Structural Morphology PDF
Structural Morphology PDF
Structural Morphology PDF
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KwangWei - An Anthology.pmd
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PREFACE
Which definition could be provided for the expression Structural
Morphology? This interrogation was the first one when Ture Wester,
Pieter Huybers, Jean Franois Gabriel and I decided to submit a
proposal of working group to the executive council of the International
Association for Shells and Spatial Structures. We were discussing about
topics of high interest for us, topics related to shapes, mechanical
behaviour, geometry of polyhedra and surfaces, design, bionicsWe had
no clear answer to our first question, but hopefully our proposal was
accepted by the executive council chaired by Steve Medwadowsky. Ture
Wester was the chairman of this working group until 2004, and I
accepted to take the baton for some years. The year after, during the
annual symposium of the association, I proposed to my colleagues to
collect some of the papers that had been published for IASS events
(symposia and seminars). Pieter Huybers suggested calling it An
anthology of structural morphology, and we agreed. This book gives our
practical answer to our initial interrogation: while reading the thirteen
chapters, a better understanding of Structural Morphology is possible.
Several aspects of this discipline, which was introduced by famous
pioneers like Leonardo da Vinci, Paxton, Graham Bell, E. Haeckel and
more recently among others R. Le Ricolais, are illustrated by the
contributions of many experts.
Next October our working group will hold its 6th international
seminar in Acapulco. This seminar will be devoted to Morphogenesis.
A new era is beginning with younger members and contemporary
problems. It is also my pleasure to know that members of the
International Association for Shell and Spatial Structures and its
President Professor John Abel are always interested in the works
produced by our group. This book simply aims to be a small milestone
on the road of structural morphology.
Ren Motro
Montpellier 27 June 2008
v
CONTENTS
Preface
15
33
4. Polyhedroids
P. Huybers
49
5. Novational Transformations
H. Nooshin, F. G. A. Albermani and P. L. Disney
63
83
109
117
vii
viii
Contents
133
145
159
173
189
CHAPTER 1
THE FIRST 13 YEARS OF STRUCTURAL MORPHOLOGY
GROUP A PERSONAL VIEW
Ture Wester
Associate Professor, Royal Danish Academy of Fine Arts, School of Architecture
Inst. for Design & Communication, Philip de Langes Alle 10,
DK-1435 Copenhagen K, Denmark
ture.wester@karch.dk
1. The Background
In this paper I will try to tell a short story about my personal
experiences/adventures as chairman for the Structural Morphology
Group, IASS working group No 15. I will describe the events as they are
in my memory and my heart. This means that this paper is not a
complete, even not a sketchy report of the many unique activities, papers,
members, research etc. It is my own impression about the absolutely
most important part of my professional life. I will go back in time and
tell a little about why it is so: I graduated from the university at the age
of 21 as structural engineer in 1963 and soon after became a teacher in
Structural Design at the School of Architecture which had almost no
tradition in research. My professor Jrgen Nielsen however was an
exception and inspired my interest for research. After researching and
teaching for 10 years in the topic of interactive dependency of shape and
structural behaviour. These studies resulted in my discoveries of the
geometrical characteristics and the necessary equation for the rigidity of
pure structures in 3D and, most important, the concept about structural
duality. At that time I had no contact with the international scientific
world, but suddenly my bubble burst and thereafter everything
developed extremely fast leading to the start of the group in 1991.
T. Wester
After a contact with Prof. Makowsky and Dr. Nooshin in Surrey, they
encouraged me to write a paper for IASS87 in Beijing. I went there and
I was totally overwhelmed. I met wonderful people everywhere. Famous
people I had read and heard about and, most important, people like me,
researching on useless topics which made them isolated and maybe
lonely in their university without sufficient local back up. As their
research and the persons behind often were extremely fascinating and
were dealing with the same interest of studying the intimacy between
structure and shape, an obvious idea began to take shape: If these people
were getting closer together they could fertilise and encourage each other
and maybe even begin to collaborate on their weird research. The
friendships made on these my first IASS events are still in the best of
health.
2. The Beginning
For the rest of the paper I must apologise for any wrong or poor memory,
any insult and all missing important information and mention - all
unintended! In order to get a report over the activities please read the
Newsletters of the group.
The IASS Working Group No 15 on Structural Morphology (SMG)
was founded during the IASS Copenhagen Symposium in 1991. There
was a relatively short period of planning before launching the group. It
must be admitted that I never expected an international group with some
of the world elite in the field as members. The first exchange of the idea
was with Huybers on a bench in a park during an excursion in ISIS89 in
Budapest. One month later at IASS89 Congress in Madrid, the gang of
4 -Pieter Huybers, Franois Gabriel, Ren Motro, myself established
an action group with the aim to form a IASS working group. The specific
name Structural Morphology was proposed by Michael Burt. Many of us
had in the beginning a problem with the word Morphology, as it
appeared to be a biological term connected to flowers and dusty natural
museums - and very far from engineering terms. But slowly it was clear
that it was exactly the right word, as all other words as form, shape,
configuration etc. indicated too narrow concepts: Morphology
simply means Study of Form and is used in many sciences. While I
3. Group Activities
In the following chapters I have tried to convey some of my
unforgettable experiences with my group. Not only professional matters
but also social and artistic events during particularly our seminars will be
discussed. I will try to describe these activities in some text and some
photos. However, I have to apologise for missing description of
important activities, experiences and events in the following chapters.
This is not because of unwillingness, but simply poor memory or lack of
awareness.
T. Wester
T. Wester
T. Wester
10
T. Wester
11
12
2.
3.
4.
5.
6.
7.
8.
9.
10.
T. Wester
13
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T. Wester
constructively discuss with each other. Thank you for the friendships
which met me across all borders that evolved inside the group. The
friendly atmosphere, the high spirit and professional level mixed with
artistic touch has been unique, everything because of the members
contribution as individuals or in groups. I bet that all who have
participated in our seminars will agree that the mixture of warm and
friendly spirit, and high professional standard, has been unsurpassed. It is
my sincere hope that this precious quality will remain in the future. It has
been a shear pleasure for me to have been the chair for this particular
group from its birth to teenager. I think many of the past activities will
continue in the future but times changes - sometimes very fast
especially for a teenager. SMG has recently included a very energetic
subgroup FFD (Free Form Design) mainly based on young researches
from TU Delft, who could fertilise our group with new energy, new
topics, young people besides, the FFD topics seems to be right in the
centre of Structural Morphology whatever the definition is! As many of
you already know, I left the chairmanship of health reasons, lack of
sufficient support from my university, but also my feeling that young
blood should be infused to the leadership. I am convinced that Ren
Motro is the best qualified to handle this process. The photos I have used
in this paper are taken by group members, but I dont remember who has
taken which photo, so I thank all who through the years have send me
these memorable photos.
CHAPTER 2
AN APPROACH TO STRUCTURAL MORPHOLOGY
Ren Motro
Laboratoire de Mcanique et Gnie Civil. Universit de Montpellier 2. Groupe
de Recherche et Ralisation de Structures Lgres pour lArchitecture
This communication aims at following several lines of questions
concerning structural morphology. This is re-positioned in the general
context of the design of construction systems to restore the full
organisational sense to the word structure. The system proposed
makes it possible to classify design parameters in four categories:
forms/ forces/ material and structure. All the data related to the four
parameters are subjected to constraints. The system which is the subject
of the design must among other things meet mechanical criteria.
The problem of design can thus be handled by ordinary systems
optimisation modelling. The position of structural morphology in
this system is at the interface between the parameters form and
structure; it expresses their interactions and meets the requirements of
the material and the balance required. A number of landmarks give
a glimpse of the potential that can be hoped for from research in
structural morphology. They are chosen from fields in which they have
already given results but without exhausting the potential: that of stone
cutting and that of geometrical studies in which stress must be laid on
both topological and dimensional aspects. More generally, structural
morphology can draw on bionics on condition that this is not limited
to surface of phenomena and that questions are raised about the
underlying principles behind them. Some classes of systems cannot do
without interactions between form and structure. This is the case of
equilibrium forms illustrated by tensegrity systems and membranes.
It can be considered that the form expresses the meaning of the
structure, that is to say the meaning of the choice of organisation of
matter decided by the designer. In fact it goes beyond the meaning of
the structure to show that of architecture, and the final coherence of
meaning should be sought.
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Introduction
The title of this communication fixes the limits - an approach rather than
complete coverage. It cannot be otherwise in a seminar where one of the
aims to draw together scattered items to give impetus to a field which has
existed since man started building structures but which is not established
as a discipline. We propose a method of approach which cannot be
considered as modelling but which makes it possible to examine in the
same way several landmarks in the past and present state of the building
sciences before asking questions on the meaning which can be given to
research on structural morphology with a view to enriching architectural
design.
1. Structural Morphology
1.1. Structure and System
1.1.1. Structure
The word structure expresses a concept which has not been the subject
of many controversies in spite of the inaccuracy associated with it. The
very aim of our work required a more accurate approach to the concept.
Systems theory and its definitions shed light on the concept of structure.
Struere, meaning build is the Latin root of the word structure.
Vitruvius used this meaning of the word in his treatise on architecture
(27-23 B.C.). For Vitruvius/ structura was brick or stone and mortar
masonry. This means that the word had a building connotation from the
outset. Archeology of the idea of structure would have to establish the
date and context of the first use of the word to indicate not just a mass of
inert masonry but the building itself with its own order - that of a
construction with both mechanical and functional determinants.
Architectural metaphor has a somewhat unsuspected role in the
archaeology of structural thinking in general. It has been the source of
numerous models, generally of the mechanistic type based on the
distinction between form and structure inherited from Viollet le Duc.17
His main contribution was the study of the science of construction, of
structure, which governs all the formal and decorative features of Gothic
17
architecture. Viollet le Duc thus showed the need for structural analysis
of architecture, implying the abandoning of the strictly descriptive point
of view. This model has been used in recent years in fields such as
linguistics which, in the structuralist field is based on the architectural
metaphor.
This structural analysis is first of all a desire to consider the
architectural phenomenon in terms of Systems, which are linked and
coherent to varying degrees, and in such a way that a change made to any
part can but be felt in other parts of the constructed organism.
This is an affirmation that architectural syntax, like the mechanisms
of articulated language, is not reduced to a combination, a coordination
of forms of equal value, but that it covers a ranked organisation of
constituent units, the latter being disposed according to a strict order but
whose subordination is variable.
It is finally my intuition that, if the architectural phenomenon has a
meaning, this should not be sought at component level but in the System
itself.4
1.1.2. System
Systems and structuralist movement theoreticians1 have drawn much of
their vocabulary from the language of architecture. The problem for
defenders of structural thinking is to go beyond the mechanistic model.
