements and rotations. He furthered his observations to the point where he started diagramming and regularising paper creases which eventually led him to produce entire tessellations composed of modules that were based on hexagonal, square and triangular geometries. Many of the patterns produced by Resch are interrelated and derived from each other.

His diagrammed tessellations were then folded using a single piece of paper- a painstaking exercise for any folder, as one must ‘fight’ with the paper to achieve the result.

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Figure 2-5 : Various examples of Active Origami Tessellations. Source: (Piker, 2009)

Resch’s work initially dealt with active paper (origami) tessellations and was driven by the desire to work towards an aesthetic form and as they happen’

‘to investigate things

(The Ron Resch Paper and Stick Film, 1992) but the need to

also find a function for these works soon began to arise. His studies on the behaviour of folding from one state to another were furthered through stick models, in an attempt to relate the folded paper models to buildings. The idea was that moveable models could be used for the form-finding process, and then lock into place and have a rigid model of the building.

Ron Resch’s approach was highly dependent on manipulating regular geometry but more recently, with the help of computational modelling some work with freeform tessellations has started to emerge, producing what is known as a ‘generalised’ version of a tessellation. This means that the regular tessellation is somewhat stretched, skewed and manipulated in such a way that it resembles a freeform surface.

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Tomohiro Tachi carried out a study on freeform origami tessellations using the Ron Resch pattern. This was re-constructed along a given surface while still keeping the origami conditions of flat foldability and developability. The result is a generalised version of the original pattern, what Tachi calls ‘freeform origami’

Figure 2-6: Regular Ron Resch pattern (top) and the process of

Figure 2-7: Origamizer and

obtaining a freeform origami tessellation (bottom).

Resch’s pattern showing tucks.

Source: (Tachi, 2013)

Source: (Tachi, 2013)

(Figure 2-6). This is achieved by introducing foldable polygons between the facets of the original pattern and mapping the original tessellation vertices onto the developable surface so that two conditions are satisfied: 1) the folding angle limitation and 2) the non-intersection of local facets (Tachi, 2013). Tachi proposed that his method can be applied to paper and metal, illustrating this with design and fabrication examples. It must be said that the use of any other material with a significant thickness may pose a problem to this method due to the complex 3D structure of the foldable polygons (‘tucks’) that need to be flat foldable for the surface to be approximated accurately, something that cannot be achieved unless they have a very thin profile (see Figure 2-7).

Active origami tessellations present themselves as an attractive solution wherever the need to employ efficient and repetitive kinetics arises. The way such tessellations have been practically utilised in the physical world so far (generally in engineering) has been limited to regular patterns. Doing anything 11

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based on freeform patterns becomes a significantly more complex challenge and although not yet realised, it is certainly possible when you take into account what recent technological advancements can provide.

2.3 Rigid Origami Further to using active origami tessellations, this dissertation explores rigid origami. Origami has found its way in computational, artistic and mathematical fields but comparatively not so much in architecture and structural engineering.

Rigid origami is that branch of origami that has the potential to make this

Figure 2-8: Rigid Origami square twist fold model simulation. Source: (Hull, 2003)

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possible. In simple terms, rigid foldable origami refers to

‘a plates and hinges

model for origami’ (Tachi, 2010) – that is, a kind of origami which is capable of continuous

transformation

without

the

deformation

of

each

facet

such

that

synchronised motion is produced (Figure 2-8). Rigid foldability can be simulated using any material that can be used to make rigid panels. Each rigid panel is connected to the adjacent panel by a rotational hinge, usually having a single axis.

The concept of rigid origami may be difficult to grasp for anybody who is not a folder- how could the condition of planar facets be limited to such an elite group of origami? And what is so rare and unique about the possibility of having a hinge replace a crease line?

The truth is that very often during the folding

process the facets of elastic (non-rigid) origami models are bent and twisted in the

process

uncommon,

of

making

often

the

requiring

product,

and

proof.

(Hull,

the

alternative

2003).

This

case proof

of

rigidity

(and

is

hence

classification) is useful when it comes to manufacturing foldable objects using sheet materials.

Tachi has reiterated that when designing kinetic architecture, using rigid origami becomes advantageous for a number of reasons: Firstly, there is potential in utilising a material surface that could provide a watertight space (subject to the success of the connection detailing); Secondly, the mechanisms presented by these

rigid

origami

structures

are

solely

based

on

geometry

and

hence

independent of the material properties and transformations therefore providing less design constraints; And thirdly, deployment of the structure can happen in an almost automatic way (Tachi, 2010).To further explain this one can say that deploying a rigid origami structure

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‘...can be considered as involving a deformation with a small energy but not

zero

energy

as

in

pure

mechanism

movements;

nevertheless,

this

stored energy may also be useful as an elastic spring acting as an actuator for desired ‘unfolding’ movement.’ (Dureisseix, 2012)

Structures derived from rigid origami have been proposed since the 1970’s by various people such as Miura, Resch, Christiansen and Hoberman who were all fascinated by the geometric relationships that exist internally. However, none of these people have worked with these moving structures specifically for

Figure 2-9: Triangular mesh model for rigid origami. Source: author.

architectural purposes. Rigid origami has also been applied in a number of engineering solutions in various industries such as deployable structures or robotically engineered materials.

Tomohiro Tachi – an architect by profession- has proposed a design methodology for rigid origami structures, which is perhaps the foundation stone for any future use and design of rigid origami in architecture. He states that the most flexible design strategy is produced from a triangular mesh where the boundary arrangement controls the kinetics of the structure. The example demonstrated by Tachi is that of a pyramid with a triangular base. Its 14

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transformations are based on the vertex positions of the

base whereby no

translational movement occurs when these are fixed. This structure is therefore statically determinate in this state, having 9 Degrees of Freedom (DOF) since the topmost vertex has 3 x 3 DOF (rotational and translational) and the base vertices are only subject to rotations.

Variations of this kind of structure can be designed easily since everything is controlled by the edge elements. However, the configuration of the structure as it moves through space from one state to another is not tackled. This can be understood by studying the behaviour at every state throughout the configuration space

using

simulation

methods

such

as

the

unstable

truss

model

or

the

rotational hinges model (Tachi, 2010).

Another design approach is to use quadrilaterals, however to do this, one must prove structural rigidity and singularity since the number of DOF is not so easily computed, which means we cannot establish whether or not it is statically determinate. A general rule for proving/disproving singularity has not been found yet. In his work Tachi therefore proposes to design a generalised version of an existing rigid origami model.

In spite of all this however, the practicality of rigid origami is highlighted through a number of very valid issues. The overall dynamic behaviour of these structures is still not fully understood, especially during the deployment stages. The fact that the quadrilateral mesh problem remains unresolved has a direct impact on the designability of rigid origami and a lot of configurations remain unclassified.

There is also a vast amount of work that can be done using rigid origami that is applied to a loop system. These structures (for example foldable cylinders) can have

a

great

impact

on

advances

in

architectural

rigid

origami.

Moreover,

making rigid origami structures mechanical is another unexplored field that will be possible to penetrate once kinetic constraints are successfully applied to rigid models.

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2.4 Origami and Mathematics At face value, the relation between origami and mathematics is not so apparent.

Realistically, origami can cover practically any form –models can range from the construction of living things to inanimate modular forms. The result incorporates several

facets

which,

when

flattened

out

reveal

the

crease

pattern

that

is

mapped onto a piece of paper, essentially comprising different polygons with various shapes and sizes. Invariably, following this logic it is not hard to imagine that there may be mathematical relationships based on geometric principles between these different polygonal patterns. Origami therefore has an intrinsic geometry, and has attracted geometrists for many years. (Schenk, 2012)

In 1971 T. Sundara Row, inspired by kindergarten teachings, published a book in which he described how Euclidian geometry can be represented more easily by using paper folding techniques rather than a compass, pen and paper. (Sundara Row,

1917).The

book

also

attempts

to

bridge

the

gap

between

geometry,

algebra and trigonometry, illustrating how they can be linked to one another. In one particular example however, it fails to show that the construction of a cube root using folding methods is possible. This was later proven by Margharita P. Beloch in 1936 (Wikipedia, 2003-2014). Eventually these observations, amongst others, led to the formulation of a number of rules which today form the basis of the mathematical principles of paper folding. (Wikipedia, 2003-2014)

Today, origami can be used in the study of calculus, geometry or even abstract algebra - several origami techniques continue to be used in the construction and analysis

of

mathematical

problems

such

as

cubic,

quadratic

and

partial

differential equations. This easy convergence between the two subjects is largely owed to the properties of a sheet of paper: stretchable, cannot

be

non-compressible, sheared

-

much

rigid

like

a

in

paper is of minute thickness, non-

tangential

plane

in

Real

directions, space,

foldable

where

all

and these

aforementioned properties can be described from a mathematical point of view.

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(Dacorogna, et al., 2010). These properties have a direct impact on the shapes that can be attained through origami folding.

A

number

of

concepts

are

important

in

this

scenario.

Gaussian

curvature,

defined at a point on a surface, is a measure of the curvature of a surface and is computed by the product of the two principal curvatures. It is an intrinsic property of any surface, and crucially, it is constant when the surface is subject to bending- in other words, when paper is subject to folding. Thus, the Gaussian curvature of flat paper is zero at every point and whenever the paper is folded.

Zero Gaussian curvature limits the geometries that the paper can assume – for example it is impossible to map a paper along a sphere or a saddle without folding or distorting it. Furthermore, because the paper surface has zero Gauss curvature, it is also a developable surface which implies that it can be flattened to a plane without distortion. All the angles at a fold vertex add up to 360° - a perhaps obvious but rather important property when computing crease patterns for an origami construction. Evidently then, such mathematical advancements in understanding the foldability of origami patterns has been, and continues to be very useful for origamists. (Schenk, 2012)

Much like geometry, origami has its own set of theorems. The most important set of axioms were discovered by Humiaki Huzita and are considered to be the basis

of

origami

mathematics.

The

Huzita-Hatori

axioms

describe

the

relationships between points, lines and folds. (Krier, 2007) Also, the concept of flat foldability is deeply embedded in these origami theorems and is a major challenge in origami mathematics. Flat foldability implies planarity when the creases

are

in

a

folded

state.

In

practical

terms

this

is

useful

where

the

objectives include minimum packaging dimensions. Theorems on this topic are available for single vertices, for example the Kawasaki- Justin theorem states that:

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‘A single-vertex crease pattern defined by θ1 + θ2 + … + θn = 2π is flat foldable if and only if n is even and the sum of the odd angles is equal to the sum of the even angles, or equivalently either sum is equal to π.’

But then for multiple vertices, this theorem falls, and even for single vertices, it (along with other existing theorems) is not enough to prove flat foldability, and is limited to local flat foldability, leaving the global scenario (where all folds and vertices in the fold pattern are taken into account) unaddressed. Of course, overcoming these limitations is a mean challenge, but it demonstrates that there are very complex rules that govern origami folding.

Rigid foldability of origami is

an important concept when dealing with structures derived from origami and will be further discussed in Section 2.5.

During

the

1980s

and

90s

a

number

of

folders

such

as

Robert

J.

Lang

demonstrated the mathematics of various folded forms visually, leading to an increase in the complexity of origami models. Lang proposed a program – Treemaker- that could create a crease pattern out of any stick model, for practically any shape (as shown in Figure 2-2). Treemaker allows the designer to set up elaborate relationships between flaps, their lengths, and their angles, resulting in far more complex relationships than what is possible using penciland-paper origami design. The resulting crease patterns are often very difficult to fold and the designer must devise his own system to tackle the physical folding process.

According to Lang, although mathematics constitutes the underlying principles of

origami,

it

may

also

pose

certain

limitations

depending

on

the

set

of

theorems chosen by the origamist. For example in some cases these theorems may restrict the action of simultaneous folding – something which is often useful in practical folding.

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In

spite

of

this,

the

mathematical

aspect

of

origami

is

essential

for

the

development of engineering related to this field. Mathematicians such as Erik Demaine continue to make this possible through their work with proving and disproving theorems and questioning motion related problems. Several ongoing mathematical problems that exist in origami are being researched some of which

include

the

problem

of

flat

foldability

for

any

given

crease

pattern,

problems associated with rigid origami, the napkin folding problem which tries to prove that a sheet of paper can be folded into a form whose perimeter exceeds the original perimeter, or the fold and cut problem which investigates the shapes that can be obtained following a particular fold and cut algorithm.

Most of the time an engineering problem related to origami also implies a mathematical problem.

2.5 Rigid Origami Mathematics Rigid origami, being a restricted form of origami folding, also adheres to a number of mathematical borders. In practical terms this means it does not follow the Huzita-Hatori axioms, it follows Kawasaki's and Maekawa's theorem and it can provide a solution to the napkin folding problem within its realm. All this is, of course, only relevant if one is familiar with the theorems themselves.

In this document, the most important rules and concepts are those that will directly influence the ability of rigid origami to materialise in the engineering world.