However, builders -engineers in particular - should take advantage of
progress in the deepening of method in the archaeology of which its
objects have a decisive position. This point led to this description of the
results of systems epistemology performed with a multi-field approach
by researchers who have deepened concepts such as structure, system
and form. Better understanding of terms, most of which are common in
builders vocabulary, can open up new pathways in structure design
related to morphology, the subject of this paper.
The notion of system, as defined by P. Delattre is a first stage:
The notion of system can be defined very generally by saying that a
system is a set of elements which interact between each other and
possibly with the external environment.6
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19
Delattre used this as the basis for listing the parts of a full definition:
affirmation of the existence of categories and relations between
categories,
the kinds of relations between categories (order, topology),
the kinds of elements in the different categories: listing of their
characteristics,
the number of elements in each category,
the analytical form of the expressions linked to the relations between
categories (assembly of the characteristics and the corresponding
numerical values).
1.1.3. Structure and System
Structure is used in the broad sense here: the manner in which the
parts of a whole are arranged. This definition serves only to affirm the
existence of an Entity, parts and organisation without establishing the
nature of the elements. This concept is therefore included in that of
system whose dynamic aspect is inherent in the very nature of the
elements which can alone account for the interactions.
It is then possible to apply a reduction process to the elements of
definition of the system to extract a definition of its structure. If one
keeps to the first two elements of definition given by Delattre the nature
of the elements is not specified. He called this a Relational Structure.
This is the meaning used in mathematics for group structure of an
ensemble, for example.
A second level of definition is obtained by adding the third element
of definition of systems - the list of the characteristics of the reference
categories grouping the elements. The definition reached corresponds to
the Total Structure of the system. Elements whose characteristics are
specified may themselves be arranged in systems; the total structure
implicitly contains the structure of subjacent levels at the description
level chosen, which does not contain the relational structure.
It is important to observe that one cannot talk in terms of relational or
total structure without defining the system to which they are related. The
level of description chosen must also be specified. The subject being
20
R. Motro
design in structures, research attention was paid to both the total and
relational levels.
1.1.4. Conclusion
Going further into the notion of structure outlined in this part would
be meaningless if the subsequent formalisation were aimed at replacing
the models well known to engineers and technicians. The aim of this
approach is to re-establish reflection on the design of structures by
removing ail unrelated information. It then remains to be seen how this
can be useful for structural design in the context of its close relation with
architecture.
1.2. Form and Structure
An attempt at defining the concept of structure soon leads to
simultaneous consideration of that of form. The two concepts are
closely related and their history displays inverse evolution. Definition of
one often makes reference to the other. Thus, the French Larousse
dictionary defines structure as follows in the psychology section:
an organic set of forms which, according to some psychologists, is
perceived directly before each detail is isolated.
On form: Form theory, a theory considering the perception of a set
of organised structures before the details are perceived and which affirms
in all domains the influence of the whole on its constituent parts.
Structure is defined as the manner in which the components of a
whole are assembled. The definitions proposed for the concepts clearly
show the existence of a whole and its parts; these notions can be related
to that of System.
In fact, the evolution of the two concepts can be clarified by noting
that both are used in a limited sense to show the spatial existence of the
object in question. Today, the concept of form is mainly used for the
limited sense of spatial configuration and that of structure is used in a
broader sense. It is known that the Ancients did the opposite - or at least
form was the broad concept for them. The word structure replaced
form little by little, leaving it only the limited sense.
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R. Motro
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R. Motro
It is in fact the direct relation between the study of form and structure
extended to cover the relational sense. This relation is affected by the
behaviour of the material and by the need to ensure the static (and
sometimes dynamic) equilibrium of the system S being designed. It is not
complete theoretical modelling but a method of approaching the
problem. Other parameters must be considered, and especially those
related to the technological facilities available for the construction
System. The cost of construction and operation is an important factor in
evaluation.
2. Landmarks
The effect of the form-structure relation as described above is present
in almost all construction systems. The importance of the role of the
two other parameters depends on the case. Four examples are chosen as
references and described briefly in the light of the conceptual procedure.
2.1. Stone Cutting
Stone cutting plays a very important role in the history of building. The
material is characterised by a dominant property of compressive strength
and gravity is the determinant element in all the actions applied. Careful
25
2.2. Geometry
Geometry plays a central role in the design of form in the broad sense. It
is related to the partitioning of space, whether this is discrete or
continuous. Designers make extensive use of polyhedral geometry in
the first case and the geometry of surfaces in the second. The two
approaches meet when the problem is not one of partitioning space but of
handling special surfaces, such as geodesic domes. Examples can be
found both in the work of Graham Bell at the beginning of the century
and in the steel framework of the Zeiss planetarium in lena in Germany
in 1923.
Researchers like Le Ricolais and Fuller have taken the question
further. The former established the basis for space-frames which were
developed very widely; his observation of Radiolaria went beyond the
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R. Motro
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R. Motro
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R. Motro
CHAPTER 3
THE STRUCTURAL MORPHOLOGY OF CURVED
DIAPHRAGMS OR THE STRUCTURAL BEHAVIOUR OF
FLORAL POLYHEDRA
Ture Wester
Associate Professor, Royal Danish Academy of Fine Arts,
School of Architecture, Copenhagen
The publication by Michael Burt The Periodic Table of the Polyhedra
Universe [Ref. 2] interpret Eulers theorem on polyhedra in three
dimensional space to its extreme, and shows by this that the classical
plane faceted convex polyhedra just are a tiny part of the full spectrum.
Burt shows that polyhedra of any genus - including infinite polyhedra
and so-called floral polyhedra where some are not imaginable with
plane facets and straight edges - are included in the Euler-polyhedral
universe. Structural Order in Space [Ref. 5] is investigating the
structural behaviour of any plane faceted so-called conventional
polyhedron, and deduces the necessary stability equation for structural
configurations stabilised by shear forces alone i.e., pure plate action and it reveals the profound interrelation between lattice and plate
structures - the structural duality - which follows the well known
geometrical duality. These statical relations for shear-stabilised
polyhedra and the structural duality turns out to be valid for any plane
facetted configuration and they can be extended from the level
of topology/stability to metric geometry/magnitude of forces and
elasticity, as shown in [Refs. 6 & 7]. It is therefore a necessity to try to
increase the statical considerations to cope with the full understanding
of Euler-polyhedra as described by Burt where the floral type is the
normal and the plane faceted type is a rare specialty. The present paper
is an attempt to approach the full spectrum of polyhedra as suggested
by Burt.
33
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T. Wester
35
Fig. 1. The five archetypal Platonic plane faceted polyhedra arranged in dual pairs. The
topological duality is extended to a structural duality, relating the stability behaviour of
pure plate and pure lattice action. If one side of a dual pair is stable as a pure plate
structure, then the other is stable as a pure lattice structure. The Platonic master-solid
the tetrahedron - is so basic that it is self-dual, hence stable as both a pure lattice and a
pure plate structure. (Courtesy of Ola Wedebrunn)
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T. Wester
37
Fig. 3. Examples of polyhedra of different genus. It also indicates the way of determine
the genus of units for infinite polyhedra. (Courtesy of Michael Burt)
38
T. Wester
This structural duality applies not only to all plane faceted polyhedra,
but to all plane faceted structures in general (Fig. 2), and performs a
method to do direct static analysis on any complicated plate structure by
transforming it to its dual, do the analysis by one of the many lattice
structure analysis software, and transform back the results. As the elastic
properties also can be dual transformed [Ref. 6], then redundant
structures may be analyzed through dual transformations.
4. Plane Faceted Polyhedra - Genus above Zero
For this case Eulers extended theorem is as follows:
V + F E = 2(1-g)
Where the number of vertices is V, the number of facets is F and the
number of edges is E while g is the genus. See Fig. 3.
As we still consider 3-dimensional finite structures, the basic stability
equations are the same as for any genus number. This means that the
stability equation for an arbitrary plane facetted polyhedron of genus g,
is:
B + SL + BU + S 3(N + P)
for a free floating polyhedron (S equal to 6)
B + SL + BU 3(N + P 2)
The redundancy is then
R = 3(E V F + 2) = 6g
This tells that the redundancy for combined plate and lattice
polyhedral structure is only dependant of the genus, and is equal to six
times the genus.
This means that a fully triangulated polyhedron of genus g has a
redundancy of 6g as a pure lattice structure and therefore 6g bars may be
removed from the structure without affecting the stability. In the same
way, a fully trivalenced pure plate polyhedron of genus g has a
redundancy of 6g, and therefore 6g shear-lines may be removed from the
structure without affecting the stability.
39
Fig. 5. Examples of floral polyhedra of genus zero. Many of these polyhedra are not
imaginable with plane facets and straight edges, if they should enclose a volume.
(Courtesy of Michael Burt)
5. Infinite Polyhedra
40
T. Wester
41
Fig. 6. We might define the structural type according to what kind of forces is transmitted
from the curved facets to the curved edge. If the edge is in tension or compression and no
shear is transferred, we talk of pure lattice action, while if it is only transferring shear
forces, we talk about pure plate action. In all other cases we talk about combined action.
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T. Wester
43
Fig. 8. Left is shown the pyramid with open rectangular bottom; inserted in a rectangular
tube, which is open in both ends. The height of the tube is H and the distance from the
base of the pyramid to the plane virtual plate with the identical stabilizing effect is called
the eccentricity e. To the right is the diagram of forces is equilibrium for the system. The
forces perpendicular to the plane of the paper are creating the moment that moves the
applied shear-force the distance e, following Bredts equation.
Fig. 9. If the pyramid is truncated, then the system is changed to plate action. The
diagram of forces in equilibrium is shown to the right. In this case e is not following
Bredts equation, as the shear stresses transferred are not of the same intensity.
44
T. Wester
e=
1
1 1
+
H h
Fig. 10. If the pyramid is rotated 45, then the system appears like this. The equilibrium
is achieved as pure plate action. The eccentricity e is found to follow Bredts equation.
45
46
2.
3.
4.
5.
6.
7.
T. Wester
47
Arts, 1991, Vol. 2, pp. 119-124. This paper, presented at the IASS
Symposium in Copenhagen, deduces the necessary stability equation
for any plane facetted conventional polyhedron as a combination of
plate and lattice action. For the structural analysis of plane facetted
polyhedra of higher genus, see also Wester, T. & Burt, M. The Basic
Structural Content of the Periodic Table of the Polyhedral Universe.
Proceedings of the IASS Symposium in Singapore 1997, pp. 869-876.
This paper is the first attempt to merge the geometric and
the structural approach, and The Structural Morphology of the
Polyhedral Universe: Preliminary Considerations. Proceedings of
the international Conference Engineering a New Architecture
pp. 197-206, School of Architecture, Aarhus 1998.
CHAPTER 4
POLYHEDROIDS
Pieter Huybers
Assoc. Professor, Delft Univ. of Technology, Fac. CiTG, 2628CN,
Stevinweg 1, Delft, The Netherlands
The title of this paper refers to the group of figures, that are related to
polyhedra, that are - so to say - polyhedron-like. They are derived
from the Platonic (or regular) and Archimedean (or semi-regular)
polyhedra, that are composed of regular polygons. They can be quite
different from the original figures, that they are based on. This paper
pays attention to a family of forms related to these so-called uniform
polyhedra. Possibilities are shown to mitigate the rigidity of the
polyhedral geometry and to make them more suitable for application in
building construction.
1. Definition of Polyhedra
A common definition of a polyhedron is [Ref. 1]:
(i) It is covered by a closed pattern of plane, regular polygons with
3, 4, 5, 6, 8 or 10 edges.
(ii) All vertices of a polyhedron lie on one circumscribed sphere.
(iii) All these vertices are identical. In a particular polyhedron the
polygons are grouped around each vertex in the same number, kind
and order of sequence.
(iv) The polygons meet in pairs at a common edge.
(v) The dihedral angle at an edge is convex. In other words: the sum of
the polygon face angles that meet at a vertex is always smaller than
360 (see Table 2).
49
50
P. Huybers
11
12
13
14
15R 15L
10
16
17
18 R18L
Polyhedroids
51
tetrahedron
octahedron
dodecahedron
icosahedron
2
1 : 2
1 : ( + 1) = 1 :
1 : = -1
1.000000
1.41421356
0.70710678
0.381996601
0.61803399
code
1
2
3
4
5
3-3-3
4-4-4
3-3-3-3
5-5-5
3-3-3-3-3
name
Tetrahedron
Cube
Octahedron
Dodecahedron
Icosahedron
Truncated Tetrahedron
Truncated Octahedron
Truncated Cube
Rhombicuboctahedron
Truncated Cuboctahedron
Icosidodecahedron
Truncated Icosahedron
Truncated Dodecahedron
Snub Cube
Rhombicosidodecahedron
Truncated
Icosidodecahedron
18 3-3-3-3-5 Snub Dodecahedron
6
8
9
10
11
12
13
14
15
16
17
3-6-6
4-6-6
3-8-8
3-4-4-4
4-6-8
3-5-3-5
5-6-6
3-10-10
3-3-3-3-4
3-4-5-4
4-6-10
Total
angle
DA* Radius
4
6
8
12
20
6
12
12
30
30
4
8
6
20
12
180
270
240
324
300
180
90
120
36
60
0.612
0.866
0.707
1.401
0.951
8
14
14
26
26
32
32
32
38
62
62
18
36
36
48
72
60
90
90
60
120
180
12
24
24
24
48
30
60
60
24
60
120
300
330
330
330
345
336
348
348
330
348
354
60
30
30
30
15
24
12
12
30
12
6
1.172
1.581
1.778
1.398
2.317
1.618
2.478
2.969
1.343
2.232
3.802
92
150
60
348
12
2.155
P = polyhedron index
Code = side-numbers of respective polygons that meet in a vertex
V, E and F = number of vertices, edges and faces
Total angle = summation of face angles that meet in a vertex
*DA, Deficient angle = angle of missing part of plane, or 360 (or flat situation) minus Total angle
Radius = radius of circumscribed sphere
52
P. Huybers
10
1
2
3
4
5
4
8
20
6
-
12
-
6
7
8
9
10
11
12
13
14
15
16
17
18
4
8
8
8
20
20
32
20
80
6
6
18
12
6
30
30
-
12
12
12
12
4
8
8
20
20
-
6
6
-
12
12
-
Volume
Area
Compactness
0.11785
1.00000
0.47140
7.66311
2.18169
1.73205
6.00000
3.46410
20.64572
8.66025
0.67113
0.80599
0.84558
0.91045
0.93932
2.71057
2.35702
11.31370
13.59966
8.71404
41.79898
13.83552
55.28773
85.03966
7.88947
41.61532
206.80339
37.61664
12.12435
9.46410
26.78460
32.43466
21.46410
61.75517
29.30598
72.60725
100.99076
19.85640
59.30598
174.29203
55.28674
0.77541
0.90499
0.90991
0.84949
0.95407
0.94316
0.95102
0.96662
0.92601
0.96519
0.97923
0.97031
0.98201
Polyhedroids
53
sphere that can be thought to pass through the vertices. The volume of
this sphere is therefore larger than that of the corresponding polyhedron.
This is also the case for the area of their envelopes. The closer these two
values are, the better is the approximation of the sphere that is reached by
a particular polyhedron. The closeness of this approximation can be
expressed in a value that is called: the Compactness
Compactness of a polyhedron.
Compactness Cp = Quotient of the area of a sphere with the same
volume as the polyhedron with the index P, divided by the surface area
of this polyhedron.
This value is given in the equation:
Compactness CP =
3 36 * Vol 2
P
AreaP
(1)
54
P. Huybers
Other solids that also respond the previous definition of a polyhedron are
the prisms and the anti-prisms. Prisms have two parallel polygons like
the lid and the bottom of a box and square side-faces; anti-prisms are like
the prisms but have one of the polygons slightly rotated so as to turn the
side-faces into triangles.
(5)
(6)
The simplest forms are the prismatic shapes. They fit usually well
together and they allow the formation of many variations of closepacking. If a number of anti-prisms is put together according to their
polygonal faces, a geometry is obtained of which the outer mantle has
the appearance of a cylindrical, concertina-like folded plane. [Ref. 6]
Polyhedroids
55
These forms can be described with the help of only few parameters, a
combination of 3 angles: , and . The element in Fig. 7A represents
2 adjacent isosceles triangles.
= half the top angle of the isosceles triangle ABC with height a and
base length 2b.
= half the dihedral angle between the 2 triangles along the basis.
n = half the angle under which this basis with the length 2b is seen from
the cylinder axis. = /n.
The relation of these angles , and n [Ref. 3]:
tan = cos cotan (n/2)
(2)
These three parameters define together with the base length (or
scale factor 2b) the shape and the dimensions of a section in such a
structure. This provides an interesting tool to describe any anti-prismatic
configuration. Two additional data must be given: the number of
elements in transverse direction (p) and that in length direction (q).
6. Augmentation
Upon the regular faces of the polyhedra other figures can be placed that
have the same basis as the respective polygon. In this way polyhedra can
be pyramidized. This means that shallow pyramids are put on top of the
polyhedral faces, having their apexes on the circumscribed sphere of the
whole figure. This can be considered as the first frequency subdivision of
spheres. In 1582 Simon Stevin introduced the notion of augmentation
by adding pyramids, consisting of triangles and having a triangle, a
square or a pentagon for its base, to the 5 regular polyhedra [Ref. 4].
Recently, in 1990 D.G. Emmerich extended this idea to the semi-regular
polyhedra (Fig. 8). He suggested to use pyramids of 3-, 4-, 5-, 6-, 8- or
10-sided base, composed of regular polygons, and he found that 102
different combinations can be made. He calls these: composite polyhedra
[Ref. 5].
56
P. Huybers
(9)
(10)
7. Sphere Subdivisions
Polyhedroids
57
All other regular and semi-regular solids, and even their reciprocals
as well as prisms and anti-prisms can be used similarly [Ref. 12]. The
polygonal faces are first subdivided and then made spherical.
8. Sphere Deformation
The spherical co-ordinates can be written in a general form, so that the
shape of the sphere may be modified. This leads to interesting new
basis but are governed by different
shapes that all have the same basis
parameters (Fig. 13A). According to H. Kenner [Ref. 9] the equation of
the sphere can be transformed into a set of two expressions, describing it
in a more general way:
(3)
(4)
n1 and n2 are the exponents of the horizontal and vertical ellipse and
E1 and E2 the ratios of their axes. The shape of the sphere can be
altered in many ways, leading to a number of transformations. The
curvature is a pure ellipse if n = 2, but if n is raised a form is found,
which approximates the circumscribed rectangle. If n is decreased, the
curvature flattens until n = 1 and the ellipse then has the form of a pure
rhombus with straight sides, connecting the maxima on the co-ordinate
axes. For n < 1 the curvature becomes concave and obtains a shape,
58
P. Huybers
The role that polyhedra can play in the form-giving of buildings is very
important, although this is not always fully acknowledged. Some
possible or actual applications are referred to here briefly.
9.1. Cubic and Prismatic Shapes
Most of our present-day architectural forms are prismatic with the
cube as the most generally adopted representing shape. Prisms are used
in a vertical or in a horizontal position, in pure form or in distorted
versions. This family of figures is therefore of utmost importance
importance for
building.
(14)
(15)
Figs. 14 and 15. Models of a space frames made of square plates or of identical struts.
Architecture can become more versatile and interesting with macroforms, derived from one of the more complex polyhedra or from their
reciprocal (dual) forms, although this has
has not often been done. Packing
of augmented polyhedra form are sometimes interesting alternatives for
the traditional building shapes.
Polyhedroids
(16)
(17)
59
(18)
Figs. 16, 17 and 18. Models of houses based on P10, truncated R7 and P10 (built in
Mali).
9.3. Combinations
Close-packing is also suitable as the basic configuration for space
frames, because of their great uniformity. If these frames are based on
tetrahedra or octahedra, all members are identical and meet at specific
angles. Many of such structures have been built in the recent past and
this has become a very important field of application. The members
usually meet at joints having a spherical or a polyhedral form (Fig. 15).
9.4. Domes
R.B. Fuller re-discovered the geodesic dome principle. This has proven
to be of great importance for the developments in this field. Many domes
have been built during the last decades, up to very large spans. A new
group of materials with promising
promising potential has been called after him,
which has molecules that basically consist of 60 atoms, placed at the
corners of a truncated icosahedron or P13 (See Fig. 20).
10. 3D-Slide Presentation
The author has shown during the conference a few applications with the
help of a 3-D colour slide presentation. Two pictures with different
orientation of the light waves are projected simultaneously on one
screen. The screen must have a metal surface which maintains
maintains these two
60
P. Huybers
Polyhedroids
61
References
1. Critchlow, K., Order in Space, Thames and Hudson, London, 1969.
2. Huybers, P., The geometry of uniform polyhedra, Architectural
Science Review, Vol. 23, no. 2, June 1980, p. 36-50.
3. Huybers, P., Polyhedra and their reciprocals, Proc. IASS Conference
on the Conceptual Design of Structures, 7-11 October, 1996,
Stuttgart, 254-261.
4. Struik, D.J., The principle works of Simon Stevin, Vol. II, Swets &
Seitlinger, Amsterdam, 1958.