Earlier, the origami

problem

means

that

of flat foldability was presented. Applying the

pattern not

only

folds

flat,

but

it

this

does

so

to

rigid

without

deformation of its facets throughout the folding process. This limits the number of crease patterns eligible to both conditions immensely. Interestingly, it has been shown that as a general rule a pattern must be made up of at least four

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creases in order to be rigid foldable. In turn, this implies that it has one DOF and that

‘a closed form solution may be found’

(Schenk, 2012). Tachi, who has also

investigated rigid flat foldability, states that: ‘in order to be able to apply rigid origami to architecture and engineering purposes

it

is

important

to

consider

the

geometry

of

origami

in

kinetic

motion and provide sufficiently generalised methods to produce controlled variation in shapes.” (Tachi, 2010)

2.5.1 Mathematical models of rigid origami Tachi’s approach is engrained in the construction of mathematical models that help gain an understanding of rigid origami. There are two ways of making a mathematical model of rigid origami – by using the Matrix Model of the Gaussian Curvature model (geometrical model). These models can be used to prove whether a piece of origami is rigid or not.

The Matrix model works on the premise that rigid motions of the plane are isometries and can be modelled with linear transformations. If we were to model

Figure 2-10: Diagrams explaining how to model a single rigid vertex mathematically using the matrix model. Source: (Hull, 2004)

a single rigid vertex, we can imagine a spider walking around the vertex on a 20

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folded paper. A number of mathematical relationships between the movement of the spider and the crease lines may be derived (See Figure 2-10). For the rigidity condition to apply, the spider must come back to its original starting point. Although the matrix model shows this mathematically, a number of problems arise – this rigidity condition does not say anything about the folding process and whether a rigid folding motion is possible, which makes it insufficient to prove rigidity. (Hull, 2004) The Gaussian curvature model deals with the Gaussian curvature of a point on a surface – a value which we can use to determine whether rigidity exists in bending particular folds. The concept of the Gauss map is related: through the projection of the curvatures of a surface onto a unit sphere, we end up with a curve that follows a particular direction, which will determine the sign of the curvature. (Wikipedia, 2013) Thus we can classify surfaces as having negative and/or positive curvatures. Interestingly, the Gaussian curvature at a point on a surface does not change if the surface is in bending. (See Figure 2-11, left)

Figure 2-11: Diagrams explaining the basic concept of a Gauss mapping in the context of folding. Source: (Hull, 2004)

Applying this to origami, we first consider the Gaussian curvature of a flat sheet of paper, which is zero, since there is no curvature, so the two principal curvatures are mapped as a parabolic point on the Gauss map. When a 21

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mountain fold is created, a positive Gaussian curvature exists, whilst when a valley fold is created there is negative Gaussian curvature. (Figure 2-11, right) Thus, as an example, we can prove that a four-valent vertex fold has zero

Figure 2-12 : A classic square twist fold, an example of non-rigid origami

Gaussian curvature through a number of equations that relate to the area of the Gauss map curve. (Hull, 2004) We can further prove that this fold can be folded rigidly. By examining the Gaussian curvature maps of several folds, it is therefore possible to know whether or not a surface can be folded rigidly or not.

(Wikipedia, 2013) An example of a non- rigid fold is the classic square twist fold as can be seen in Figure 2-12.

2.5.2 Kinematic Models of rigid origami When it comes to modelling the kinematics of rigid origami, there are two methods of representation that can be used – the Unstable Truss model (UTM) and the Rotational Hinges model (RHM). Both methods adopt certain constraints in order to measure the variation that occurs over the rest of the model. The UTM represents a rigid origami structure using the positions of the vertices, i.e. where the rigid bars meet along the crease lines of the pattern. The lengths of all the ‘plate edges’ of a rigid origami model are therefore preserved. In the RHM, constraints are applied on closed loops (such that they do not open) and the configuration is represented using the rotational angles of each edge. (Tachi, 2010)

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2.6 Rigid Rigid Folding in Nature A number of systems that utilise folding can be seen in nature. Natural systems (both flora and fauna) use this principle repetitively to achieve opening and closing movements or to provide structural stiffness. Folded structural systems may be observed in broadleaf-tree leaves, flower petals, carnivorous plants (such as the Venus fly trap), insect wings as well as many other species.

To illustrate this by example, the leaf of Chamaerops humilis sports a radial zigzag corrugation emanating outwards in order to provide it with its stability, especially against alternating wind loads (Trautz & Herkrath, 2009). This system is highly optimised, with the frequency of the fold decreasing closer to the edge of the leaf in correspondence to the decreasing load.

Similarly, it is thought that the blood vessels of a dragonfly and butterfly’s wings are not orthogonal: most cases in origami folding are not orthogonal, and also

Figure 2-13: Folded plate structure in Chamaerops humilis. Source: (left) (Trautz & Herkrath, 2009); (right) Author.

in nature. The dragonfly develops from a larva, metamorphoses into an adult insect, and the wings open from a closed condition, so in this process there is some relationship with origami folding. The straight lines we learn about in mathematics, circles, 2nd order functions, as well as curves and so on are simple because they are artificial. These kinds of curves rarely exist in the natural world. (Only the complex ones commonly exist) 23

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The structural characteristics of kinematic folding are analogical to the ones of the rigid folding. Natural folding as lightweight structure is highly interesting for engineering and architecture. It is a material-saving and efficient method of construction because it supplies the load bearing structure and the building envelope at the same time.

Nature’s ability to deal with folding so seamlessly has not gone unnoticed.

Experiments with thin walled cylinders in paper or polypropylene have revealed that a natural twist pattern exists under torsional loading. Highly regular self organising folds appear when a cylindrical sheet is rotated in two opposite directions

at

both

ends.

Although

these

patterns

have

been

designed

and

produced it is interesting to observe that they can be naturally derived. The material is perhaps the main reason behind this result – but a material that behaves in this way is undoubtedly present in nature itself.

Modern

bio-inspired

engineering

(BiE)

tries

to

imitate

natural

systems

and

biological structures for technical solutions and has so far had a fruitful history.

The

principle

biological

structures

environmental conical

behind

BiE and

pressures.

shaped

is

that

the

functions (Jenkins

members,

&

folding

evolutionary that

are

processes

highly

Larsen,

2008).

circular

sheets,

have

optimised Foldable 3D

produced to

diverse

cylinders

cores

based

and on

honeycomb patterns, foldable elliptical spheres, wrappable spiral models et al are

being

investigated

with

the

aim

of

producing

functional

structures.

Application exists in foldable commercial products and aerospace engineering.

2.7 Origami in Engineering Engineering Engineers have turned to origami as a source of inspiration for a diverse number of applications. Not surprisingly, the most successful area has been deployable space structures. The renewed interest in origami folding throughout the 20

th

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century has had enormous potential. This borrowing of origami principles is convenient for many reasons: firstly, scale is not an issue, changes in volume before and after packing are substantial, there is efficiency in mechanics and often

the

model can

go

from

a

flat

plane

to

a

flat

foldable

state,

easier

fabrication techniques are on the rise and in general the aesthetical result is appealing.

Throughout the

world, the

origami revolution has been transforming

several

research fields so much so that many of the world’s best Universities have set up their own research groups which focus solely on exploiting origami principles.

Funding designed

specifically

for such initiatives also exists in the US. This

sheds light on exactly how serious the international origami community is about making origami inspired products and smart solutions for the future.

Most of the research being done is structured around the automotive industry, nano-materials, self-assembling systems, aerospace solutions and product or pop-up manufacturing. The potential of this research is widespread, therefore the

aims

are

diverse

and

not

necessarily

defined

by

a

particular

solution.

Already, there exists a solid market for origami inspired medical devices, optimal packaging structures, toys, impact resistant materials and deployable space structures.

A concise review of the work achieved in each major field follows below. It sheds light on the usefulness and creative potential that these structures have.

2.7.1 Aerospace Engineering Due

to

the

huge

expense

incurred

by

space

missions,

the

most

important

concern when building something that needs to be launched in space is how to keep it lightweight and compact. The need for these types of structures has

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opened up vast opportunities for origami applications within the aerospace industry.

Most deployable solar panel arrays are intended to expand once the spacecraft has finished launching. The way they are packed prior to expansion has a significant effect on their suitability for such missions, which is where origami has proved to be so useful. The most famous solar panel array was designed by Koryo Miura in Japan using a new pattern which eventually became known as Miura-ori. This was used to deploy a large solar panel array from the SFU (Space flight unit) research vessel. (Trei, 2013). A recent collaboration between Brigham Young University (BYU) students, NASA and Robert Lang has led to the design of a solar array similar to an origami flasher, which is 25m in diameter when opened and 2.7m when folded.

Figure 2-14: (Left) The Eyeglass Telescope by Robert Lang. Source: Laurence Rivermore National Library. (Right) The Miura-ori solar panels array shown in space. Source: (Miura, 1994)

Further developments include the ‘Eyeglass telescope’, designed by Robert Lang. This is a 25m diameter foldable telescopic lens that can be packed into a space shuttle taking up a minimal amount of space (Fei & Sujan, 2013).

However, a full scale model is yet to be built.

Other developments in origami-based deployable structures for the aerospace industry include solar sails, inflatable origami booms, highly rigid extendible masts, space lenses and more.

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2.7.2 Medicine In medicine, the negative Poisson’s ratio exhibited by certain origami patterns has been exploited to create new tools. This property enables folded papers and cylinders to exhibit a transverse expansion when stretched An origami stent graft based on the waterbomb pattern has been developed to enlarge clogged veins and arteries, treating diseases such as artery stenosis and

Figure 2-15: An origami stent. Source: (You & Kuribayashi, 2006)

oesophageal cancer. The stent is in a compact state when it is travelling throughout the body, and then deploys at the site where the clot is encountered.

(Fei & Sujan, 2013) Deployment occurs through the super elasticity of the material or the SMA (shape memory alloy) effect at a particular body temperature.

The cylindrical surface fold is achieved by dividing a flat sheet into a number of rectangular units; each one being further divided between the neighbouring units.

Two edges of the flat sheet are subsequently joined to form the deployable cylinder shown in Figure 2-15. This process is, essentially, the reverse of the common folding process, achieving a negative Poisson’s ratio structure through expansion in both longitudinal and radial directions. A helical distribution of folding patterns is created to synchronise the expansion as well as to increase the radial stiffness after full deployment; this is achieved by shifting the two edges longitudinally by one unit.

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One major problem with previous stent designs was the fact that once opened, the artery becomes blocked again due to tissue in-growth through the opening of the meshes. The origami stent graft prevents this from happening through an integrated enclosure, without additional cover. (Kuribayashi, et al., 2006) Apart from the stent, origami has been used to make what is known as DNA

Figure 2-16: The Origami Paper Analytical Device (oPAD). Source: (Liu & Crooks, 2011)

origami and Cell Origami both of which adopt folding techniques. Fast and easy diagnosis on patients is now also possible using an ‘Origami Paper Analytical Device’ (oPAD) which can analyse body fluids with no laboratory analysis nor technical skills. The concept is to place all the samples on a sheet which is folded into a number of layers. Once the samples penetrate al the layers the oPAD may be unfolded and analysis is produced via colour changes. (Origami Resource Center, 2014)

2.7.3 Material science Folded meta-materials are materials that have been engineered to behave in a particular way – essentially a fancy name for a sheet of paper that has been folded to an origami pattern. These meta-materials can be made useful as folded shells or as impact resistant structures.

Folded shell structures exhibit the important property of auxeticity- i.e. a negative in-plane Poisson’s ratio, and a positive out-of-plane Poisson’s ratio.

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This is a useful property for morphing structures which can change shape whilst maintaining a continuous surface. Morphing wings is a typical application.

Stacking a number of these folded layers forms a cellular meta-material that

Figure 2-17: Cellular meta-material made from stacking a number of folded sheets.

Source: (Schenk, 2012)

can be used when it comes to blast impact mitigation, packaging material or safety crash-boxes in cars. The fold patterns in such arrangements are capable of withstanding higher buckling modes and therefore absorb more energy, providing more safety. (Schenk, 2012). This property of origami structures is being researched further and will potentially be integrated into more applications such as micro-structural units for ultra-light materials with superior thermal and mechanical properties. (You, 2007) Other work with origami in the field of materials includes the folding of microscopically thin silicon into different three dimensional shapes by means of water droplet actuators which, when entering the silicon capillaries induce forces that pull the edges of the foil together, causing the 2D to 3D transformation. (Fei & Sujan, 2013)

2.7.4 Nanotechnology Nanotechnology and Self folding Origami The field of self actuation is a vast topic that encompasses robotics, nanotechnology, programmable folding and more. In this scenario origami is 29

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advantageous because it provides a readily configured pattern that is awaiting kinematic manipulation. Some recent work revolves around self folding polymer sheets which gain motion when exposed to light, vacuum powered rigid foldable structures, origami DNA nano-robots, hydro folding- using water and ink to actuate self folding, and programmable joints that flex in response to electric warming.

The application of origami in these fields is not so different in principle from the

Figure 2-18: Programmed Origami Folding. Source: (Hawkesa, et al., 2010)

rest of the engineering world. For example, to simulate origami dynamics each origami segment is modelled as a rigid body and the creases are represented mathematically by revolute joints. Origami segments should remain rigid in order to maintain device integrity, meaning that a stiff material has to be used for the structural membrane. The creases only allow one rotational degree of freedom and the revolute joint model completely describes the relative motion of the segments. Screw calculus is then used to parameterize the centre of motion of each segment based on their centre of mass with respect to an inertial frame of reference. The equations of motion are formed using the relevant kinematics, material properties and constraints from which the trajectory is computed.

(Stellman, 2006) 30

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2.7.5 Other Applications Origami has also been used to make foldable paper Lithium-Ion Batteries for space and energy efficiency, an antenna made from nano-paper that can send and receive a wide range of frequencies, cosmic origami- dealing with the study on

how

dark

matter

folds,

origami

grocery

bag,

pollen

origami,

ultra

high

resolution origami lens, airbags, and basically anything else that needs to be folded and/or unfolded. (Origami Resource Center, 2014)

2.8 Origami rigami in the Building Industry: Industry: Architecture Architecture and Engineering Following from what we can observe in nature, the principle of folding may be used as a tool to develop structural systems. Yet while this has been widely exploited in other industries (automotive, aerospace etc.), the building industry stands in contrast and folding has so far been given a secondary role. And evidently there is scope to stop doing so, as self supporting walls and slab elements made from a flat thin sheet of metal and able to withstand a high load capacity have already been achieved in the aforementioned industries.

The approach so far has taken two distinct forms, namely Static and Non-Static architecture. Static origami structures are interpreted primarily as folded plate structures whilst non-static ones come in various forms ranging from roofs to kinetic building elements.

2.8.1 Static interpretation of Origami in Architecture Over the years the interpretation of origami in architecture has largely taken the form of structures that are shaped in a way that simply copies active origami tessellations at instance- static structures (like practically everything else in the building

industry).

These

structures

have

been

termed

as

‘folded

plate

structures’.

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However, in colloquial terms ‘folded plate structures’ can mean two things which appear to be identical from an aesthetical perspective. They are actually similar in principle but somewhat different in engineering terms and it is important to clarify this here. In rigid origami we speak of facets that are joined together at crease

lines

and

are

able

to

transform

without

facet

deformation.

One

interpretation is an exact scaled copy of this model – purely a ‘folded plate structure’ where rigid panels made from the material of choice would replace the facets and the physical connection would exist along the crease lines only.

Another interpretation is to use a ‘truss folded structure’ where the crease lines are

replaced

additional

by

structural

member

members

inserted

to

and

counter

the

facet

shear

is

and

either

increase

a

void

or

rigidity.

an The

connection exists at the vertices, similar to how it would exist in space frame structures.

For

this

dissertation

we

will

only

consider

the

former

interpretation

in

our

exploration of building with origami structures. First, the principles of folded plate structures will be described briefly. Following this we can then divide static folded

structures

according

to

the

material

as

constructions

of:

reinforced

concrete, wood, metal, glass and plastic materials (polycarbonate, synthetic resin reinforced with glass fibres, polyester resin, etc.).

2.8.1.1. Principles of folded folded plate structures Folded plate structures (FPS) can be useful and economical in a number of situations. Certain characteristics have a direct influence on the efficiency of a folded plate structure. These include: the shape of the folding pattern and the relationship between the mountain and valley folds; the base within which the folds lie – for example a flat plane, a parabola or a dome; the properties of the chosen material for the folded plate; the connection between each plane and the boundary conditions and bearing design at the perimeter.

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Figure 2-19: Image showing load distribution in folded structures. Source: (Trautz & Herkrath, 2009)

FPS transmit external forces along groups of planar surfaces (plates or slabs) which are connected though a shear connection (the ‘crease’ line). Structures of this type are stiff and robust, having relatively thin sections. External forces entering a folded plate are first transmitted through the plate towards the shorter edge of the fold such that the resulting axial forces are then divided along the neighbouring elements, so that forces are in turn transmitted down to the ground or supporting structure. (Trautz & Herkrath, 2009) The folding system adopted affects the transmission of load and the major direction of span. We can divide these folding typologies into three types- linear, spatial and radial typologies. Linear (or Prismatic) folding systems are characterized by a series of mountains and valleys in continuous folding edges and may be single or double-layered. The direction of the fold need not be straight but can be curved or skew. The main principle is that they do not

Figure 2-20: Image showing different types of folded plate structures. Source: (Trautz & Herkrath, 2009)

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intersect themselves and the parallel repetition is retained. Spatial (or facet) folding, is characterised by a number of intersecting faces that meet at several vertices.

The

resulting

pyramidal

folds

can

be

derived

from

triangular

or

polygonal shapes, as displayed in Buckminster Fuller’s Kaiser Hawaiin Geodesic Dome, or

quadrangular

shapes.

These

types

of

systems

are

more

akin to

space-frame structures. Radial typologies are similar to linear types, but the corrugation intersects at a single point.

The base plane of these regular structures influences the overall performance and the kind of fold that is adopted. Folded structures ‘in the plane’ are those whose highest and lowest point lies on two parallel surfaces. There are also structures which take the fold along a complete frame and others which adopt dome-like shapes and characteristics (these may be pyramidal or polyhedral or a combination). Combinations of all these base plane types may exist. Today, research is also focusing on FPS based on freeform or double curved surfaces.

Tessellating these shapes in a reasonably for production purposes is the initial hurdle that could lead to the realisation of irregular FPS, possibly also double layered ones.

The folding texture of the planes results in a structure that is stiffer than most regular structures as a whole and hence has greater load capacity for the same material. Using this simple principle, it is possible to work with extremely thin sections based on plate-like elements which both define the building envelope and provide main structural support. Folded structures are therefore also costeffective as they use less material because of their high stiffness in the plane of maximum bending in the direction in which they span.

The possibilities of folding the structure typology, the scale, the span and the bearing capacity are directly related to the characteristics of the material being used.

It

is

important

to

compare

previous

work

in

concrete,

metal,

and

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synthetics in order to determine the most economical solution. This can be demonstrated through buildings by Nervi, Fuller and Piano respectively.

The principles of lightweight engineering are synonymous with these folded structures – there is high efficiency in terms of use of material and building process due to modularity, subject to regularity in geometry.

2.8.1.2. Folded plate structures in Concrete Concrete is the most commonly applied material for these fold structures, having been used as far back as the mid 20 century. Although elegant and th

architecturally appealing, these structures have been adopted due to the structural integrity which they provide.

Concrete structures were the first type of folding structures to be realized and

Figure 2-21: (Left) Eugene Freyssinet’s Orly Hangar in pre-stressed concrete. Source: (Anon., 1916) (Right) ‘Folded plate structure’ concrete roof. Source: (Anon., n.d.)

they often took the form of a V shaped linear, parallel corrugation, whose height depended on the span and the load. These folded structures were built on a large scale, made of in-situ concrete and with a constant frequency and based on a regular plan. With cast in situ concrete, difficulty is envisaged with making the formwork and placing the reinforcement in order to adhere to the tessellated shape. Ultimately, the structure behaves monolithically, although care must be 35

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taken to ensure that the thickness of the section is adequate enough to ensure monolithisation.

The formwork was generally made of wood, and in some

cases where the angle of the fold was quite steep, a double layer of plating would be required.

Other than using in-situ concrete it is possible to produce such structures with pre-cast elements. One of the first folded concrete roofs was the remarkable Aircraft Hangar at Orly Airport in Paris in 1923, which was designed by Eugene Freyssinet. Here, pre-stressed corrugated concrete was used to make a 70m span

structure

out

of

pre

fabricated

elements.

(Šekularac,

et

al.,

2012)

Freyssinet also applied the same principle of corrugated shell roofing for the construction of two other airplane hangars spanning 55m at Villa Coublay in 1924.

One of the main advantages for this type of construction is the thickness of the concrete section. Although having a thin section folded plate concrete roofs behave differently from shells because they do not benefit from the properties of curvature

and

nor

do

they

exhibit

full

membrane

behaviour.

According

to

(Cassinello, 1974):

‘The earliest folded plate solutions […] drew from their close resemblance to corrugated plate and cylindrical shells. The underlying idea is quite simple: longer spans can be accommodated with relatively small increases in weight by enlarging the lever arm of the structure; the top and bottom chords of each slanted slab house the main reinforcements while the shear stresses are absorbed across the sloping sides.’

The majority of erected FPS in concrete adopt simple, regular forms although a number of complex shapes have been available for a long time. In some cases such as chapel ceilings, more complex shapes were realised.

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In

the

Netherlands,

FPS

steadily

gained

popularity

following

the

Third

International symposium in Shell Structures in 1961, with various examples of short,

medium

representative designed

by

and of

Van

large

these den

scale

structures

structures

Broek

include

where

the

being the

folded

built.

Some

University structure

of

Building is

the

most

at

Delft,

composed

of

a

cantilevered overhang, and is spread over 32m.

2.8.1.3. Folded plate structures in Timber Over the years, only a few structures of this type have been realised in timber, with other materials taking precedence mainly due to the world’s rapid reduction in timber reserves, the limited size of the elements available and the reduction in the achievable spans. Another reason why FPS have been relatively unachieved is because of the complexity in making the joints. Today, with the onset of digital fabrication and CNC milling techniques, manufacturing can be made more accurate and efficient.

In principle, the process of building with timber is not a complicated one – all that is necessary are the precisely cut polygonal plates and then it is a question of

joining.

Transportation,

handling

and

assembly

are

also

not

an

issue

especially if prefabricated folds are used.

Research being done by EPFL has questioned which origami patterns would be more suited for building such structures. The principle in FPS is that the higher the amplitude of the valley and mountain folds, the stronger the resistance of the structure. Thus to solve the problem of overall weakness at the free edges, an increase in amplitude can be applied.

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The Chapel of St. Loup in Pompales, Switzerland, is an example of a folded plate timber structure based on origami (See Figure 2-22). It was built as a temporary structure using cross laminated timber and folding was applied on

Figure 2-22: (Top Left) Chapel of St Loup, Switzerland. Source: Milo Keller. (Top Right) Timber pavilion, Osaka. Source: Beijing International Design Triennial. (Bottom Left & Right) Prototype testing of folded plate structure made from cross-laminated timber panels. Source: (Buri & Weinand, 2008)

both the walls and the roof. This structure was completed as part of a research initiative at the EPFL in Lausanne, where Weinand, Buri and Haasis have been investigating the potential of timber in such origami inspired structures. Their research also investigates the appropriate detailing that is required for such structures to retain their stability. In addition to virtual simulations, much of this research is done using real scale models and experiments and attempts to find better jointing methods in order to improve the load bearing capacity of such structures. Jointing between wooden elements are articulated which means it may be necessary to install stiffeners.

(Buri & Weinand, 2008) (Haasis &

Weinand, 2008) 38

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The other known folded plate timber structure of this sort is also temporary and very similar to the chapel at St. Loup. It was constructed in Osaka in 2009 by RAA using veneer Kishu panels.

2.8.1.4. Folded Plate Structures using Metals For roofs with materials like metal, aluminium or steel, it is more common to find truss structures rather than those using sheeting only. This is especially true for larger spans, as metal sheeting may easily warp under variable loads such as wind loads. Very few buildings of this type exist, although it is possible to work with this technique.

Figure 2-23: Detail of the Hinzert Museum exterior. Source: Robert Miguletz

An example of a case study is the Hinzert Museum in Germany, a 42m long structure whose roof and faรงade are one and made out of 12mm triangular Corten steel plates which were welded together off-site and then transported.

(Figure 2-23) The museum was brought to Hinzert in 12 pieces. Apart from an exhibition space the volume also houses offices, conference hall and other habitable rooms. The interior is composed of timber triangular panels and is made to mimic the exterior steel surface. A carefully designed insulation system had to be used between the two layers in order to comply with habitable standards. Several triangular openings also exist. (Ana, 2013) 39

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Although often associated with FPS, the real structure underneath Walter

Figure 2-24: US Air Force Academy Chapel during construction.

Source: The Denver Post Library Archive

Netsch’s Air Force Academy Chapel in Colorado in 1963 is a truss system of tubular steel elements which are clad in aluminium plates. As mentioned previously, although these structures are derived from origami the load takedown is unlike that of plate structures, and hence unlike rigid origami models, so they are not the purpose of this discussion. Similarly, Zaha Hadid’s Transport museum in Glasgow and FOA’s Yokohoma International Port Terminal are also steel trussed fold structures.

2.8.1.5. Folded Plate structures using other materials Folded plate structures in glass, synthetics and PVC are difficult to construct without incorporating some additional supporting system. Perhaps it can be done in the future using multi layered system folded sheets although no examples have been found to be built so far.

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J. Gattas proposes a temporary shelter made out of honeycomb paper (similar to cardboard) that follows UNHCR regulations and could be deployed in relief situations. (See Figure 2-25) Apart from structural stability this proposal deals

Figure 2-25: Plate House Model by Joe Gattas. Source: www.joegattas.com

with a new jointing scheme: a ‘non-connection’ approach is used to solve the problem of multiple complex joints that were having a negative impact on the efficiency of production. These connectionless joints are incorporated into the structural plates and, apart from eliminating the need to produce s separate component, they save on fabrication time.

2.8.1.6. Folded Folded plate structures as building elements Using the increased bending stiffness property of folded sheets, miniature FPS, also similar to impact resistant meta-materials, sandwich panels have been made for flooring and wall construction. These increased-stiffness products are available in timber, steel sheets or reinforced concrete. Timber structures of this type are constructed using a horizontal top and bottom sheet with a linear corrugation in between (commonly a ‘V’ pattern). Metal or steel products are usually made from trapezoidal steel sheets and can independently take loading.

Uses for these structures include mezzanine flooring such as Trofdek and 41

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Kielsteg, profiled metal sheeting, interior partitions, roofing systems and more.

Insulation and other services can be incorporated easily in some systems.