5. Emmerich, D.G., Composite polyhedra. Int. Journal of Space
Structures, 5, 1990, p. 281-296.
6. Huybers, P. and G. van der Ende, Prisms and antiprisms, Proc. Int.
IASS Conf. on Spatial, Lattice and Tension Structures, Atlanta,
24-28 April 1994, p. 142-151.
62
P. Huybers
CHAPTER 5
NOVATIONAL TRANSFORMATIONS
1. Introduction
Configuration processing is concerned with computer aided creation
and manipulation of configurations and the programming language
Formian provides a suitable medium for configuration processing
[Ref. 1]. Configuration processing activities are performed through a
variety of conceptual tools. The tools include a number of families of
transformations that allow the body of a configuration to be deformed in
various ways. One such family of transformations is discussed in the
present paper.
63
64
2. An Example
Consider the configuration shown in Fig. 1(a). This is a spatial
configuration consisting of 17 line elements that are connected together
at 12 nodes. The nodes are numbered in the manner shown in Fig. l(b).
Now, suppose that the configuration is required to be deformed so that
it fits in a particular location and suppose that the manner in which the
configuration is to be modified is given by a number of specifications
regarding nodal positions, as follows:
(1) Nodes 1, 6 and 8 are to be moved to the positions shown by little
circles, as indicated by arrows in Fig. l(b).
(2) The positions of the other nodes may be altered without any
restriction except for nodes 3 and 9 that are to remain in their
original positions.
A possible modified shape of the configuration that satisfies
conditions (l) and (2) is shown by dotted lines in Fig. 1(c) together with
the original configuration. The modified configuration is also shown in
Fig. l(d). It is seen that in addition to satisfying the specified conditions
for nodes 1, 3, 6, 8 and 9, the modified configuration involves
movements of the other nodes. These additional nodal translocations
(movements) have the effect of bringing the nodal positions throughout
the configuration into harmony with the specified nodal positions. In
other words, a node whose position is not directly dictated is moved in a
manner that conforms with the trend of the specified nodal movements.
Such a nodal translocation is referred to as a conformity translocation.
(a)
(b)
(c)
(d)
Fig. 1. An example of novational transformation.
Novational Transformations
65
66
Figs. 2(a) and 2(b). Thus, the position of a node remains unchanged
unless it has a directly specified translocation.
(a)
(b)
Fig. 2. An example of a sharp novation.
Novational Transformations
67
68
decay of the curve. In other words, the higher the value of C, the less the
effect of TSk will be on the coordinates of the nodes of the configuration.
An example of the application of the ED novation is shown in
Fig. 5(a). The planar configuration shown by full lines in Fig. 5(a) has
8 line elements and 6 nodes. There is only one specified translocation
which is indicated by an arrow at node 5. The components of this
translocation in the first and second directions are TS1 = 2 and TS2 = 1,
respectively. The configuration is subjected to an ED novation with
C = 1. The resulting configuration is shown by dotted lines, with the
numerical values of the nodal coordinates and translocation components
given in Fig. 5(b). The values given for node 5 in the last two columns of
the table in Fig. 5(b) are the components of the specified translocation
but the values for the other nodes in these columns are the components of
conformity translocations.
Node
U1
U2
3
4
Translocation
T1
T2
0.32394
0.16197
0.63246
0.31623
0.39258
0.19629
0.88608
0.44304
0.63246
0.31623
(b)
Novational Transformations
69
Node
U1
Translocation
U2
T1
T2
0.36917
0.67178
0.17541
0.49637
-0.17541
1.00980
-1
(b)
70
where d12 is the relative distance between the positions of the first and
second specified translocations. The first equation states that the
combination of
the first component of working translocation at node 5 and
the translocation in the first direction of node 5 caused by the first
component of the working translocation at node 6
must add up to the first component of the specified translocation at
node 5. The remaining three equations have similar implications.
The above equations in matrix notation will assume the form:
Novational Transformations
71
where
and since
then
72
where
D12 is the distance between the point that has the first specified
translocation and the point that has the second specified translocation.
D13 is the distance between the point that has the first specified
translocation and the point that has the third specified translocation,
etc and
D is the length of the diagonal of the box frame of the configuration.
The number of coordinate directions in the above matrix is taken
to be three. However, the concept of novation is applicable to
configurations with any number of coordinate directions.
Novational Transformations
73
When there are two or more specified translocations then the control
parameter C is required to be nonzero. This is due to the fact that when
C = 0 then the interaction matrix A will be singular and the system of
simultaneous equations ATW = TS will not have a unique solution (With
C = 0, all the nonzero off-diagonal elements of A will be equal to 1 and
the matrix will have a number of identical rows and columns and a
vanishing determinant). However, when there is only one specified
translocation then the control parameter C may have a zero value without
any problem. In this case every node of the configuration will undergo a
translocation identical to the specified translocation.
When the system of simultaneous equations ATW = TS has a unique
solution then an efficient way of obtaining this solution will be to solve
the following equivalent system of simultaneous equations:
where
is the kth component of translocation at a typical node j,
is the number of points that have specified translocations,
is the kth component of the ith working translocation,
is the relative distance of node j from the position of the ith specified
translocation and is given by the ratio Dij/D, where,
Dij is the distance of node j from the position of the ith specified
translocation,
D is the length of the diagonal of the box frame of the configuration and
C is the control parameter.
Tk
n
TiWk
dij
74
Novational Transformations
75
76
Novational Transformations
77
where L is the span of the arch (16 in the present example) and h is the
height of the arch at the middle (4 in the present example). The actual
specification of the translocations for this example is given in section 9.
The configuration of Fig. 8(f) is obtained in a similar manner except
for the position of the arch which is along a central line. The same
approach is used for the creation of Fig. 8(g) that involves three arches.
Finally, the configuration of Fig. 8(h) is obtained by creating arches
along all four edges.
8. Indirect ED Novations
In the case of a sharp novation, the specified translocations are required
to be at nodal positions of the configuration. However, as far as an ED
novation is concerned, the specified translocations need not necessarily
be at nodal positions. An ED novation in which all the specified
translocations are at nodal positions is referred to as a nodal ED
novation or a direct ED novation. In contrast, an ED novation that
involves one or more specified translocations at non-nodal positions is
referred to as a non-nodal ED novation or an indirect ED novation.
The process of ED novation as described in the paper will be identical
for both direct and indirect ED novations except for a minor difference
that for a direct ED novation the relative distance dij is always in the
range 0 to 1 but in the case of an indirect ED novation the value of dij
may be greater than 1.
78
Novational Transformations
79
80
Novational Transformations
81
CHAPTER 6
SOME STRUCTURAL-MORPHOLOGICAL ASPECTS OF
DEPLOYABLE STRUCTURES FOR SPACE ENCLOSURES
Ariel Hanaor
Sr. Research Associate, National Building Research Institute, The Technion,
Israel Institute of Technology, Technion City, Haifa 32000, Israel
The paper is a review of deployable structural systems that have
been proposed recently for the purpose of space enclosure. The
structural systems are characterized and classified by their structuralmorphological properties and by the kinematics of deployment. Some
retractable and dismountable configurations are also reviewed. The
systems are evaluated in terms of their structural efficiency, technical
complexity and deployment/stowage efficiencies. The paper includes
an extended list of references.
1. General Principles
1.1. Definitions and Scope
Deployable structures are generally used in two types of applications:
a) as temporary structures; b) in inaccessible or remote places, such
as outer-space. The first application implies a reversible process of
deployment and undeployment, while the second may not. Two extreme
types of deployable structures can be distinguished: Fully deployable
structures are fully assembled in the stowed state, and the deployment
process involves no component assembly. In fully dismountable
structures (assemblable according to Ref. 9), on the other hand, the
structure is stowed as separate components (at member level). It is
assembled on site from these components and can be disassembled back
into the stowed state.
83
84
A. Hanaor
The paper deals primarily with structures that are fully deployable or
composed of large deployable sub-assemblages, and with the specific
application of space enclosure. However, some reference is made to
some types of dismountable structures and also to certain types of
retractable roofs. Although outer-space applications are excluded from
the scope, some technologies that have been proposed for outer-space
applications are adaptable to terrestrial space enclosures and are included
in this review and in the reference list.
1.2. Classification
The purpose of a classification system is to highlight in a hierarchical
fashion the principles governing the set of objects under discussion. The
present paper is concerned with the structural-morphological properties
as the primary interest. The overall classification system is presented in
the chart in Fig. 1, in the form of a two-way table. The columns of the
table represent the morphological aspects and the rows the kinematic
properties, which are of primary significance in the context of deployable
structures.
Two subcategories are considered for each of the main classification
categories. The two major morphological features are lattice or skeletal
structures, and continuous or stressed-skin structures. It should be noted
that in the context of space enclosures, all structures have a functional
covering surface. The difference between the two classes of structures
mentioned above is that in skeletal structures, the primary load-bearing
structure consists of discrete members, whereas in continuous structures
the surface covering itself performs the major load-bearing function. A
third class, namely hybrid structures, combines skeletal and stressed-skin
components with approximately equal roles in the load-bearing
hierarchy, but in the present classification each of the components is
dealt with in its respective class. The two major kinematic subcategories
are systems comprised of rigid links, such as bars or plates, and systems
containing deformable or soft components, which lack flexural stiffness,
such as cables or fabric. In general, the deployment of structures
composed of rigid links can be more accurately controlled than that of
85
101
19
70
16
22
51
71
93
96
78
88
56
80
85
111
115
64
65
83
86
A. Hanaor
87
88
A. Hanaor
Fig. 3. Basic scissors units: a) Peripheral scissors triangular and rectangular prisms22;
b) Radial scissors triangular antiprism, rectangular prism22; c) A clicking unit.43
89
90
A. Hanaor
91
92
A. Hanaor
93
c
Fig. 4. a) Basic minimal SKDOF mechanisms for generating folded plate structures5;
b) Four-fold folded surface94; c) Merging of four-fold to six-fold pattern.
94
A. Hanaor
folded surfaces are usually generated with six folds, rather than four,
meeting at a point (see, for instance illustration of linear deployment in
Fig. 1101). This configuration is, in fact, a merging of two four-fold
vertices Fig. 4c. This merging enables increasing the surface curvature
but results in increasing also the number of KDOFs. This is not a major
drawback, since definition of the boundaries uniquely defines the
geometry and provides control of deployment.
The majority of folded plate structures proposed to date have a linear
(or curvilinear) deployment and they fold into a compact slab of stacked
plates. Some configurations have been proposed in the literature that fold
to a compact cylinder and deploy radially out of the cylinder to form a
folded disc.5,97 The illustration in Fig. 15 shows such a disc with
circumferential folds in the deployed state (domical configurations are
also possible). To fold it, a radial cut is made and the edges overlap
(several turns, theoretically) until a compact spiral cylinder is produced.
To realize such configurations with curved folds in practice, the material
has to be very thin and flexible. A configuration with spiral folds
that folds to a compact cylinder has been proposed97 originally for
deployment of membrane in outer space, but it can also be realised as a
folded plate. This pattern does not involve overlap and is therefore more
suitable for implementation as deployable structure.
Structural efficiency and weight: Folded plate structures inherently
possess high structural efficiency. This inherent efficiency, however,
does not automatically translate into light weight, since the plates
themselves, which are subject to compression and flexure, require
minimum dimensions. The resulting overall weight may be higher than
in structures surfaced with membrane.
5.2. Curved Surfaces
Doubly (and singly) curved surfaces can be discretised into plate
elements. In general, these surfaces cannot be folded and deployed as
one unit. Unfolded planar patterns can be obtained by disconnecting
certain joints between plates and these patterns can, in principle, be
further folded into compact stacks. The concept is more suitable for
95
96
A. Hanaor
97
hence the need for high pressure. Stiffness is generally very low and is
often a limiting factor in their application. The thickness and toughness
of the membrane is higher than for low-pressure structures and, in
addition, the total membrane surface area is large due to the closed cell
requirement. Consequently, stowage efficiency is reduced compared to
low-pressure structures and to fabric structures.
7. References
7.1. General
1. Astudillo, R. and Madrid, A.J., Eds., Shells and Spatial Structures:
From Recent Past to the Next Millenium, Proc. IASS 40th
Anniversary congress, Madrid, 1999, V II, Section D Retractable
and Deployable Structures.
2. Escrig, F. and Brebbia, C.A., eds., Mobile and Rapidly Assembled
Structures II, Proc. 2nd Int. Conf. MARAS 96, Seville, Spain,
June, 1996, Computational Mechanics Publications, Southampton,
UK, 1996.
3. Escrig, F., General Survey of Deployability in Architecture,
Mobile and Rapidly Assembled Structures II, Proc. Intnl
Conf. MARAS 96, Computational Mechanics Publications,
Southampton, 1996, 3-22.
4. Hernandez, C.H., New Ideas on Deployable Structures, Mobile
and Rapidly Assembled Structures II, Proc. Intnl Conf. MARAS
96, Escrig F. and Brebbia, C.A., Eds., Computational Mechanics
Publications, Southampton, 64-72.
5. Kent, E., Periodic Kinematic Structures, Ph.D. Dissertation, The
Technion, Israel Institute of Technology, Haifa, 1983.
6. Kent, E.M., Kinematic Periodic Structures (KPS), Innovative
Large Span Structures, Concept, Design, Construction, Proc.
IASS-CSCE Intnl. Congress, Canadian Society of Civil Engineers,
Montral, 1992, 485-496.
7. Kronenburg, R., Ephemeral/Portable Architecture, John Wiley &
Sons, New York, 1998.
98
A. Hanaor
99
100
A. Hanaor
31. Gantes, C.J., Logcher, R.D., Connor, J.J. and Rosenfeld, Y.,
Geometric Design of Deployable Structures with Discrete Joint
size, Int. J. Space Structures, Special Issue on Deployable Space
Structures, 8, 1-2, 1993, 107-117.
32. Hoberman, C., Art and Science of Folding Structures, Sites,
V24, 1992, 34-53.
33. Kwan, A.S.K., A Parabolic Pantographic Deployable Antenna
(PDA), Int. J. Space Structures, 10, 4, 1995, 195-203.
34. Langbecker, T., Kinematic Analysis of Deployable Scissor
Structures, Int. J. Space Structures, 14, 1, 1999, 1-15.
35. Pellegrino, S. and You, Z., Foldable Ring Structures, Space
Structures 4, Proc. 4th Int. Conf. on Space Structures, University
of Surrey, September, 1993, Thomas Telford, London, V1,
783-792.
36. Piero, E.P., Teatro Ambulante, Arquitectura, N. 30, Junio,
1961, 27-33.
37. Piero, E.P., Three Dimensional Reticular Structure, U.S. Patent
3.185.164, 1965.
38. Piero, E.P., Estructuras Reticulantes, Arquitectura, 112, April
1968, 1-9.
39. Piero, E.P., Estructures Reticule, LArchitecture daujourdhui,
141, Diciembre 1968, 76-81.
40. Piero, E.P., Teatros Desmontables, Informes de la Construccin,
231, 1971, 34-43.
41. Puertas del Rio, L., Space Frames for Deployable Domes, Bull.
IASS, 32, 2, 1991, 107-113.
42. Rosenfeld, Y. and Logcher, R.D., New Concepts for Deployable
Collapsable Structures, Int. J. Space Structures, 3, 1, 1988, 20-32.
43. Rosenfeld, Y., Ben-Ami, Y. and Logcher, R.D., A Prototype
Clicking Scissor-Link Deployable Structure, Int. J. Space
Structures, 8, 1-2, 1993, 85-95.
44. Snchez-Cuenca, L., Geometric Models for Expandable
Structures, Mobile and Rapidly Assembled Structures II, Proc.
Intnl Conf. MARAS 96, Computational Mechanics Publications,
Southampton, 1996, 93-102.
101
102
A. Hanaor
103
63. Pedretti, M., Smart Tensegrity Structures for the Swiss Expo
2001, Lightweight Structures in Architecture, Engineering and
Construction, Proc. LSA98, Sydney, 1998, V2, 684-691.
64. Pedretti, M., Internet site http://www.ppeng.ch/base_expo.htm,
July, 1998.
Others
65. Kwan, A.S.K. and Pellegrino, S., A New Concept for
Deployable Space Frames, Int. J. Space Structures, 9, 3,
153-162.
66. Kwan, A.S.K. and Pellegrino, S., Design and Performance
Octahedral Deployable Space Antena (ODESA), Int. J.
Structures, 9, 3, 1994, 163-173.
Large
1994,
of the
Space
104
A. Hanaor
105
106
A. Hanaor
104.
105.
106.
107.
108.
107
108
A. Hanaor
CHAPTER 7
PHANTASY IN SPACE: ON HUMAN FEELING BETWEEN THE
SHAPES OF THE WORLD AND HOW TO LOOK ON
NATURAL STRUCTURES
Michael Balz
Dipl.-Ing. (SBS), Stuttgart, Germany
109
110
M. Balz
Phantasy in Space
111
When we open our eyes it is nice to have a reason not close them
immediately again.
Therefore, what do we like to see doubtlessly?
There are the difficult questions on beauty, harmony and safety by
which our decisions are influenced, but they are certainly made in
agreement with a situation or a certain picture.
Human eyes stimulate phantasies in our mind. Therefore bare rooms
and bleak landscapes can hardly evoke enjoyable atmospheres.
Originally the human brain is used to enable surviving. For this aim it
is very important to define the spaces that surrounds us
To develop a sense of direction we need to locate
locate our individual
position.
To understand space it must be measured and must be brought in
relation with the individuum. For this measuring we need rather solid
fixpoints or lines as well as edges and plains and other surfaces.
Between sharp edges or corners orientation becomes possible.
112
M. Balz
Human mind always tries to define a sense behind the pure optical
impression. Therefore it is logical that chaotic looking picture rarely give
a satisfaction in terms of harmony.
Our mind together with the eye always seeks for understandable
structures which show or seem to show sense in themselves.
This sense does not always have to be completely understood by the
observer often the pure rhythm of the object is enough to stimulate us
Especially the shapes of natural
natural developed creatures, plants or
landscapes stimulate human feeling in a positive way.
All shapes in nature are the result of physical and chemical processes
which go on since the beginning of life and even before that date.
Human being is a result of this endless process as well: the evolution.
Consequently it seems obvious that man prefers to have a sight on
natural looking objects and structures when looking around.
Phantasy in Space
113
114
M. Balz
Tubularid Hydroids
Polyp-like creatures between plant and animal. They can only exist in the sea on rocks
where the sun provides energy and where the water supplied nourishing value.
Between submarine gardens with phantastic colours.
Phantasy in Space
115
All life comes out of the water. Here is demonstrated how an idea
becomes a very tender veil and starts to move and to be.
116
M. Balz
Reference
1. Haeckel, Ernst: Kunstformen der Natutur, Neudruck der Erstausgabe
von 1904, Prestel Verlag, Mnchen, 1998.
CHAPTER 8
AN EXPANDABLE DODECAHEDRON
1. Introduction
Certain viruses having the shape of a truncated icosahedron [Ref. 6]
expand under the effect of pH change. The pentamers and hexamers
depart from each other while rotating but remaining in contact by
protein links. This discovery called our attention to expandable
polyhedra.
The characteristic of the motion of these viruses is different from that
of Hobermans popular toy, the expanding globe. Hobermans globe is a
polyhedron, represented by its edges, in which the faces preserve their
orientation while expanding. The basic transformation is a uniform
enlarging where all the edges are increased in the same proportion. In
the viruses, however, the faces preserve their size in principle but
are subjected to a translation-rotation along their symmetry axes; and
simultaneously, interstices appear between faces. So, the motions of
these viruses are similar to a special type of deformation of honeycombs
[Ref. 3] and to the motion of Fullers jitterbug [Ref. 2], but the actual
117
118
An Expandable Dodecahedron
119
2. Physical Model
120
An Expandable Dodecahedron
121
prisms are taken into account (Fig. 4). Let the edge length be denoted by
a. Since the rotation of two adjacent pentagons about their own axis
requires the rotation of the linking element (digon) as well and the digons
also rotate about their radial symmetry axis, we can define for the sake
of simplicity this angle of rotation () as a function of the radius (R) of
the circumscribed sphere.
Fig. 4. Simplified network of two pentagonal members (inner edges and vertices only).
122
Now the arc drawn from the vertex V to the mid-point of arc XY (M)
is perpendicular to arc XY. The cosine theorem for the spherical triangle
VMX shows that:
cos XVM = cos VMX cos MXV + sin VMX sin MXV cos XM ,
where XM = i/2, XVM = /3, VMX = /2, MXV = /5.
From this equation
cos
i
3 = 1
cos =
2 sin 2 sin
5
5
is obtained.
Looking at Fig. 4 again, another formula of the same theorem for
triangle PSV says:
cos VP = cos PS cos SV + sin PS sin SV cos PSV ,
__
where sin VP =
PV
__
VO
sin SV =
a
2R
r
a
=
, PS = i/2,
R 2 sin R
5
and PSV = .
cos =
substituting
5 5
for sin we obtain that
5
8
2
cos =
5 5 R
R
2
1 4 1
2 a
a
3 5
2
(1)
An Expandable Dodecahedron
123
R
=
a
5 + 9+3 5 2
Rmin
1.401,
a
Rmax
2.478.
a
The quotient of these values shows that the circumradius of our
model can increase by about 77% during the expansion.