In general the structural benefits are similar to what has been so far described in general for FPS. Another benefit of folded plate panels over similar products such as honeycomb cores include the fact that these structures can be designed with an inherent global curvature which can be manufactured continuously with minimal material deformations.

Figure 2-26: Mezzanine ceiling called "Kielsteg", made from timber. Source: (Ĺ ekularac, et al., 2012)

A study on timber curved origami beams is also ongoing at the Laboratory for Timber Constructions in Lausanne. The idea is to use origami principles and study the geometric considerations required of such structures so as to carry out an experimental investigation that is backed up by numerical studies which conclude whether these beams are fit for use. Although they physically do not appear to be the same as FPS, these beams are derived from origami principle of reverse folding. They also have potential from an architectural point of view.

(Buri, et al., 2012)

2.8.1.7. Irregular folded plate structures Recently a number of projects and/or research initiatives have been investigating freeform folded plates. From the buildability aspect these projects are generally expensive, material consuming and very complex. So, as (Trautz & Herkrath, 2009) rightly stated, in spite of the rising interest it is clear that: 42

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‘... there is an immense gap between the conventional building system and the innovative design strategy. There is often no coherent general design concept which includes the manufacturing. For these design projects it is therefore

necessary

material-consuming

to

develop

construction

very

complex

methods

and

which

thus seem

expensive to

assure

and an

effortless manufacturing because of their two dimensional shape.’

Figure 2-27: Design of an irregular folded plate structure. Source: (Trautz & Herkrath, 2009)

Ongoing research work includes the development of software tools that will enable more rational form generation, increased optimisation; the tessellation of freeform shapes; structural analysis and improvement in manufacturing and production techniques. Future aims also include sustainability into the equation.

New ideas for the building process of freeform shapes share also on the agenda.

2.8.2 Non –static interpretation of Origami Origami in Architecture and Structural Engineering By non-static origami structures in the building industry we refer to folded plate structures that have the capacity to move so that their geometry can be repetitively altered from an extended to a compact state whilst fulfilling the basic requirements of architecture. They combine some of the benefits of FPS with the 43

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ability to have a reversible building element. Such structures have only recently gained significant attention and have so far been realised in the form of temporary lightweight pavilions, roofs, building elements and mobile shelters.

These structures are not to be confused with mechanical linkage models such as those produced by C. Gantes, S. Pellegrino and C. Hoberman. Such linkage models have been around for quite a while and have been useful in the development of deployable structures and rapidly assembled structures, machines, tensegrity structures and robotics. The main issue in the context of architecture is that the watertight enclosure would have to be thought of as a separate thing that can somehow integrate with the linkage structure. This may not always be the neatest or best solution.

It could be beneficial to work with dynamic FPS in certain situations in the building industry. Realising this can however be quite tricky as will be seen from the examples discussed below. This topic is subject to ongoing research.

2.8.2.1. Pavilions and Temporary Structures In the architectural field, folding techniques were experimented with in Josef Albers’ class at the Bauhaus early as 1927. To reproduce such paper models on a human scale and without additional support systems can be done relatively

Figure 2-28: Temporary structures built using thin profile materials

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easily with materials such as cardboard, polypropylene sheets, aluminium sheeting and other such materials with a thin, rigid profile. This has been achieved on various independent occasions. Most of these structures are temporary and only able to deploy with the aid of additional moving supports (usually humans). The trend seems to be that these structures are used in their deployed (open) state and closed occasionally at the end of a cycle of use or for transportation purposes. When static, they are stable because the material is stiff enough to transmit in-plane forces to the ground, and because they are very well supported at all boundaries. Their nature is temporary either because the material is not durable enough or because the significant technology to make them so has not been developed. Examples of such structures can be seen in Figure 2-28. These structures are often built due to their attractive appearance and the interesting design-build exercise that they provide.

These structures provide little challenge when it comes to jointing as the

Figure 2-29: Arch model of a potential temporary structure modelled using thick aluminium sheets. Source: (Gioia, et al., 2011)

structure is manufactured from a sheet material through folding, much like a piece of paper.

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A similar study dealing with a temporary structure based on the Miura-ori model was recently completed using thick aluminium metal sheeting by researchers at the SLA Laboratory in Montpellier. The prototype opens up to a 2.78m span arch with the help of two people as shown in Figure 2-29 although there is sufficient stiffness for the structure to stand alone at any point during deployment. Further possible

investigations

are

envisaged

for

locking

the

unfolded

configuration

using hydraulic jacks and deploying devices allowing the displacement of the structure’s pinned joints. A virtual model of this prototype was also performed using

finite element

methods. The

results

indicate

a number of ‘free

edge

effects’ which can influence the stability. (Gioia, et al., 2011)

2.8.2.2. Foldable plate structures as building building elements and extensions Foldable plate structures in buildings can exist in a variety of ways – in fact we can consider windows, shutters, cladding and such elements to be part of this group. However, we are more interested in those elements that are somewhat more structural such as roofs, walls or beams and columns. Not surprisingly, some work on all these examples has been carried out so far.

In 2000, a folding aluminium sheet roof for an existing stadium was proposed in 2

Caracas, Venezuela to cover an area of 850m . The deployable aluminium sheeting was used to seal and work with an existing trussed structure as shown Figure 2-30 which ultimately meant that it was supported at various points.

However, this structure has achieved a fairly interesting deployment and jointing scheme. The roof cover is composed of six sheet modules which are wired up to a system of motor-driven cables that enable them to transform from a 24m length to a 4.6m length. The pulling system was developed to be able to withstand the rotational movements that were created upon displacement of the sheets. (Hernandez & Stephens, 2000) The structure was designed to fold as a

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parallel corrugation and the geometry chosen was based on a simple triangular prism that differs in scale at the ends.

For the ‘mountain’ joint, a continuous hinge was used to combine the folded modules together while at the ‘valley’ joint an aluminium section running continuously along the edge was fixed. The main problem is making this joint water tight while allowing the roof to remain rigid when necessary and permitting transformation at the same time. The roof surface is made from a large number of relatively small components, which further enhances these problems. The possibility of using flexible materials (rubber, textile membranes, silicon rubbers, thin metal sheets, etc.) which allow movement whilst still having closed joints was also considered although it did not materialise because the forces between the different components were not adequately transmitted which meant that the

Figure 2-30: (Top) Images showing the foldable aluminium structure and the existing supporting structure. (Bottom) Details of the ‘mountain’ and ‘valley’ connections.

Source: (Hernandez & Stephens, 2000)

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holistic structural behaviour would be lost.

Another jointing issue was the fact that the joint tends to become fatigued with repeated use and in the case of rubber and plastic this process is accelerated because the UV radiation breaks the polymer links. Thus rigid hinges were used, also because they can transmit forces whilst allowing movement. For the valley joints, a connection similar to a door hinge was used and measures were taken to exclude dust and dirt from entering the joint by gravity.

Further to the jointing and sealing system, an intricate system for deployability had to be developed to execute the overall movement. This was done using cables and a pulley system that was intricately designed to move alternately and in opposite directions to allow gradual movement to occur. This system is powered by a DC motor. (Hernandez & Stephens, 2000) The essential factors that led to the success of this structure were the simple choice of geometry and the appropriate jointing system coupled with a compatible deployment system. Ultimately, the project proved to be very complex to implement and to date, it is one of the only ones of its kind.

Figure 2-31: Design of a foldable passage between two buildings (left) and its construction details (right).

Source: (Tachi, 2010)

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Another building element that has been studied in connection with foldable plate structures is Tachi’s passage between two buildings (see Figure 2-31). The project has not been realised however it provides a number of relevant observations. Tachi’s design exercise presents a scenario of two misaligned openings that need to be connected using a rigid foldable mechanism. Tachi speaks about how to come up with the geometry for such a problem. For example the structure must be developable, flat foldable, planar for each folding edge, have a fixed boundary and firm ground conditions. He starts from a regular origami vault that satisfies most of these conditions and then optimises them to find the best solution. This configuration yields a number of options from which the designer can choose the final shape. (Tachi, 2010) The problem of thickness is also discussed briefly, as the structure will not be able to be fully folded.

With regards to other structural elements like columns, slabs and beams, ongoing research by J. Gattas is dealing with morphing origami building elements under different types of loading (lateral, axial loading, out of plane and longitudinal). Prototypes are made from steel sheeting and are similar to the impact resistant sandwich panels (meta-materials) discussed previously such that they employ a layered system of origami panels. A sandwich panel has also been produced. It functions just as the single layer pattern would, except that the secondary layer provides overall structural rigidity during, after and before

Figure 2-32: Collapsible beam prototypes built using aluminium sheeting. Source: www.joegattas.com

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deployment.

The panel has improved load bearing properties – the faces mostly carry in plane loads while the core takes the shear. Other properties such as developability are retained. The

constructability of the

structure still remains an issue. Gattas

explores the constructability of the structure through the optimum core and face geometry and the hinge connections between layers and between the individual panels.

This

is

explored

for

three

materials

(timber,

steel

and

paper

honeycomb), by subjecting a number of connections to tension, shear and peel tests and by running a number of numerical models. Physical tests are being carried out to further support this research.

2.8.2.3. Mobile Shelters Mobile deployable structures have the advantage of speed, ease of erection and dismantling compared to conventional building forms. Although they can easily be designed using linkages and scissor like elements (SLEs), this method fails to provide a waterproof solution and often a ‘casing’ must be incorporated.

Foldable plates could be a neat solution for mobile homes if the technology is developed well and if a lightweight, simple model is adopted. A number of proposals exist although for the most part these are fictitious projects. The main difficulty

repeatedly

revolves

around

figuring

out

the

complexity

of

the

mechanical joints and the deployment process.

Previous studies by Foster and Krishnakumar have shown that origami derived foldable plate structures can exhibit both single and double curvature yet it is necessary

to

study

the

simpler

genre

first,

as

these

structures

are

usually

required to be as simple as possible. In his work on deployable mobile shelters, Temmerman discusses the impact of choosing a particular fold pattern over another as well as the great influence of the apex angle on the geometry of the structure such that it could constitute a habitable shelter. A design process is 50

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Figure 2-33: Foldable bar structure based on the geometry of foldable plate structures. Source: (Temmerman, 2007)

also developed to parametrically design regular structures of single curvature and then adapted for double curved structures and variants of the regular one.

(Temmerman, 2007) It is evident at an early stage that the plate thickness has a direct impact on the compactness of the stowed configuration, or whether this is possible at all.

Temmerman uses foldable plate structures to derive a linkage structure that makes use of the one rotational DOF as a hinge between the plates. Ultimately, this solution borrows the neat vertex connections that exist within foldable plate structures, discards the extra weight of the plates and exploits the structure as a foldable plate truss system. The interior is created using fabric which is attached to the outer bar structure. ‘Although

in

the

end

underlying geometry

is

a

foldable

bar

structure

is

to

be

designed,

no different from that of the plate structure

the it is

derived from.’ (Temmerman, 2007).

However, whether such a structure functions in reality is still to be determined.

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2.9 Concl Conclusion clusion This chapter has provided a sound introduction to the principles of rigid origami and its relation to various engineering topics.

In

origami-inspired

static

structures,

it

has

been

shown

that

buildability

is

possible through folded plate systems. Although examples are plentiful many a time these are mainly constructed in concrete, meaning that there is still some work to be done with other materials such as timber, aluminium, plastics, and perhaps

tensegrity

advancements

in

systems.

technology

In it

is

light now

of

the

more

fabrication

relevant

to

revolution

continue

and

directing

research towards irregular surfaces with rigid origami tessellations. This area of study has the potential to provide architecture where the design component is respected without compromising structure, or vice versa.

For

non-static

FPS,

origami

models

remain

simple

in

form but

difficult

to

construct. Various attempts at overcoming this have been made and continue to be studied as has been shown in Section 2.8.2. To round up what these case studies have shown, we can conclude that it is desirable to have a material that can perform structurally during and after deployment and that the connection will significantly affect the ease with which these structures can be transformed from one position to another. Agreeably, most of the technological issues that exist in a real scale system are a result of the inferiority of the material and components available

when

compared

to

the

abilities

of

paper

in

small

scale.

Further

investigation into these limitations is therefore necessary.

This review has also shown that the current research on the use of origami structures as a basis for design in the building industry is scarce. The benefits of a dynamic origami structure are easier to apply in situations where the problem parameters are more directly related to the qualities of origami, such as where deployment is absolutely necessary, or where packing for transportation is an

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issue.

Because of the transformation aspect we tend to associate origami

structures

with

emergency

shelters

and

structures

with

less

weight

and

permanence. This may be one reason why this topic hasn’t really taken off yet, although

a

considerable

amount

of

origami

pavilions

have

been

built,

and

research on transformable structures is gaining popularity.

This literature review precedes further investigation into the technological and detailing issues that are ultimately the main drawback in a realistic scenario.

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CHAPTER 3. BUILDING WITH RIGID FOLDABLE ORIGAMI STRUCTURES

3.1 Introduction Working with deployable folded plate structures on a larger scale than paper and with a different material presents a number of challenges. Using paper on a small

scale,

these

structures

have

the

ability

to

transform

themselves

and

remain geometrically stable after being pinned at the boundary. This often does not remain true for larger scale structures in other materials, and they become structurally flexible even if they are geometrically fixed at the boundary.