As to the volumes of the polyhedra, another calculation can be carried
235 + 105 5
7.6631a 3 ,
8
truncated icosahedron is
3. Numerical Models
In the previous chapter we analysed a special motion of the expandable
dodecahedron. It is still not known, however, whether the structure is
able to move only in that way or it has other independent motions. In
order to answer this question, it is necessary to execute a detailed
analysis of the compatibility matrix of the structure.
124
An Expandable Dodecahedron
125
ten nodes are excluded from the free ones, consequently, the total
number of free nodes will be equal to:
n = (12-1) 10 + 30 (5-2) = 200
and
c = 3n = 600.
Under the same conditions, r can also be determined as follows:
r = (12-1) 24 + 30 9 + (2 30) 1 = 594.
Since c represents the number of unknowns, r that of the conditions,
we have got an important information about the kinematical behaviour,
namely, c - r = 6 means six independent infinitesimal motions for the
structure.
This calculation, however, still does not give the final answer for the
question that how many independent motions the structure has it can
only be decided by investigating the rank deficiency of J. It implies
another problem, because the computational process (especially if we
want to get information about the characteristics of the detected motions)
is very sensitive to the size of the matrix. It was this recognition that led
us to a new approach of numerical modelling.
(x
xi
) + (y
yi
) + (z
zi
lij = 0
(3)
126
F b ij xi x j
.
=
xi
lij
Now let us consider again the simplified network of Fig. 4, that is, all
nodes and bars are ignored except those on the lower shell of the
structure. Our task is now to determine additional types of constraints
(and unknown pointers if necessary) that are sufficient to make the
elements of the structure rigid.
A pentagon can be made rigid in its plane by two additional
bars but it is not enough for complete rigidity. If we look at four points
(Pi, Pj, Pk, Pl) in the space, the following expression can be formulated:
F v ijkl =
xi
yi
xj
yj
xk
yk
xl
6Vijkl = 0
yl
zi
zj
zk
zl
(4)
where the determinant equals six times the actual volume of the
tetrahedron spanned by the four points, while Vijkl is its original volume
[Ref. 4]. The non-zero derivatives can be written as a sub-determinant
like this:
1
1
yj
1
yk .
zi
zj
zk
F v ijkl
= yi
xl
An Expandable Dodecahedron
127
(5)
(ii) v must keep its inclination angle i/2 to the normal vectors of
the adjacent pentagons. In Fig. 7, points B, B1, B2 and B3 span a
tetrahedron. Since each coordinate of B3 is the sum of that of B and
vBE, the volume can only change if the i/2 inclination angle changes
as well. This fact implies a constraint function Faijkl similar to Fvijkl
(here the index l belongs to the vector vl that points from the third
node Pk):
F a ijkl =
xi
yi
xj
yj
xk
yk
xk + xl
6Vijkl = 0
y k + yl
zi
zj
zk
z k + zl
(6)
128
(iii) The previous statement is true only if the length of vl does not vary.
If the original length was vl then:
2
F l l = xl + yl + zl vl = 0 .
(7)
In this new model c will involve not only nodes but vector
coordinates as well. Keeping a supposition that one of the pentagons is
fixed and calculating with 7 bars per pentagon and one per digon the
following values are obtained for c and r (the terms are in the order of
constraint functions):
c = 3 ((12-1) 5 + 30 1) = 255,
r = ((12-1) 7 + 30 1) + (12-1) 2 + 30 1 + 2 30 1 + 30 1 = 249.
It is seen that c - r = 6 is unchanged, so this model gives the same
result with a less than half-size matrix.
(8)
An Expandable Dodecahedron
129
(9)
(10)
(11)
130
5. Conclusions
Verheyens dipolygonid can be physically realized as an expandable
dodecahedron, but the model shows some extra degrees of freedom. It
remains to be seen whether they are finite or only infinitesimal. The
actual physical meaning of the additional degrees of freedom should be
identified, that is, it should be shown how the free motions, in addition to
the expansion, look like.
The difference between the number of columns and the number of
rows of the compatibility matrix of the model is 6, but the number of
degrees of kinematical indeterminacy (the infinitesimal degrees of
freedom) is 7. Therefore, due to the generalized Maxwell rule, the model
has a one-parameter state of self-stress. The properties of this state of
self-stress should be determined in the future.
In the expandable viruses, between two adjacent morphological units
(pentamers and hexamers) there is a double-link connection, contrary to
that of our model, where there is a single-link connection. According to
some pilot studies it seems to be possible to construct an expandable
dodecahedron model with double links between the pentagonal units. In
this case, however, the rectangular units in the links should have a free
rotation about an axis parallel with their longer sides.
Acknowledgements
The research reported here was done within the framework of
the Hungarian-British Intergovernmental Science and Technology
Cooperation Programme with the partial support of OMFB and the
British Council. Partial support by OTKA Grant No. T031931, and
FKFP Grant No. 0391/1997 is also gratefully acknowledged.
An Expandable Dodecahedron
131
References
1. Fowler, P. W. and Manolopoulos, D. E. 1995. An atlas of fullerenes.
Oxford University Press.
2. Fuller, R. B. 1975. Synergetics: exploration in the geometry of
thinking. Macmillan, New York.
3. Kollr, L. and Hegeds, I. 1985. Analysis and design of space frames
by the continuum method. Akadmiai Kiad, Budapest/Elsevier Sci.
Pub., Amsterdam, p. 146.
4. Kovcs, F., Hegeds, I. and Tarnai, T. 1997. Movable pairs of regular
polyhedra. Proceedings of International Colloquium on Structural
Morphology, Nottingham, pp. 123-129.
5. Kovcs, F. 1998. Foldable bar structures on a sphere. Proceedings of
2nd International PhD Symposium, Budapest, pp. 305-311.
6. Speir, J. A., Munshi, S., Wang, G., Baker, T. S. and Johnson, J. E.
1995. Structures of the native and swollen forms of cowpea chlorotic
mottle virus determined by X-ray crystallography and cryo-electron
microscopy. Structure, Vol. 3, pp. 63-78.
7. Verheyen, H. F. 1989. The complete set of jitterbug transformers
and the analysis of their motion. Computers Math. Applic., Vol. 17,
pp. 203-250.
CHAPTER 9
EXAMPLES OF GEOMETRICAL REVERSE ENGINEERING:
DESIGNING FROM MODELS AND/OR UNDER
GEOMETRICAL CONSTRAINTS
Klaus Linkwitz
Geometrical reverse engineering comprises a number of techniques
applied in cases, when the (geometrical) design of a structure is so
complicated that neither conventional CAD nor numerical formfinding
by figures of equilibrium are adequate means to transform the ideas and
visions of a designer into reality.
133
134
K. Linkwitz
135
136
K. Linkwitz
Stuttgart in the sixties, it was applied for the first time for the
construction of the Caroni-River-Bridge in Venezuela. Stimulating the
introduction of the new method was the fact, that the Caroni-River varies
about 10m in height between low and high waters, has a swift currency
and is notoriously susceptible to quick changes in water level caused by
unstable weather condition in its region of inflowing rivers. Thus the
erection of a strong and reliable scaffolding for the construction of the
bridge body would have been complicated and expensive.
The - at that time revolutionary - -idea of Bauer was, to have the
whole bridge body of about 200 m length constructed off the river on one
of the banks; then lying there on rails after termination of these partial
works. Simultaneously, in the river, the pillars of the bridge were to be
constructed. Then, in the final step of construction, the bridge was to be
pushed from the bank over the pillars with the aid of hydraulic pressure
gadgets, gliding on its path from the original construction site to its final
position, first on the rails and then on the pillars. To lower friction during
this process the pillar bearings would be coated by Teflon, also newly
invented at that time.
The CEO Lenz of the executing Company Zblin, also from Stuttgart,
highly attached to new progressive if economic methods of execution
and not afraid of calculated risks, and after promising to nail and hang
Fritz Leonhardt, Senior CEO of Leonhardt & Andr to the first pillar
if the construction should fail, got the contract and succeeded with
execution.
Ever since then bridges have been constructed using the B&PMethod. First only bridge designs of simple geometries were considered
for pushing: Elevation and plan of the bridge are straight lines. Then
combinations of straight lines and circles in elevation and plan were tried
and successfully executed. An attempt in Switzerland of building and
pushing a bridge the geometry of which consisted of both circles in
elevation and plan failed. During midway construction, the bridge being
pushed progressively encountered ever increasing friction on the pillars
and finally seized to a standstill like the piston in a cylinder when oil is
drained out.
137
138
K. Linkwitz
reach from the abutment on the one side of the river/valley to the
abutment on the other side of the river/valley if the bridge were
constructed in one piece and then pushed from abutment to abutment.
During pushing each bridge segment has to glide from its original
construction site to its final position of the bridge in situ. (Fig. 7)
Imagining the most favorable case of a bridge being prefabricated in two
pieces and then pushed from the abutments on the banks on either side,
the front segment of the bridge has to glide from the abutment unto the
middle of the bridge, the second segment has to glide until it reaches
its location behind the middle segment, and so on. As each segment
has to glide smoothly it has to fit perfectly into the form of the gliding
track - i.e. the bearings on the pillars - in its transition from site of
premanufacturing to its final location in the bridge.
(8)
(9)
139
140
K. Linkwitz
141
142
K. Linkwitz
143
144
K. Linkwitz
CHAPTER 10
CRYSTALLINE ARCHITECTURE
Arthur L. Loeb
Senior Lecturer and Honorary Associate on Visual and Environmental Studies
and Member of the Faculty, Graduate School of Education, Harvard University,
Cambridge MA 02138, USA
Architects such as Ton Alberts and R. Buckminster Fuller have used
the adjective organic to indicate that their designs were based on forms
found in nature. Specifically, whereas traditional twentieth-century
architecture tended to make use of stacked two-dimensional designs,
the new millennium is beginning to derive structures from stacked
three-dimensional polyhedra. The right angle, which is rare in natural
forms, and unstable, is eschewed by these architects. Crystallographers
and solid-state scientists, attempting to understand the reasons why
crystalline forms are the way they are, and working in the same
Euclidean space on a much smaller scale than architects, tend to have
their own perspective on that space. We present here a number of
concepts from crystallography and molecular spectroscopy relevant to
architecture and vice versa in the hope that the spatial repertoire in both
disciplines will be enriched.
1. Introduction
This paper is dedicated to the memory of the late Dutch architect Ton
Alberts, who gained special celebrity through his ING Bank headquarters
in southeastern Amsterdam, and the Gasunie building in Groningen.