At this point, we discuss one DOF mechanisms and their realisation as it is futile to attempt to go further before we are able to create at least real, working prototypes based on the simplest forms. This scenario is also true for static folded plate structures and as Schenk rightly points out, the success stories in origami

engineering

are

all

based

on

the

simpler

forms,

especially

due

to

difficulties in manufacturing. (Schenk, 2012)

In order to be able to make use of the properties of foldable rigid origami tessellations we must solve all the technical problems that large scale brings with it. This chapter is meant to highlight these problems and to illustrate, compare and discuss any proposed solutions so far.

3.2 Basic forces in active rigid origami origami tessellations As

mentioned

kinematics

of

in

Section

an active

2.5.2,

rigid

there

origami

are

two

methods

tessellation.

While

of

the

analysing

the

kinematics

are

important to understand how the model behaves, it also becomes relevant to discuss the flow of forces as progress is made through the deployment path.

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Although this would require proper computational analysis, some simple deductions can be made.

The way forces are transmitted through an object changes according to the material stiffness and thickness. As with most materials, when paper is bent, tension is induced at the top face and compression on the bottom face. (See Figure 3-1) The bending stresses are carried throughout the plane of the paper and travelling between one facet and another until they reach the support point.

When the model is given a thickness, the tension- compression scenario still exists however there is a greater distance between these stresses. In effect, the section can now take more moment however its geometric properties are weakened because it is no longer thin. For example the strain energy needed to accommodate the movement is much higher and unless the material is uncut,

Figure 3-1: Diagrams showing the behaviour of forces inside a thin paper model (left) and a thick material model (right). Image source: Author.

the ease of connection between plates is reduced.

When the folded plate model is in a compact state, the bending stresses are highest and the shear stresses at the vertices are at its minimum. As the model expands the opposite is achieved: its bending stress is least in the deployed state and shear stress is highest at this instance. The forces in a dynamic 55

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origami model are therefore changing at different deployment stages and the section must be able to take all worst case combinations. This change is critical to the main components of the structure, particularly to the connections.

3.3 Existing problems The problems that exist when working with large scale prototypes include: choice of material, issues of thickness, connection, issues regarding the state of the structure

during

conditions

the

necessary

deployment to

facilitate

and

contraction

movement. All

process these

and

the

factors are

boundary not

only

directly related to each other but they also influence each other substantially, having an overall effect on the performance. Material properties are fundamental in dealing with the said problems. In a study on a fictitious large scale origami roof that paid particular attention to the issues of material and scaling it was stated that:

“If the material thickness, t, increases linearly with the other dimensions, then the flexural rigidity of the edges will increase as t3and the subsequent ability to fold the large-scale sheet will be dramatically reduced if not eliminated.� (Gentry, et al., 2013) In no gravity situations scaling has little effect which is why deployable rigid origami structures in the aerospace industry are far more successful. The main issues that come with gravity are tackled with in more detail below.

3.3.1 Thickness

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The issue of thickness of structural elements becomes very relevant in the context of practical foldable plate structures. One of the main reasons why these structures are so attractive in an architectural and engineering scenario is because they have the ability to transform from a large volume to one which is drastically smaller. The thickness of the plates will affect how compact the structure can become when it needs to be packed, and if this is unsatisfactory it becomes relevant to question the logic of using these structures in the first place. The thickness of the plate section is largely influenced by the material

Figure 3-2: Diagrams explaining the basic problem of thickness. Image source: Author.

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used, as it needs to be something rigid, able to take and transmit loads yet with the minimum profile possible.

As will be seen in the following paragraphs, although the position of the hinge affects the problem of thickness, it may not always be ideal to alter it as a solution. In fact, in most models shifting the hinge position still causes global thickness problems.

The thickness problem has been studied in some detail over the last two decades. One proposal patented by Charles Hoberman addresses symmetric degree 4 vertex model using a variation of the Miura-ori fold to create a number of linearly deployable structures. The diagrams marked as FIG 32, FIG 33 and FIG 34 in Figure 3-3 constitute the basic element of the structure, whereby two

Figure 3-3: (Left) Diagrams showing Hoberman’s proposal for thickness. Image from (Hoberman, 1988) (Right) An example of sliding hinges model proposed by Trautz and Kunstler. Here the sliding value is accumulated along the hinges on the right. Image from (Tachi, 2010)

triangular panels are hinged to the main body. A component, made from two elements is shown in the diagrams marked FIG 35, FIG 36 and FIG 37. The structure proposed by Hoberman is composed of a series of adjacent components with degree 4 vertices at the most. (Hoberman, 1988) A void is 58

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created within a thicker panel so as to enable the triangular elements to fold in such a way that the structure may be packed as shown. In this way any collision is prevented and the structure can function using hinges connected on the exterior or interior face of the panel, depending on the fold. (This approach of placing the hinge on different panel faces is termed ‘axis shift’ and will be referred to further on). The structure is flat foldable and constitutes the sum of thicknesses of the panels when in the fully folded state.

Another

method

by

Trautz

and

Kunstler

proposes

sliding

hinges

and

can

accommodate more than one DOF, unlike Hoberman’s proposal (see Figure 3-3). This is done by allowing the panels to slide along their rotational axes (i.e., the hinged edges). The panels in the Sliding Hinges Model now have 2 DOF – sliding and rotation. The amount that the structure can fold is related to, and often limited by the amount that the hinges slide. Since edges are common to more than one panel, the sliding ‘space’ is shared, and it is easy to end up with global folding problems in spite of the success of the model on an isolated module. Thus, while it may be possible to use for some patterns, and while it allows symmetrical 4-valent vertices to be folded, (Trautz & Kunstler, 2009) it cannot be used as a general method for all models. The motion contributes to the increased complexity of the model and compaction is ultimately constrained by neighbouring plates. (Tachi, 2010)

Generalised origami tessellations may also be built with a thickness following the Tapered Hinges Model. This option applies the motion of thin, ideal origami to a thickness model by avoiding the use of axis-shift system (see FIG (a) and FIG (b) of Figure 3-4). Axis shift can be exemplified through a door hinge which in normal circumstances is fixed on one side of two plates. When applied to origami,

the

hinge

position

alternates

between

valley

and

mountain

side,

changing the position of the rotational axis with it. A problem arises because the kinetic motion of origami is influenced by the position of the interior vertices

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which become over-constrained in an axis shift system since each vertex has 3 rotation and 3 translation degrees of transformation.

Thus, Tachi proposes that the rotational axes lie exactly on the edges of typical paper origami by bisecting planes of dihedral angles between adjacent facets, in process tapering the panels as shown in Figure 3-4. Flat foldability cannot be

Figure 3-4: (Top) Two approaches for enabling thick panel origami. (a) Axis-shift (b) The proposed method based on trimming by bisecting planes. Red path represents the ideal origami without thickness. (Bottom) Freeform origami with modified panels using the Tapered Hinges Model.

Both

images from (Tachi, 2010)

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achieved because otherwise half of each panel volume would intersect, so there is a limit on the maximum folding angle (Tachi, 2010).

The tapered panel model is not infallible as globally, it is not guaranteed that the structure will not collide although it is possible to prevent this by predicting the location of collision and tapering even further.

Another proposal, also suggested in (Tachi, 2010), is to have constant thickness panels that are offset away from each other by a certain distance (see Figure 3-5). This model is proposed for situations where the thickness to panel length ratio is not too high when compared to the dihedral angles used in the tapered

Figure 3-5: Image showing a model using the Constant Thickness Method. Source: (Tachi, 2010)

hinges model. The space created between the panels could be a customised hinge or a flexible material such as fabric. A 2.5m x 2.5m model using cardboard panels and fabric joints was constructed by Tachi to support this theory. Due to the nature of the fabric there may be too much flexibility at the interface between the panels which may be disadvantageous for a large scale system.

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3.3.2 Connections The connection refers to the area where two rigid faces meet and are subsequently joined to form a unified structure. This may be a point, i.e. a vertex at the interface between several plates or an edge between panels forming a mountain or valley.

In a paper model, the edges are allowed to rotate fully if the model is flat foldable which means that a simple hinge having 1DOF can be used to achieve this in real life. The vertex connection is far more complex: with 9DOF, it connects a number of facets which are respectively moving in different directions. The edge connection is more relevant because it has more control over the plate since it is connected a larger section of it. While the edge joint may exist without the vertex joint it is usually difficult to have the vertex connection without the edge connection. More often, the vertex joint is ignored and a simple hinge is used between each facet edge.

The primary function of the connection is to facilitate movement. It is also envisaged that a degree of rigidity is offered by the connection at instances when the structure is in a static state, between deployment and contraction stages so as to contribute to the overall stability of the structure. This means

Figure 3-6: Doubly expandable shell with reinforcement. Image from (Resch & Christiansen, 1970)

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that a ‘lockable hinge’ system may provide the solution. The need for this was demonstrated by Resch and Christiansen who built a doubly expandable shell which proved to be flexible under self weight and imposed loads, requiring reinforcement bars in order to achieve full stability. (See Figure 3-6) In the process however, the structure became non-transformable. Thus, what is required is a temporary stiffening solution when static, but one which does not interfere with the movement when required. (Resch & Christiansen, 1970) The most conceivable bi-functional solution (one that can allow and hinder movement) may be derived from nature. It is composed of fluid controlled joints that use a pressure system (Figure 3-7). A system using vacuumatics was proposed in (Tachi, et al., 2012), using a double layered skin containing trapped air between rigid panels. The stiffness, movement and structural behaviour of

Figure 3-7: Vacuum hinges as proposed by (Tachi, et al., 2012)

multi DOF rigid foldable models were investigated. In general, it is difficult to control a vacuumatics system globally, so pressure control was only used for the jointing areas. Also, a system with equal mountain and valley fold vertices was chosen to equalise the deformation that it would undergo when pressurised (both positive and negative curvatures are present in this way). The connections are flexible during deployment and regain stiffness through added pressure when the desired configuration is reached. The overall automated nature is seemingly convenient.

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In another case study PETG sheets with living hinge connections embedded within the material sheet were used for a roof study. When thermoformed, the

Figure 3-8: (Left) Typical example of a living hinge made from PETG. Image from http://www.sdplastics.com; (Right) Diagram explaining the concept of internal and external trusses used for stiffening connection zones and facilitating deployment. Image from (Gentry, et al., 2013)

sheets have the ability to be bent to strain ranges of around 200% of the original. Under gravity, the 92m spanning structure was initially stiff until a major deflection occurred due to weakness in the configuration and the jointing.

Subsequently a strategy comprising of an internal and external origami truss was developed so as to reinforce the failure zones. The internal members, located at the edge of each panel would function as reinforcing ‘battens’ embedded in the plates. When modelled, this system yielded a stable and elastic structure. More information about this can be seen in (Gentry, et al., 2013) Connection points at vertices may be replaced with a void, as has been done in models produced independently by Tachi and Gattas (see Figure 3-9). Having a joint at the vertex complicates the connection issue drastically as it needs to cater for four plates that are moving in different directions in all of the x, y and z directions. Further to this, dealing with thickness at the vertex becomes another issue. Most likely, the vertex joint alone is not enough to secure the panels to each other, especially if large spans are used so in addition, the hinge joint would still need to be used. The drawback that connectionless joints bring with them is lack of water tightness, although there may be ways of dealing with this 64

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Figure 3-9: Images showing the principle of ‘connectionless’ joints at the vertices. Left image from (Tachi, 2010) and right image from (Gattas, 2012)

by patching up the area with a flexible material or using the structure as an internal skin.

3.3.3 Deployment The deployment aspect refers to the process and ease with which the structure transforms from one geometric configuration to another without compromising the overall structural stability. It also deals with systems of actuation.

So far, poor mechanical performance has been consistently observed upon the deployment of large scale foldable plate structures. For example a Miura-ori model has the ability to move freely as long as the boundaries are not restrained. According to (Gentry, et al., 2013) the effective flexural stiffness is notable between 30 and 80 degrees because outside this zone the model is unstable and unlikely to function as a rigid structure. Since the properties of paper cannot be so easily assimilated in large scale it is not easy to predict how the origami structure would behave based on what we know about how paper behaves (Gentry, et al., 2013). Thus, a large scale structure would require either some additional reinforcement that could somehow provide support during the process of transformation (without compromising the structure’s ability to move) 65

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or be made of a material that can take the stresses at every stage of deployment.

A number of ways to facilitate deployment have been proposed. The most intuitive solution would be to use a looped cable system powered by a motor at the boundary. The integrated cable and pulley system would be somewhat similar to what was proposed by (Hernandez & Stephens, 2000) in Caracas, designed specifically to work with the particular fold pattern. A cable system will become more complex if the structure deploys in more than one direction at once.

A fluid pressure system such as the vacuumatics example proposed by (Tachi, et al., 2012) is another option. Whatever the mode of actuation, it is important to facilitate movement in a uniform and controlled manner because otherwise the structure will be unstable. In the vacuumatics proposal it becomes apparent that at any point in time, the state and strength of every connection may influence the overall form and stability of the structure. The shape of the proposed shelter is induced by the amount of negative pressure provided, which is increased for the purpose of strengthening the structure when the desired shape has been configured. The negative pressure also induces a small moment

Figure 3-10: Diagram showing the construction process of a deployable structure powered by vacuumatics. Image from (Tachi, et al., 2012)

at the hinge lines which helps to remove overall instability caused by the singularity in the unfolded state. (Tachi, et al., 2012) Whilst actuation is clearly the forte of the vacuumatics system and could potentially be adopted in other models, the way the deployment issue is dealt with is particular.