In the impersonal wasteland of the Bijlmermeer, Albertss ING
headquarters and the adjacent town square De Amsterdamse Poort
constitute an oasis of humanity. Characteristic of Albertss architecture
are the undulating and inclined surfaces of his structures, and the use of
145
146
A. L. Loeb
Crystalline Architecture
147
(1)
148
A. L. Loeb
Roger Penrose had already demonstrated8 that the Euclidean plane can be
tiled by two rhombuses, one having angles 36 and 144, the other 72
and 108, such that the symmetry appears to be five-fold locally, but
does not extend throughout the pattern. Analogously, three-dimensional
space may be filled by two kinds of parallelipipedeons, yielding models
for alloys whose symmetry has no five-fold rotational symmetry, but
whose X-ray diffraction pattern does. Such crystals are called
quasicrystals. Linear quasi-symmetrical strings are of interest to the
designer.
Crystalline Architecture
149
150
A. L. Loeb
C
R
Q
O
A
S
Crystalline Architecture
151
152
A. L. Loeb
6. Buckminsterfullerene
In 1985, Robert F. Curl, Harold W. Kroto, and Richard E. Smalley5
discovered a new form of elemental carbon, having the chemical formula
C60. Kroto was working in microwave spectroscopy when he became
interested in long chains consisting solely of carbon and nitrogen. At this
time, Smalley was studying cluster chemistry. Smalley had designed a
cluster beam apparatus which was able to vaporize almost any known
material. On September 1st, 1985 the three got together in Smalleys
laboratory; in trying to devise the structure of the new carbon molecule,
Smalley recalled having bought a Fuller dome kit for his young son, and
was able to use this kit to construct a model for the new carbon molecule,
which was accordingly named Buckminsterfullerene.
It is possible to tile the plane with regular hexagons, but a sphere
needs some polygons having fewer sides. When a sphere is tiled by a
combination of hexagons and pentagons, regardless of the number of
hexagons the number of pentagons must be exactly twelve. A soccer ball
has twelve pentagons and twenty hexagons; many domes have more
hexagons, as do carbon molecules found subsequently, for example C70.
Crystalline Architecture
153
7. Tensegrity
Buckminster Fuller and Kenneth Snelson invented a class of structures
comprising compression members which do not touch, but are held in
equilibrium by tension members; each compression member is
surrounded by a loop of tension members into which other compression
members hook. Particularly stable is the six-strut tensegrity, of which an
octant is shown in Figure 6. The other octants are reflections of adjacent
octants across the cartesian planes. Three pairs of mutually parallel
compression members lie parallel to the cartesian axes; the tension
members lie along the edges of the shaded triangle. The distance between
parallel compression members equals exactly half of their lengths at
equilibrium, and any attempt to alter that distance, either by pushing
compression members together or by pulling them apart, will increase
the length of the tension members, which will resist these attempts,
holding the structure in equilibrium. When one pair of compression
members is made to approach each other, the other ones will also
approach each other; when they are pulled apart, the others will again
follow suit.
Once more, these architectural inventions have influenced molecular
science. Donald Ingber and his associates4,13-17,57,58 have applied the
tensegrity concept to molecular biology with much success.
154
A. L. Loeb
8. Conclusions
We note that the fundamental equations governing these examples from
the grammar of Design Science are quite simple and elegant. Although
upon initial analysis natural structure might appear to be complex, it
will be understood if perceived as generated from simple modules
by a simple generating rules or algorisms.36 Alberts, for instance,
generated his building designs by a dynamic process, working with the
client on a three-dimensional model while the design work was in
progress. Our three-dimensional space is not a passive vacuum, but poses
constraints. Only when these constraints are well understood, can one
build the extensive repertoire permitted by these constraints.
References
1. Adams, M. & Loeb, A.L.: Thermal Conductivity: Expression for
the Special Case of Prolate Spheroids. J. Amer. Ceramics Soc. 37,
73-73, 1954.
2. Baglivo, Jenny & Grtaver, Jack A.: Incidence and Symmetry in
Design and Architecture, Cambridge University Press, Cambridge,
U.K., 1983.
3. Cahn, J. & Taylor, J.: An Introduction to Quasicrystals, Contemp.
Math. 64, 265-286, 1987.
4. Chen, C.S. & Ingber, D.E.: Tensegrity and mechanoregulation: from
skeleton to cytoskeleton. Osteoarthritis and Articular Cartilage,
1999; 7/1: 81-94.
5. Curl, Robert F., Kroto, Harold W. & Smalley., Richard E. cf the
Intrnet under Kroto.
6. Edmondson, Amy: A Fuller Explanation, the Synergetic Geometry
of R. Buckminster Fuller, 1986.
7. Fuller, R. Buckminster: Synergetics Macmillan, New York, 1975.
8. Gardner, Martin: Penrose Tiles to Trapdoor Codes, W.H. Freeman,
New York, 1988.
9. Goodenough, J.B. & Loeb, A.L.: A Theory of Ionic Ordering,
Tetragonal Phase Formation, Magnetic Exchange and Lamellar
Crystalline Architecture
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
155
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A. L. Loeb
23. Loeb, A.L.: The Subdivision of the Hexagonal Net and the
Systematic Generation of Crystal Sructures Acta Crystallographica,
17, 179-182, 1964.
24. Loeb, A.L.: The architecture of Crystals, in Gyorgy Kepes, ed.:
Module, Proportion, Symmetry, Rhythm, Vision and Value Series,
Braziller, New York, 1966.
25. Loeb, A.L.: A Systematic Survey of Cubic Crystal Structures J.
Solid State Chemistry, 1, 237-267, 1970.
26. Loeb, A.L.: Color and Symmetry John Wiley & Sons, New York,
1971.
27. Loeb, A.L.: Preface and Concluding Section to R. Buckminster
Fullers Synergetics Macmillan, New York, 1975.
28. Loeb, Arthur L.: Space structures, their Harmony and Counterpoint,
1991 (Originally published Addison Wesley, Reading MA 1976).
29. Loeb, A.L.: Color Symmetry and its Significance for Science, and
other contributionsin Patterns of Symmetry, ed. G. Fleck &
M. Senechal, U. of Massachusetts Press, 1977.
30. Loeb, A.L.: Algorithms, Structure and Models in Hypergraphics,
David Brisson, ed., AAAS Selected Symposium Series, 49-68,
1978.
31. Loeb, A.L.: Sculptural Models, Modular Sculptures in Structures of
Matter and Patterns in Science, ed. Marjorie Senechal & George
Fleck, Schenkman, Cambridge Mass., 1979.
32. Loeb, A.L.: A Studio for Spatial Order, Proc. Internatl Conference
on Descriptive Geometry and Engineering Graphics, Amer. Soc.
Engineering Education, 13-20, 1979.
33. Loeb, A.L.: Structure and Patterns in Science and Art, Leonardo, 4,
339-345, 1971 Reprinted in VisualArt, Mathematics and Computers,
ed. Frank Malino, Pergamon, 1979.
34. Loeb, A.L.: Natural structure in the Man-made Environment. The
Environmentalist, 2, 43-39, 1982.
35. Loeb, A.L.: Vector Equilibrium Synergy. Space Structures, 1,
99-103, 1985.
36. Loeb, A.L.: Symmetry and Modularity J. Computers and
Mathematics with Applications 128, 63-75, 1986. Reprinted
in Symmetry, Unifying Human Understanding, ed. I. Hargittai,
Crystalline Architecture
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
157
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A. L. Loeb
CHAPTER 11
FLAT GRIDS DESIGNS EMPLOYING THE SWIVEL
DIAPHRAGM
Introduction
Scissor structure is a generic name given to certain types of kinetic
systems that make use of a particular mechanism where two rigid
components are connected by a rotational hinge or pivot and move freely
159
160
161
Angulated elements
Straight elements
External fixed pivot/support
162
(1)
Hexagon
Heptagon
Decagon
Octagon
Hendecagon
Nonagon
Dodecagon
163
Fixed pivot
2. Flat Grids
2.1. Diaphragm Morphology
For physical and practical applications the morphology of the swivel
diaphragm may vary depending on the use, the scale, and/or the loading
of the structure. The members that constitute the diaphragm can adopt
nearly any shape as long as the relationship between the pivots is
preserved and overlapping of the elements is avoided. As may be seen in
Figure 1, each angulated element compromises three pivots that form a
triangle; the fixed pivots transport the loads to the external support, and
the other two help to hold up the straight elements that link each module
to the proceeding. The load conditions differ when the diaphragm is
placed vertically, horizontally or tilted. The diagrams in Figure 4 show
the loading distribution in the horizontal and vertical positions. Precise
loading and actions in Figure 4 will depend on the detailed structural
design of the diaphragms and link bars. Using the appropriate bending
moment diagrams as guidelines the members of the SD can be sculpted
into more efficient shapes eliminating any unnecessary excess of
material. This is of central importance for this type of structure, since
164
165
166
167
168
the grid will always be the same as the polygonal figure used for the first
central ring. The hexagonal grid shown in Figures 9 and 10 is of third
frequency; the angulated elements spin around fixed pivots linked by
straight bars. Groups of straight bars operating over the same axis move
in unison, thus the bars can be fused into one longer bar connected to
various angulated elements. The following equations serve to calculate
the number of components needed for the grid according to its polygonal
shape and frequency:
Number of angulated elements = p* 0.5f * ( f+1)
(2)
Number of straight bars = p * f
(3)
Number of joints in the straight bar = f+1
(4)
f = order of the grid, p = number of sides of the desired polygonal shape
Fig. 9. Hexagonal SD grid generated to third order using the fractal method.
Fig. 10. Model of a hexagonal SD grid generated to third order using the fractal method.
169
170
Pivot 2
Pivot 1
Pivot 2
Pivot 1
Conclusion
The Swivel Diaphragm system shows potential to be used with in flat
grid configurations. It offers important structural advantages due to
the use of fixed supports. Additionally, it allows very different
configurations of grids, providing the designer with a very wide range of
options to choose from. Further research needs to be carried out
regarding the capabilities of these types of grids with in practical
171
CHAPTER 12
FORM-OPTIMIZING IN BIOLOGICAL STRUCTURES THE
MORPHOLOGY OF SEASHELLS
Edgar Stach
University of Tennessee, College of Architecture and Design,
1715 Volunteer Boulevard, Knoxville, TN 37996, USA
Fax 001- 865 9740656
stach@utk.edu
The purpose of this case study is to analyze, the structural properties of
natural forms in particular seashells based on digital methods. This is
part of a larger architectural study of lightweight structures and formoptimizing processes in nature. The resulting model was used to
visually display the internal structure of a seashell and also as input to
structural analysis software. The generated renderings show the
algorithmic beauty of sea shells.
1. Introduction
Henry Moseley1 started in 1838 the study of seashells and in particular
the mathematical relationship that controls the overall geometry of shells.