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Another approach stems from the fact that these structures have so far always been composed of single layer sheets which are unable to take much load and exhibit extreme stress concentration at sharp joints and free edges (Gattas, 2012).The idea is to provide a kinematic second layer, hence deepening the section when static and facilitating movement when transforming.

This study proposes to add a collapsible sheet atop a semi-deployed Miura-ori structure so as to reinforce it. The added patterned sheet is compatible with the movement of the structure and in its fully deployed state is serves as a

Figure 3-11: Morphing panels using the Foldcore concept described in Section 2.8.1.6. Images by (Gattas, 2012)

constraint for further expansion. It is then able to contract when necessary by displacing itself upwards. In this manner, the assembly becomes like a collapsible space frame system made from panels, although issues of how to achieve this in practise still exist. This system could be very effective because the added panels provide increased rigidity against bending and when static, the 67

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sandwiched Miura-ori structure provides sufficient shear resistance. However with

regular

hinge

connections

the

system

is

only

stable

in

the

horizontal

configuration, and when the panel is rotated vertically to mimic a wall panel it becomes kinematically unstable and folds into its compact state. Using selflocking hinges could solve this problem and make the panels useful for any building

part.

compared

to

Although straight

no

extension

extensible

is

occurring,

members

that

the

facilitate

concept

could

movement.

be

Further

research into this work (especially on the connections) is currently being carried out by (Gattas, 2012).

The

Foldcore

model

is

also

the

more

appropriate

one

for

the

issue

of

compactability; however the main problem of deployment is still to be resolved.

The hinges (or any other devices) must be capable of controlling the movement during

deployment.

extensible

members

In

principle

that

the

facilitate

concept

deployment

may

be

except

compared that

in

this

to

having

case

the

extensible members are local to each module and there is nothing that keeps the whole system together. The Foldcore system is also easy to manufacture as it only requires 2D panels.

In

the

study

by

(Gentry,

et

al.,

2013),

external

trusses

are

proposed

as

deployment aides in the final stage of the investigation. These members would be attached to the vertex location (as can be seen in Figure 3-8) and would run in the x and y direction due to the nature of the deployment path of the Miura Ori pattern. They are proposed to be extensible and act as actuators to deploy the system.

3.3.4 Boundary As stated, the boundary refers to the position and condition of the reaction points that will facilitate the load takedown or provide reaction forces for a real origami structure. The boundary conditions are greatly influenced by the kind of 68

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structure

being

dimension,

proposed

pattern

and

and

the

direction

way

of

it

deploys.

deployment

it

Due

is

to

difficult

differences to

propose

in a

general approach. In theoretical studies relating to geometry and kinematics the issue is often not considered because the research remains on a conceptual level.

In most case studies relating to real life design projects, support is always present at every position that the structure assumes or is usually provided with a flat plane such as the floor. In the Caracas canopy case study (Hernandez & Stephens, 2000), an existing criss-cross system of trusses was present for the moving structure to rest on (See Figure 2-30). However, it is better to provide a solution that can move along with the structure otherwise there is almost no point of having a deployable structure in the first place.

If

the

processes

described

in

Figure

3-10

are

adopted

for

a

vacuumatics

system, the boundary supports must be provided at all times, and this may not always

be

possible,

for

example

where

the

deploying

structure

is

a

roof.

Constructing a moving support is not an ideal solution due to the complexity that it creates spatially, and mechanically. A tension system with tightening and release

capabilities

could

provide

the

moving

support

the

structure

needs,

however there would probably be issues of sagging, or multiple cables would have to be used and tightened at different rates depending on distance.

3.4 Discussion This section is meant to analyse the facts presented in Section 3.3 and discuss those

methods

that

could

be

used

on

a

human

scale.

Active

origami

tessellations in the building industry are envisaged as a temporary skin that separates different environments. This could be an external wall, a floor, roof, canopy, an internal partition or combinations. It is difficult to pinpoint the most

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appropriate methods to achieve this prior to digital and physical testing, however certain proposals presented in this chapter may be eliminated at this stage.

While the thickness model proposed by Hoberman is both tangible and realistic, the solution is very particular to the fold and although it is possible to extend this proposal to other linearly deploying structures this model is unsuitable for bidirectionally

deployable

structures.

Real-scale

origami

structure

models

are

generally not complex and have so far been linearly deployable, yet finding a method that can suit various different prototypes is most ideal.

Certainly, bi-directional folding is possible using the Tapered Hinges Model which was ultimately designed to enable generalised rigid origami structures to have a thickness. Because it follows the ideal path of a thin origami structure, this model can allow almost anything to fold. However, although at first glance it may seem to be an optimal and flexible approach one must recall that the system is not infallible and that there is certainly a limit to the amount of volume that can be tapered. For simpler and more realistic models this approach seems to

be

rather

appropriate,

and

as

stated

in

(Tachi,

2010)

this

method

is

‘applicable for human scale structures.’ If

the

panel-width

another

method

ratio

by

criterion

Tachi.

The

is

satisfied,

constant

once

again

thickness

we

may

method

consider

requires

no

modification to the panel and it seems that the model can be achieved using readily available materials of a lightweight nature, such as plastics.

Regarding 1DOF connections, the first logical step would be to use those that already exist and are used. One such connection is a typical door hinge. It has also

been

proven

that

the

gear

hinge

as

shown

in

Figure

2-30

may

be

successfully implemented. However, with these hinges we remain with two main problems

–

the

first

is

deployment. Perhaps a

waterproofing

and

the

second

valid solution to waterproofing

is

control

may be to

during

integrate

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rubber within the connection, or to incorporate fabric so as to seal the joint. A mechanical connection could potentially be more durable because movement is facilitated by way of its geometric nature and is not so much dependant on material performance. This may be an advantage in some cases as wear and tear can undermine the system, unless of course the material is good enough. A fabric joint would need to be stiffened in some way, if not at the vertex then as close to the vertex as possible.

Facilitating deployment can become quite complex if the structure moves in more

than

one

direction

at

the

same

time.

Although

not

impossible,

the

solutions would vary depending on the deployment path of the structure. With certain active tessellations it is very easy to produce relevant structures based on a generalisation of the original, and their corresponding pathways may range from an arch, circular, curved, linear and others. Perhaps with cables it is easiest to assimilate any of these paths due to the flexibility offered. However one must keep in mind that cables alone are not enough, and an additional anchor component is required with any cable system.

Although it is very probable that solutions to both aforementioned problems may be found, it is easy to end up with a mix of systems that may not work together properly, which makes one question whether a more integral approach would be preferable.

The vacuumatics proposal seems to offer just this, although in practice one would need to have a mechanical system or a motor in order to actuate and control

the

pressure

changes.

Undoubtedly

also,

if

the

pressure

is

to

be

controlled as demonstrated in Figure 3-7, a series of pipes would need to be attached to every panel, which may cause packing problems and look unsightly if placed beneath the structure and may be subject to the elemental weathering if placed above. In such a system it is paramount that the air from each panel is removed simultaneously otherwise the structure may become unstable.

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It may be said that every system described in this chapter seems to have its own set of advantages and disadvantages, and thus finding a single solution may be tricky. Ultimately it is more important to remain true to the principles of rigid origami

and

the

objectives

of

the

exercise

or

to

work

on

aspects

that

systematize the way all the different components come together. Producing a well-functional, organised end result will undoubtedly affect whether origami will be used as a building typology or not.

Without a doubt the deployment stage is one of the most important stages in the life of an origami structure and dealing with it using a practical system is necessary

if

these

structures

are

to

become

a

building

tool.

Keeping

the

structure as lightweight as possible plays a part in this although this is something directly related to the material. Probably, such structures would be made from a material

like

aluminium,

plastic,

carbon

fibre

reinforced

polymer

or

less

favourably wood.

It is important that the structure remains easily deployed by a cohesive method which is not so difficult to control. The state of the connections should be allowed to control this.

Conceptually we may imagine boundary support points to be located at every free edge vertex. Is this necessary? If not, will there be significant free edge effects as resulted in (Buri & Weinand, 2008)? How can we further prevent this?

The

approach

of

stiffening

the

edges

after

they

have

been

weakened

substantially through fatigue is a patchy solution. The answer is not to keep reacting to all the issues that crop up, but rather to deal with the problem from an initial standpoint, choosing the right materials and conditions from the outset.

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3.5 Conclusion In this chapter we have reviewed several existing solutions and ongoing research projects regarding issues that affect building scale origami structures. In addition we have also highlighted the more relevant ideas and discussed how they can be integrated. However, without applying these proposals to a case study and testing them out through prototypes and models it is difficult to say what will actually work or not. While some solutions can be used to realise particular structures they may not be suitable for a different tessellation.

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CHAPTER 4. CASE STUDIES STUDIES

4.1 Introduction This chapter demonstrates and discusses the solutions presented at the end of Chapter 3 through two main building element typologies- a deployable surface as a roof or canopy structure and the foldable cylinder as a habitable volume.

Several

observations,

advantages

and

disadvantages

are

pointed

out

and

geometrical considerations are explored. The study is an investigation of the most realistic method available for building an origami structure in real scale, based on the literature reviewed and on the author’s own analysis.

4.2 Deployable Roof Roof Structure As an example, a 14m x 28m rectangular space requiring a cover (roof or canopy) was considered.

4.2.1 Pattern Selection Miura-Ori is one of the most widely used tessellations in origami. Invented by Koryo Miura, the pattern has been successfully implemented in solar panels in space because of its simple form, efficiency and rigid folding properties. A single module is composed of four identical parallelograms mirrored in the X and

Z

planes

(Figure

4-1).

The

module

transforms

in

all

three

directions

simultaneously when active, with the most notable movement being in the Y direction. As described by (Gentry, et al., 2013):

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“As this patch of tiles goes from fully deployed (a flat sheet of paper....) to fully folded (a thin strip of paper...), the overall x dimension goes to zero, the overall y dimension reduces slightly, and the overall z dimension reaches its maximum.”

In this document we define the different stages of deployment as θ = 90° when fully deployed, θ = 0° when fully folded. The diagram in figure 4-1 shows a module when θ = 50°.

Several variations of the structure may be created, depending on the shape of the

parallelogram

in

the

crease

pattern.

Based

on

a

similar

design-a-roof

exercise by (Gentry, et al., 2013), the selected tessellation for the roof structure is

composed

of

Miura-ori

parallelograms

with

internal

angles

of

45°,

45°,

135°,135°. The panel dimensions were kept tangible: length (l) =2m, width (w) = 2m,

diagonal

(d)

=

2.83m.

The

maximum

height

(h)

=

1.415m

when

the

structure is in the fully folded state (θ = 0°).

The material is unspecified, although a working thickness of 0.1m was used as a realistic option.

More information about the geometric structure of the Miura-ori fold may be found in (Stachel, 2009). Furthermore, the Miura-ori crease pattern may be found in Appendix II.

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Figure 4-1: Detail of a Miura-Ori module, Source: Author

4.2.2 Usability of Structure Different stages of deployment may be seen in Figure 4-2. As mentioned, the structure can travel from one flat state to another flat-foldable state. However, 76

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we cannot make use of it at the maximum and minimum angles of θ. Previous studies (as discussed in Section 3.3.3) suggest that the effective flexural stiffness of the structure is at its best between 30° < θ < 80°. Beyond this range the structure is not likely to behave as a rigid structure. This condition is also related to the fact that the Miura-ori pattern is a developable surface and thus possesses no out of plane stiffness when deployed. Therefore in order to effectively satisfy both conditions the structure should ideally be modelled with a customized geometry that prevents its developability whilst keeping its kinematic efficiency and therefore obtain a non-developable and flat foldable structure as proposed in (Gioia, et al., 2011).

However, since no numerical analysis will be carried out in this investigation, and since the geometric challenges remain practically the same in any case, we will consider the flat Miura-ori roof structure and follow the 30° < θ < 80° range for this exercise.

Figure 4-2: Deployment stages of a Miura-ori tessellated canopy. Source: Author

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4.2.3

Connection and Thickness

In order to realise this structure, thickness must be accounted for and the connections must be able to work with the structure. The possibility of having a watertight roof is also considered to be the preferable option.

Initially, an attempt was made at using Tachiâ&#x20AC;&#x2122;s proposed Tapered Hinges Model in order to prevent panel collisions. The assumed uniform thickness of each panel prior to tapering was 0.1m. At the maximum folding angle (θ= 80°) the amount of intersecting volume is at its maximum and the tapered surfaces of each panel may be derived as instructed in (Tachi, 2010).

If done manually, (as

was done in this case) the process is both tedious and tricky since every face on every panel must be considered separately.

A single Miura-ori module was modelled in isolation, producing two unique panels (see Figure 4-3). At certain corners the panel tapered to a point, hence undergoing

a

drastic

reduction

in

section

size

and

therefore

becoming

a

potential failure zone. At most instances the overall thickness of the panel was reduced to half the original size, meaning that if this were a real life model, one would have to start with double the original thickness in order to be end up with the desired structural strength. However, when the panels are thickened, more collision is induced and hence more taper would be required, ultimately leading to a weakened panel nonetheless.

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Figure 4-3: Model of a single Miura-ori module modelled with thickness (top left) and with tapered panels (top right) .Source: Author

Thus, although this model can geometrically evade thickness collisions too

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much tapering is not practical as it can significantly reduce the structural capacity. As can be seen in Figure 4-4 the panels will have different thickness when placed adjacent to each other in a flat state (when θ = 90°) which makes it harder to connect mechanical hinges should they be the preferred option.

Although it may seem that the hinge could be affixed on the flat face of the panel, more often than not this is not possible due to the way the model is conditioned to move â&#x20AC;&#x201C; i.e. the face on which the hinge is to be affixed is predetermined by the kinematics of the rigid origami model. The red zones highlighted in Figure 4-4 show the places where a hinge could potentially be

Figure 4-4: Plan view and detail of a Miura-ori module modelled using the tapered hinges model showing the zones where hinging can occur. Source: Author.

placed. As can be seen, these zones have been reduced quite significantly even at such a small thickness-width ratio which in this case amounts to 0.05.

Furthermore, confining the hinges to the central parts of an edge may weaken 80

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the extremities especially if this is as drastic as shown in Figure 4-4 which is a proportional, scale diagram.

Following this exercise the Constant Thickness panel method was investigated.

Using two uniform thickness elements for each panel (which are identical to the original panel in form but slightly smaller in size) are offset a certain distance

Figure 4-5: Figures showing a series of Miura-ori modules as modelled using the constant thickness panel method. Source: Author

away from each other. This distance is based on the minimum dihedral angle θ.

Figure 4-5 demonstrates the model θ = 80°. When testing it out at the minimum angle θ = 30°, using panels with the previously selected overall thickness (0,1mm) an offset distance of 0.2m it resulted that the panel components collided. This meant that a greater offset distance was required or a higher minimum dihedral angle θ would need to be accepted as the limit state meaning

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that the model would be less compact. Thus, a number of limitations may also be pointed out with this model

In an ideal world, the connection system of choice would ensure single DOF movement but also have a considerable amount of tensile strength, folding endurance and stiffness. For a real scale build, fabrics or plastics may be the closest options although usually in the case of fabric, not enough stiffness is present

while

with

plastics

the

folding

endurance

is

generally

lower.

A

combination of both may be an option.

On the other hand another option that does away with many of these conditions is

to

use

a

mechanical

system

which

allows

the

structure

to

exercise

its

kinematic properties through force transfer and geometry. Mechanical systems such as a typical door hinge or a gear hinge are the ultimately the more reliable options as they have been â&#x20AC;&#x2DC;tried and testedâ&#x20AC;&#x2122; on numerous occasions.

The only

issue in the context of an origami structure is that the structure must not only be capable of behaving as a mechanism but also exist in a static state at any instance during deployment.

Perhaps a system of lockable hinges may provide

the solution, although existing solutions are individually controlled and for such a structure they will need to act in synchrony with each other. Of course this is easier said than done, and it is also difficult to do without aides, actuators or additional support systems. There is a certain sense of stability and integrity that a paper model provides which all these connections being discussed are not able to provide.

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A possibility for the connection could be to use a system that is conceptually similar to rods in a fabric pocket (Figure 4-6). The difference would be that the internal rods or cables would be rigid panels, with fabric encasing them all

Figure 4-6: Physical model using fabric sticks and thread to assimilate and test out the concept of a fabric pocket. Source: Author, 2013

around. The panels could also be external, with fabric sandwiched in between.

This setup could be used with the Constant Thickness Panel method whereby the connection system will provide a solution to waterproofing, which is difficult to solve in any other way due to the little room available between the folding panels. A series of fabric patches at the vertices is probably the only way to seal the vertex connections that are generally left open.

4.2.4 Boundary and Deployment In this case said boundary is a rectangular perimeter measuring 14m x 28m and the structureâ&#x20AC;&#x2122;s deployment path is a linear one (although not uni-directional). It must be mentioned that this need not always be the case and that with some small variations of the Miura-ori pattern we can achieve a curved deployment path to create an arched structure. This system was employed in (Gioia, et al., 2011) who used an arch so as to end up with a non-developable surface. Due to its strength in compression, an arched system may be more stable at instances when structure is in a static state.

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The

support

points

envisaged

for

the

roof

structure

will

be

the

most

straightforward option in terms of force transfer, i.e. a series of moving supports at all four edges wherever the structure touches the roof perimeter.

Figure 4-7 shows our linearly deployable roof structure at different stages of deployment. In this approach deployment is made possible by fixing two out of four edges so as to provide some anchorage and having two free edges that regulate the movement. In this case when the roof is in the fully folded state (θ = 0°) the structure sits in the corner of the rectangular plan. Another option (as shown in Figure 4-8) is to deploy the structure symmetrically from all four sides such that the structure offsets within the rectangular space. In this scenario the aides (possibly cables) would be acting on all four edges.

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Figure 4-7: All deployment stages of the Miura-ori roof showing the position of the fixed perimeter edges. Source: Author

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Figure 4-8:

All deployment stages of the Miura-ori roof showing

compaction towards the centre. Source: Author

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It is envisaged that in both cases, keeping the support system taut is necessary to prevent overall sagging of the structure. If the deployment aides are cables, this may be possible, although a degree of deflection will always be present. At this stage it becomes increasingly evident that trying to make a structure of this sort behave like a rigid origami paper model is going to be very difficult with a combination of structure)

and

mechanical hinges, cables (essentially supporting the a

motorised

pulley

system.

This

kind

of

structure

whole

does

not

embody the cohesiveness and versatility proposed by origami but is more of a bulky mechanism hanging from a set of cables.

The approach of having a single material within which the connections are embedded whereby the deployment system simply aids movement rather than provides too much support is more true to what an origami model is. This was proposed

by

(Gentry,

et

al.,

2013)

using

PETG

sheeting

and

living

hinges

however the system failed under gravitational loading. Part of this reason was due to the inferiority of the material and jointing system and to the 92m span that was chosen as a design constraint.

In any case apart from a guiding rail system, a deployable folded plate structure of this sort also requires support points along all its edges so as to transfer loads to the supports. Providing these moving support points at all edges is complex because they would have to move in two directions at the same time in order to follow the structure.

4.3 Foldable Cylinder Structure It is possible to create a fully enclosed volume or extensible module using a foldable cylinder.

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4.3.1 Pattern Selection In the process of selecting a suitable retractable cylinder model a number of tessellations were initially investigated. The main criteria of selection were rigid foldability, ease of deployment and practicality of volume enclosure.

The most conventional examples such as the helically triangulated cylinder,

Figure 4-9: A series of non-rigid foldable cylinders and their different deployment states. (From Left) Cylindrical bellows cylinder, classic bellows cylinder and deployable cylinder based on the Yoshimura pattern. Source: Author.

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cylindrical or square bellows corrugation or the Yoshimura pattern proved to be non-rigid foldable due to warping of the panels at the interface between the edges during the later stages of deployment. These structures could only be useful up to a certain point when the dihedral angle between the panels reaches a certain value â&#x20AC;&#x201C; beyond this they start to deform so in a real life scenario they are unable to utilise the volume encompassed by a paper model example. Thus they do not make an ideal practical solution.

Another issue that played a part in pattern selection was to utilise a tessellation that could deploy in either a linear, uni-directional fashion, or at most, in a bidirectional fashion. In practise, having a structure that requires a number of random transformations is not only impractical but it increases the time needed for assembly.

Lastly, the pattern must be able to produce a volume that can serve as an

Figure 4-10: Deployment stages of a Zig Zag cylinder from (Tachi & Miura, 2013). Image source: Author

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architectural space at any instance during the deployment process otherwise there is ultimately no use for it.

Since the cylinders had to be rigid foldable, a number of models proposed by (Tachi

&

Miura,

2013)

were

studied.

The

first

cylinder

model,

derived

by

mirroring 4 parallelograms of a standard Miura-ori pattern was eliminated due to the impracticality of the internal space.

From a flat state in the XY plane, the

structure rises to its maximum height at 50% deployment before starting to compact

in the

ZY

plane. (See Figure

4-10) At this

maximum height, the

cylinder ‘wall’ panels are at their maximum angle, which, in any case is still a very acute angle. Thus, the internal space created within the cylinder is nonusable since the walls are too heavily inclined. The lack of usability is still present even if the parallelogram panel edges are very slightly angled (as shown in Figure 4-10 which is based on parallelograms with edges inclined at 5°).

Further to this, the cylinder ‘base’ is not a flat plane which means that additional supporting systems would have to be integrated and a flat floor would have to be added inside, further decreasing the available volume.

A

more

complex

version

-

the

Zonogon

cylinder

-

was

derived

from

this

example (Tachi & Miura, 2013) although similar but milder issues of practicality may still be observed.

The Tachi-Miura polyhedron (TMP) may be used to make the most practical rigid foldable cylinder in terms of the aforementioned criteria. During use, this structure would expand primarily in one main direction although until it eventually reaches a flat foldable state. Until a particular angle the model appears to be expanding linearly before it starts to drastically decrease in height and flatten as shown in Figure 4-11.

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Figure 4-11: (Top) Images showing the TMP rigid foldable cylinder crease pattern and deployment stages as proposed in (Tachi, 2010). (Bottom) Diagram showing a typical TMP cylinder highlighting Miura-ori fold within. Source: Author

The TMP cylinder geometry is derived from a Miura-ori module as can be seen in Figure 4-11, so typically it transforms in a similar way.

The crease patterns for this cylinder are attached in Appendix II.

4.3.2 Size and Proportion The dimensions of a TMP Cylinder may be adjusted such that different interior volumes may be created depending on the intended use. Figure 4-12 shows how the width dimension can be adjusted to accommodate different functions.

The height may also be controlled in a similar way and increased if the design requirements call for it or if there is the intention to add more floors within.

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Figure 4-12: Varying widths are possible by simply modifying the crease pattern dimensions.

Source: Author

Thus, as can be deduced, the interior of this cylinder is flexible because the dimensions can be changed easily without compromising usability. In a static state, the roof of the cylinder is essentially a corrugation which means that it behaves in the same way as a folded plate structure. Thus if the wall-ceiling connection is properly fixed the cylinder will remain stable in spite of potentially large internal spans.

4.3.1 Building components components It is also possible to integrate typical building components such as floors and openings within this rigid foldable cylinder. This may be done easily by replacing a number of panels with glass or openings lie within a typical panel dimension.

Figure 4-13: Schematic diagram depicting the insertion of clear glass panels. Source: Author

Typically the floor will act as a diaphragm, bracing the structure in the lateral direction. It would make sense to design this floor such that it unfolds from 92

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within the cylinder itself once the rest of the structure is well in the process of deployment.

Since

this

sort

of

structure

is

a

very

plausible

option

for

an

emergency shelter or rapidly deploying structure, it must be ensured that with regard

to

material

choice,

those

with

the

best

durability

and

thermal

performance are to be preferred.

4.3.2 Thickness, Thickness, Connection and and WaterWater-tightness Given the evident similarities with the previous pattern, the issues of thickness, connection

and

water-tightness

are

not

explored

in

detail

so

as

to

avoid

repetition.

It is possible to create this cylinder using the constant thickness panel model and a series of waterproof hinges. Similar to the canopy, these hinges would either be mechanical or a high strength fabric sandwiched in between panels. In any case the structure must remain watertight in order to ensure that it is fit for habitation.

The

ends

of

the

rigid

foldable

cylinder

would

also

have

to

be

enclosed using lightweight panels.

Such a structure possesses some inherent stiffness due to its closed cylindrical form which means that it may require less flexible connections than the canopy.

However, some critical connections exist at the interface between the ‘wall’ and ‘roof’ element of the cylinder. This connection must be able to lock itself when the cylinder is in a static state. This is completely essential due to lateral loading on the structure, which could cause it to sway and ultimately undermine the system.

4.3.3 Boundary and Deployment The cylinder requires a flat path as its base in order to be allowed to deploy back and forth. Continuous moving supports at the base would need to be 93

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provided as is evident from Figure 4-13. The structure possesses some inherent stiffness and strength in compression due to its form but similarly to the canopy it would be deployed using guide rails or cables.

4.4 Conclusion In conclusion, we have seen how typically two case studies would best be produced

using

constant

thickness

panels,

considering

that

it

is

generally

possible to achieve low thickness to width ratios. The amount that the structure will be allowed to compact will however be reduced slightly when compared to the tapered hinges model. Depending on the use, this limitation might not be so relevant.

The ability of the connection to become stiff when necessary has also been repeatedly highlighted.

Possibly one of the main reasons why difficulty at the deployment stage is so significant is because the structure we are trying to build is composed of a number of separate panels and a number of separate connections. The model we produce at an abstract stage is a single piece of paper which adapts itself according

to

the

tessellation

we

apply.

Ultimately,

with

a

combination

of

excellent properties and with the arrangement of paper fibres in a particular way an underlying network structure is created and the paper can take the repeated folding

and

unfolding

movements.

To

try

and

produce

a

connection

that

behaves in this way from scratch is a difficult task, which is why often the focus is shifted to deployment aides and boundary conditions.

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CHAPTER 5. FINAL DISCUSSION

5.1 Research Questions Given the technological difficulties that exist, is the adoption of foldable origami tessellations realistic in todayâ&#x20AC;&#x2122;s building industry? How and why is this so?

What can we say about the direction being taken from the way research has dealt

with

this

topic

thus

far?