He was followed by many researchers such as Thompson,2 Raup,3,4
Cortie,5 and Dawkins6 and others.
The shape of seashells is caused by a logarithmic natural growth
(Fig. 4-5). There are three basic shapes: the Planispirally coiled shell
(Fig. 1), the Helically coiled shell (Fig. 2), and the Bi-valve shell
(Fig. 3). Environmental factors such as the availability of raw materials,
the type of substrate, the amount of calcium present, as well as many
other factors contribute to deviations from the three basic forms.
173
174
E. Stach
2. Growth
Fig. 4. Computer simulation of a planispirally coiled shell and of three helically coiled
gastropod shells (Harasewych p. 17).
175
176
E. Stach
Fig. 6. Seashell geometry (Kamon Jirapong and Robert J. Krawczyk, Illinois Institute of
Technology).
Fig. 7. An electron microscope photograph of the shell of Busycon carica broken parallel
to the edge of the shell. Three different crystal layers can be seen (Harasewych p. 16).
177
Fig. 8. A cross-sectional view of the mantle and shell at the growing edge of a bi-valve
(Harasewych p. 16).
178
E. Stach
Fig. 12. Computer wire frame model of nautilus planispirally coiled shell.
179
Fig. 13. Elevation and X-ray view of helically coiled gastropod shell (Conklin p. 45).
180
E. Stach
Fig. 17. Elevation and X-ray views of helically coiled gastropod shell (Conklin p. 153).
Fig. 19. Computer wire frame model of helically coiled gastropod shells.
181
Bi-valve shell
Fig. 22. Elevation of computer wire frame model showing exponential curve of shell
growth.
182
E. Stach
183
Fig. 27. Finite Element analysis of sea urchin shell, color coded stress analysis [Process
und Form, K. Teichmann].
184
E. Stach
Computer-compressed evolution
Design space and finite elements
Computer-compressed evolution like the SKO method (Soft Kill Option)
(Fig. 28) follows the same construction principle that nature employs to
promote for example the shell growth of a sea urchin (Fig. 26/27) or the
silica structure of sea shell (Fig. 1/3). Building material can be removed
wherever there are no stresses, but additional material must be
used where the stresses are greater. This is the simple principle that
evolution has used for millions of years to produces weight optimized
components. Using computer programs based on computer-generated
genetic algorithms like the SKO method, scientists are now able to
simulate this evolution and compress it into a short time span.9
Fig. 28. SKO method (Soft Kill Option). [Hightech Report 1/2003, pp. 6063].
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resulting force exerted on every one of the finite elements. The FE model
shows exactly where there is no load stress on a component and in turn
shows where it is possible to make savings with regard to the materials
used. On the other hand, for areas that bear heavy stress the simulation
program indicates the need to reinforce the construction material. Like
nature the computer repeats this finite element cycle several times. As a
result, they can refine a component repeatedly until the optimal form
one that evenly distributes the stresses within a component is found.
Conclusion
The abstract geometrical properties of seashells can be described
by there mathematical relationship. The translation of abstracted
nature in mathematical terms and by applying prerequisite architectural
considerations is the fundamental concept of form and structure analyses.
The value of this research is to develop mathematically definable models
of structure systems in nature. The goal is to define a set of structural
principles, and to make those principles applicable for architects and
engineers.
Seashell structures are perfect study models for self- organization
structures in nature because of there relatively simple physical and
morphological principle and geometry based on four basic mathematical
parameters. Self-organization it the defining principle of nature. It
defines things as simple as a raindrop or as complex as living cell simply a result of physical laws or directives that are implicit in the
material itself. It is a process by which atoms, molecules, molecular
structures and constructive elements create ordered and functional
entities.
Engineers are using this concept already successful for optimization
processes in a white range of applications starting in mechanical-,
medicine-, air and space engineering. Architects are only one step away
adopting the same technique for designing in a macro scale buildings and
structures. Material scientists are already designing and producing
new materials or smart materials in a Micro scale using the self
organizing principles. In the future, the material engineers will develop
constructions out of self-structuring materials that consciously use the
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E. Stach
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CHAPTER 13
EXPANDABLE BLOB STRUCTURES
1. Introduction
This paper is concerned with the geometric design of expandable
structures consisting of rigid elements connected through cylindrical
hinges or scissor joints that only allow rotation about a single axis. The
authors have been interested in developing stacked assemblies formed by
rigidly interconnecting the expandable plate structures that they had
previously developed.4 The connections between individual plates can
themselves be volume filling, and so the stacked structure can also
become an expandable three-dimensional object. As the plate structures
from which one starts can have any plan shape, and only simple
kinematic constraints have to be satisfied in order for them to maintain
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their internal mobility in the stack configuration, nearly any shape can be
generated, including so-called free-forms or blobs.
The approach presented in this paper starts by developing a method
for rigidly connecting two identical and individually expandable plate
structures, such that the assembled structure only possesses a singledegree-of-freedom. The kinematic constraints that must be satisfied by
the connections and the connected plate structures are derived, and are
shown to allow the stacking of non-identical plate structures as well,
thereby allowing free-form profiles to be obtained for the assembled
structure.
This paper is presented as follows. The following section briefly
reviews the kinematic properties of the expandable plate structures that
are to be connected, and their underlying bar structures. Then, the section
An Expandable Sphere describes a method for connecting the plate
structures such that they can form an expandable sphere. The next
section is then concerned with the kinematic constraints that have been
satisfied by these connections and the formulation of a general set of
rules for such connections. These are then used in the following section
to design an expandable free-form or blob structure demonstrating the
possibility of designing vivid and exciting expandable structures using
this method. A brief discussion concludes the paper.
2. Background
Simple expandable structures based on the concept of pantographic
elements, i.e. straight bars connected through scissor hinges have been
known for a long time. One of the simplest forms of such pantographic
structures is the well-known lazy-tong in which a series of pantographic
elements are connected through scissor hinges at their ends to form twodimensional linearly extendible structures.
More sophisticated expandable or deployable structures have been
developed over the last half century.1,2,12 Many of these solutions are
based on the so-called angulated pantographic element.3 In its simplest
form, shown in Figure 1, it consists of two identical angulated elements
each composed of two bars rigidly connected with a kink angle . Unlike
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CE AE
AC
tan( / 2) + 2
EF
AC
(1)
Fig. 1. Pantographic element consisting of two angulated elements, each formed by two
bars.
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clockwise as can be seen in the figure. The dashed-line layer, on the other
hand rotates counter-clockwise. Note how the multi-angulated elements
form three concentric rings of rhombus-shaped four-bar linkages all of
which are sheared as the structure expands.
closed open
(2)
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Fig. 4. Model of non-circular structure where all plate boundaries are different; both
layers are shown.
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3. An Expandable Sphere
To investigate the possibility of creating three-dimensional expandable
structures by stacking identical plate structures, a simple design for the
plate structures was chosen. The design, shown in Figure 5, consists of
16 identical plate elements of which 8 form the bottom layer and 8 the
top layer. As for the bar structure shown in Figure 2, the 8 plates forming
the top layer move radially outwards while rotating clockwise, while the
plates forming the bottom layer (of which only small parts can be seen in
the figure) rotate counter-clockwise. From symmetry it can be concluded
that all plates in the same layer rotate by identical amounts; the rotations
of plates in different layers are equal and opposite.
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made perpendicular to the original plate, and follow the periodic edge
shape of the plates.
Figure 7 shows such a spherical model. It was constructed using
identical plastic plate structures of which four were trimmed so that their
outer boundaries form circles of different radii. The connections were
made from identical blocks of light foam board, cut using abrasive waterjet cutting. The blocks were then glued to the back of the individual
plastic plates that form the plate structures and the spherical profile
obtained by removing the excess foam board material.
4. Stack Structures
To rigidly connect two expandable plate structures the motion of the
individual plates being connected must be identical. Earlier, the motion
of each plate was described as the combination of a radial motion, i.e. a
translation and a rotation. Kassabian et al.5 found that, as the rotation in
the two layers is equal and opposite, imposing an additional rigid body
rotation to the whole structure, equal to the rotation undergone by one of
the layers, the motions of the two layers become a pure rotation and a
pure translation, respectively. If, for example, the imposed rotation is
such that the plates in the top layer of the bottom structure undergo a
pure rotation, then each plate in this layer rotates about its own fixed
centre.5
Hence, consider two plate structures that are to be connected.
Following the above approach impose the same rigid body rotations on
both structures; clearly the plates in the two connected layers must rotate
about the same axes of rotation and by the same amounts. Since the
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shown. It can also be seen in the figure that the allowable rotation of the
stacked structure is limited by contact at DII in the extreme closed
position.
For an assembly where all the plates in each layer are to be
connected, the axes of rotation for all elements of the two layers must
coincide. Hence, the polygons defining the axes must also be identical.
As these polygons also define the open plan shape of the structures,
when scaled to double size, the two layers must therefore be formed from
identical polygons. However, the location of the origin for the two
structures need not coincide, as was the case for the spherical assembly
where they were both located on the central axis of the expandable
sphere. In Figure 8 note the different origins OI and OII.
Note in Figure 8(a) that the axes of rotation could be chosen to be the
same also for the elements in the dashed-line layers but this is only the
case when the structures can be expanded fully, i.e. the dashed- and
solid-line layers coincide when fully expanded. This is not normally the
case as the physical size of hinges and plates will prevent this and thus
different axes have to be identified if another structure has to be added to
the stack, by connecting two facing dashed-line layers.
Fig. 8. Stacked bar assembly in three different configurations; (a) fully open;
(b) intermediate; (c) closed.
5. An Expandable Blob
Having determined the overall kinematic constraints that must be
satisfied for two plate structures to be connected together, a more
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6. Conclusion
An investigation into three-dimensional expandable shapes made by
rigidly interconnecting individually expandable plate structures has been
presented. It has been shown that it is possible to create structures with
highly irregular shapes; the internal mobility of the plate structures is
preserved if simple kinematic constraints are satisfied. Several models
have been designed and constructed to verify and demonstrate this
finding, of which two have been shown in this paper. The two models
presented show that it is possible to create such expandable assemblies
with almost any plan and profile shape. They can hence be visually
pleasing and attractive for applications in architecture and design.
References
1. F. Escrig (1993), Las estructuras de Emilio Prez Piero, In:
Arquitectura Transformable, Publication de la Escuela Tecnica
Superior de Architectura de Sevilla, pp 1132.
2. F. Escrig (1993) and J.P. Valcarcel, Geometry of Expandable
Space Structures, International Journal of Space Structures, Vol. 8,
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3. C. Hoberman (1990), Reversibly expandable doubly-curved truss
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4. F. Jensen and S. Pellegrino (2002), expandable structures formed
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