Is

it

leading

somewhere

solid

and

are

the

applications justifiable?

5.2 Why foldable origami structures in the building industry?

When human civilisation started, temporary shelters were used. This function of temporality fit the need that man had as a hunter gatherer: an entity that was always alert and on the go, following his source of life. With changing trends and the exploitation of land through agriculture, the nature of shelter began to become

more

permanent.

The

eventual

discovery

of

new

materials,

in

particularly concrete, changed the building tradition completely. Concrete is one of the most permanent building materials in the industry to date. Anything that is module or component based can, in some way or another, and for the general case, be dismantled or deconstructed in a relatively straightforward way. When concrete is cast-in-situ however, this degree of flexibility is eliminated.

The idea of permanence and impermanence in the building industry is therefore very much tied with the use of material. The provision of certain properties through engineered materials can provide the functions required by a building typology. Thus, the

rather unachieved transformable origami structure could

partially be due to the lack of technological advancements that have been made thus far.

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However in truth is there a need to have temporary buildings? Certainly, there is a need to provide shelter which incorporates agility, flexibility and adaptability as can be proven by global issues such as constantly changing climate conditions, mass displacements and rising emigration, war or natural disasters. Being able to

transport,

set

up

or

remove

shelters

in

the

least

time

possible

in

such

scenarios is essential to keep these situations under control. However we are also implying that our structures of choice for these temporary buildings should be derived from geometries that are inspired from origami. Why should this be the case? Temporary structures do in fact exist as functional, highly rationalised systems which most often incorporate linkages and fabric skins. Why should we make them any different?

Perhaps the strongest reaction to these questions comes from the fact that origami structures- if successfully implemented- would provide an all inclusive solution to the number of design constraints that exist within the environment they are suited for. By this we mean that in theory they would be able to address issues

revolving

around

openings,

volume

containment,

sealing,

ease

of

deployment, growth, architectural design, ease of assembly, transportability, and others in a holistic manner using a unified approach given that the material choice does not prohibit this. The set geometries and relationships that exist within a piece of origami condition many of its behavioural properties, and in certain case – i.e. for certain tessellations – these relationships can produce quite a successful result.

As

architects

and

engineers

whose

work

is

a

constant

play

between

both

disciplines we should, at all times, aspire to achieve forms that possess a degree of value in both worlds. As Theo Jansen rightly said that the

‘walls

between art and engineering exist only in our minds’. There should be no reason why temporary structures cannot be based on intricate origami tessellations. The eventual comparisons against current temporary shelter solutions may be drawn

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up once both systems are up and running, but until then one cannot condemn a system before it can be made to work.

Perhaps these structures may also become useful within a more permanent building fabric, where a degree of flexibility may be desired on a smaller scale: for example the need to provide a changing interior structure on a common basis to accommodate multiple functions throughout the day and night, or the need

to

control

environmental

and

seasonal

changes

through

mechanical

response, or simply to reorganise spaces so as to demonstrate the beauty of motion.

Making our building fabric more adaptable through transformable structures is something that could potentially contribute to the â&#x20AC;&#x2DC;lack of spaceâ&#x20AC;&#x2122; problem. The concept

may

also

be

extended

to

the

afterlife

of

a

building,

rather

than

deconstructing and rebuilding, transformability could be the solution. On a more social level, different activities within a building often do not happen at the same time and providing the option to transform the interior could be an economical prospect if the adjustment is well thought out. How much the exterior fabric and hence interior volume is likely to transform in a 20

th

century city scenario is

questionable. Apart from continuous structural stability, which is virtually still an unresolved area, a number of other issues would arise, for example the question of territory ownership would no longer be defined in the traditional sense.

If the nature of buildings is to change then the nature of resident manâ&#x20AC;&#x2122;s lifestyle must change with it. In the civilised 21

st

century world, nomadic lifestyles are

anything but the norm. A life of settlement offers concrete jobs, income and family growth, perhaps only prone to occasional political instabilities. However, in post war and disaster zones or in developing countries the situation may be quite different. The building industry is generally a source of economic growth as it

creates

jobs,

enabling

previously

uncertain

families

to

begin

to

stabilise

socially and slide into the life of permanence.

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These

societies

are

slowly

converted

into

stable

communities

with

regular

lifestyles that continuously evolve and grow. This dynamic has created cities, metropolises, villages and other settlements: systems of consistency, growth, trade and routine. This is the way man has lived for centuries, and perhaps it is too ambitious to want to change this simply because we have discovered a way of building attractive temporary shelters. Realistically, and as satirical as it may sound, these temporary structures do not have the potential to permanently transform our built society that has solidly embodied the idea of developing one’s own territory for centuries.

These structures do however have the potential to exist within this fabric – as extensions, modifications, interior dividers, skins or installations. Whether their existence in this form is significant and successful is for society to decide.

5.3 The Concept Concept of Transformable Architecture In today’s day and age, the concept of architecture that embodies some sort of reaction or kinematic response to its environment is a leading approach in design.

Simulating,

studying

and

producing

dynamic

elements

within

architectural spaces is not only becoming easier to achieve but also extending the purpose and concept of buildings. Transformability concepts have had a colourful history and are continually being redefined through various forms and scales: modular housing, cities on wheels, kinetic facades, adaptable interiors, urban interventions, sculpture and much more.

This idea of change, although often attributed to a client’s or society’s changing needs, actually stems from something much deeper. The implementation of parametric

design

and

computation

in

architectural

thinking

has

a

major

influence. This approach towards design is also an instigator to change the way we perceive components and the spaces within which they exist. In terms of

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spatial quality, there is something inspirational and powerful happening when the fourth dimension is integrated.

More often than not, clients want buildings to last forever but then they want them to be flexible enough to meet their changing needs. Yet to what extent does one make a building flexible? Peopleâ&#x20AC;&#x2122;s needs are always changing and there are different ranges of mobility and flexibility.

5.4 Usability and Performance It has been established that active origami structures are worth pursuing due to their prospects as temporary structures. Although a number of technicalities still persist in terms of buildability, once achieved these structures may become pleasant solution to temporality in architecture. It is highly probable that origami structures will be applied as extensions or flexible bridges between existing buildings in which case an appropriate watertight detail will need to be provided at the interface. Their use in disaster and hostile territories is also very plausible as they incorporate solid panels, which is a more durable and less susceptible to damage than fabrics (which is a common current option). Further to this some insulation

may

also

be

achieved

through

the

panels

apart

from

an

overall

increase in robustness and stability.

Another positive

aspect that these

structures have

over other transformable

enclosures is that there is a very short assembly stage prior to deployment (In fact these two stages are somewhat merged). Deployment may occur in a linear, circular or curved manner.

Designing these origami mechanisms is essentially based around the study and design of their relationships in time and space. There are two aspects - when static, the main aim of the structure is to transmit forces to the supports while when mechanical, these forces are converted into motion. When designing these 99

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structures one is essentially making something that has to negotiate between its motion and

its

structural support. It

certainly leaves us wondering

how the

structure is capable of doing both, negotiating all these forces through a stable process.

Because paper is so unique it its performance, we are generally led to believe that this can be assimilated in its true form in larger scales. When it comes to using another material the situation is consistently paradoxical â&#x20AC;&#x201C; one needs a material that is capable of taking loads, but it also has to be weakened at the crease lines (creating a hinge reduces the section) in order to enable the folding motion.

Since

most

materials

cannot

perform

as

a

result

there

must

be

alternative ways to stiffen these crease-lines so that it does not buckle locally and to support the structure holistically. As a result, the material is being reengineered by adding new components. Thus it can be said that material choice is just as important as solving the geometric limitations.

5.5 Research trends in the field Research in rigid origami is necessary to continue reinforcing and extending the applications

of

a

topic

with

great

engineering

potential.

Unconditionally,

geometry plays a very relevant part in all this. Reinforcing or extending previous theories,

venturing

into

more

complex

geometries,

working

with

existing

underdeveloped patterns and creating new relationships is without a doubt the main fuel of all the possibilities that can emerge. This is why a lot of research being done concerns topics such as freeform origami, curved folding, elastic origami, cellular structures and so on.

However, most of this research remains on a conceptual level and does not genuinely

seek

to

explore

potential

applications

in

much

detail.

In

the

engineering world origami has most certainly been proven useful for a number of

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things, and it is today solidly viewed as a subject with great potential. Investment into this topic is also growing and is being given more and more attention, as can be

revealed

by

the

substantial

amount

of

funding

reserved

to

origami

research, especially in the US. This research has no tangible, single purpose but that any result is welcomed and exploited.

In aerospace engineering these structures have always had great potential not only because of their ability to be transported easily into space but also because they are well suited to be self-deploying especially if equipped with robotic actuators and sensors which is an ideal requirement in space. Thus, research in the field of origami structures in space â&#x20AC;&#x201C; even as future habitable spaces- is certainly a way forward. Transformable origami structures in space will most probably be subject to fewer gravitational forces.

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CHAPTER 6. CONCLUSION

6.1 Conclusions This text has shown that while these structures are slowly gaining some standing in various engineering fields, there is still a lot of work to be done. Much like in others, the economical aspect of any development within the building industry has a significant degree of influence when it comes to the success of new structures, methodologies or products.

The ongoing work on active rigid origami tessellations has great potential to inspire structural engineers to further investigate these structures and seek to find creative ways of integrating them within the built environment. While this will come as a challenge, the long term flexibility prospects of future builds may be substantial. The less permanence in our approach to future buildings, the more flexible sustainable we can be.

6.2 Recommendations for Future Work During the course of this dissertation there were several instances where the study

could

not

be

taken

further

due

to

the

lack

of

available

research

or

solutions regarding a topic.

The recommendations for future work are the following;

-

Solving the 9 DOF vertex connections for a Miura-ori pattern so that from a practicality aspect, there would not be too much reliance on fabric to seal the joint.

-

Studying the possibility of having a connection developed specifically to work

with

the

structure

in

its

different

forms,

ideally

possessing

the

capabilities to lock and unlock such that stability is achieved mid-way 102

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through deployment if the structure is required to stop at a particular angle.

-

A detailed study regarding the physical state of paper during all stages of deployment and thus the development of a material composite that can better assimilate these properties on a large scale.

-

A study of the moving boundary supports of an active origami tessellation.

-

A study on how to determine the ideal tessellation density for the spans and form of the structure being investigated.

-

A study relating to efficient control of the structural stability during the deployment process

-

Water-tightness, physical flexibility and durability as the ideal material properties for such structures.

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APPENDIX I â&#x20AC;&#x201C; Software Description RHINO V5 Version of Rhinoceros; Version 5.0

Rhinoceros (Rhino) is a standalone, commercial NURBS-based 3-D modelling software, developed by Robert McNeel & Associates. The software is commonly used

for

industrial

design,

architecture,

marine

design,

jewellery

design,

automotive design, CAD / CAM, rapid prototyping, reverse engineering, product design as well as the multimedia and graphic design industries

Rhino

specializes

modelling.

in

Plug-ins

free-form developed

non-uniform by

McNeel

rational includes

B-spline

(NURBS)

Flamingo

(raytrace

rendering), Penguin (non-photorealistic rendering), Bongo, and Brazil (advanced rendering).

Over

100

third-party

plugins

are

also

available.

There

are

also

rendering plug-ins for Maxwell Render, V-ray, Thea and many other engines.

Additional plugins for CAM and CNC milling are available as well, allowing for tool path generation directly in Rhino.

Like many modelling applications, Rhino also features a scripting language, based on the Visual Basic language, and an SDK that allows reading and writing Rhino files directly. Rhinoceros 3d gained its popularity in architectural design in part because of the Grasshopper plug-in for computational design. Many new avant-garde architects are using parametric modelling tools, like Grasshopper.

Rhino's

increasing

popularity

is

based

on

its

diversity,

multi-disciplinary

functions, low learning-curve, relatively low cost, and its ability to import and export over 30 file formats, which allows Rhino to act as a 'converter' tool between programs in a design workflow.

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GRASSHOPPER Version of Grasshopper 0.9.0014

Grasshopperâ&#x201E;˘ is a visual programming language developed by David Rutten at Robert McNeel & Associates. Grasshopper runs within the Rhinoceros 3D CAD application. Programs are created by dragging components onto a canvas. The outputs to these components are then connected to the inputs of subsequent components. Grasshopper is used mainly to build generative algorithms. Many of Grasshoppers components create 3D geometry. Programs may also contain other types of algorithms including numeric, textual, audio-visual and haptic applications.

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APPENDIX II – Crease patterns patterns TACHITACHI- MIURA POLYHEDRON CYLINDER

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MIURA ORI TESSELLATION

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APPENDIX III â&#x20AC;&#x201C; Grasshopper Definitions This definition was used to produce a Miura-ori tessellation with the ability to transform from the flat paper state to the flat folded state. The model is based on the path traversed by each vertex between the two states. The form and position of these paths and their relation to each other in time and space were deduced

through

continuous

observation

of

a

paper

model.

The

travelling

vertices were then combined to form new surfaces that define the panels of the active tessellation. By recording the measurements of each panel during the deployment stages it was then ensured that they remained the same size at every point during deployment. This proved that the model was acting as a rigid foldable structure.

The initial surface was a single parallelogram that was modelled in Rhino V5.

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APPENDIX IV IV â&#x20AC;&#x201C; Documentation of physical models

122