Thesis Report

FRACTALS

An exploration and abstraction.

Chan Min Yun U086646R

Thesis Submission for Bachelor of Arts in Industrial Design National University of Singapore 2011/2012

"Clouds are not spheres,

mountains are not cones, coastlines are not circles, and bark is not smooth,

nor does lightning travel in a straight line."

Benoit Mandelbrot, who coined the term fractal.

ABSTRACT

We live in a world of fractals. From coastlines structures to the graphs of stock exchange, our lives are governed by fractal geometry in both the natural world and the man-made world.

While the geometry of fractals presents fertile opportunities to

industrial design, it has been under-utilised. This is despite the recent proliferation of additive manufacturing *.

My thesis sets out to change this, to introduce fractal geometry to the field of industrial design currently dominated by Euclidean geometry.

By employing fractal geometry in the design of everyday objects across different scales, this thesis seeks to ignite our imagination beyond our perceptions and to create an inquisitive attitude towards fractals.

* Such as selective laser sintering and fused deposition modeling.

PREFACE

With some of the basic definition of fractals unsettled, and various differences in opinion among leading players, the field of fractals underscores the human aspect of science. No longer a crystalline image of deductive perfection, mathematics is revealed to be full

of guesses, mistakes, and room for discovery as any other creative

activity; in stark contrast to much of the mathematics that we were exposed to in mainstream education, ancient knowledge perfected by the Egyptians and the Greeks.

These, in tandem with my belief that art and science should be

embracive instead of contrasting, ground my passion for the thesis proposal.

ACKNOWLEDGEMENTS

I would like to extend my deepest gratitude to my family, for unfailing financial support and faith in my pursue.

Faculty staff, who provided motivation and critics, without which the thesis would not be what it is now, and for having confidence in this relatively unconventional topic and approach to industrial design.

Friends, for the encouragement in the ups and downs, amplifying the joy and fragmenting the sadness.

My thought process, a chaos induced logarithmic spiral.

CONTENTS

PRECEDENT STUDIES

7

WHAT IS FRACTAL

16

WHY FRACTALS?

20

QUESTIONING EUCLIDEAN GEOMETRY AROUND US

23

MAPPING FRACTAL PROPERTIES TO DESIGN

29

DIRECTIONS AND REFINEMENT ER ASER

40

HUMIDIFIER

44

REFERENCES

50

APPENDIX

52

SPIR AL JOINT

42

PRECEDENT STUDIES Ongoing Research on Fractals

Topological weather forecast.

Statistics.

Urban mobility.

Turbulence.

Heartbeat analysis.

Image compression.

Volcanic eruption.

Viral infection.

8

Tangible Fractals Applications

ANTENNA

High performance, compact fractal antennas enable unique mobile

device configurations not previously possible. Antenna performance is attained through the geometry of the conductor, rather than with the accumulation of separate components or elements that increase complexity and potential failure points.

COOLING NETWORK

Engineers at Oregon State University have developed fractal pattern

that can be etched into a silicon chip to allow a cooling fluid (such as liquid nitroge) to uniformly flow across the surface of the chip and

keep it cool. The fractal pattern above derived from our blood vessels

provides a simple low-pressure network to accomplish this task easily.

MILITARY CAMOUFLAGE

Fractal tactical camouflage, digital camouflage patterns generated with fractals. Understanding how camouflage works in the visual spectrum depends on tracing the braided contributions of such

seemingly unrelated threads as the biophysics of visual processing,

the distinction between detecting a target and recognizing it, signal detection theory, biomechanics and perceptual vector analysis.

FLUID MIXERS

The standard way of mixing fluids, stirring, produces turbulent

mixing, which is unpredictable, so two different batches may not end up identically mixed. Secondly, turbulent mixing is energy intensive and is disruptive to delicate structure. The engineered fractal fluid

mixers provide a different solution to the problem of mixing. Here, there is an actual physical branching fractal network of tubes that

can distribute fluid thoroughly into a chamber containing another

fluid. This system provides a low-energy reproducible way of mixing chemicals.

9

Fractals in African Culture Fractals appear in a widespread components

is the sacred altar. As a logician would put it,

hairstyles and kente cloth to counting systems and

as the altar is to the house. They view this as a

of indigenous African culture, from braided

the arrangements of homes and settlements.

The Ba-ila settlements of southern Zambia are enormous rings. They are made up of smaller

rings, which are livestock pens; and those are

made up of smaller rings which are single cylindrical houses and storage rooms. It is a ring of

rings of rings. Toward the back of the village is a

minature village; that of the chief ’s extended family. Toward the back of each coral is the family

living quarters, and toward the back of each house 10

the chief ’s family ring is to the whole settlement recurring functional role between different scales within the settlement. The chief ’s relation to his

people is described by the word kulela , a word we would translate as ‘to rule’. However it has this

only as a secondary meaning—kulela is primarily

to nurse and to cherish. The same word applied to a mother caring for her child, making the chief

the father of the community. This relationship is

echoed throughout family and spiritual ties at all scales and is structurally mapped through selfsimilar architecture.

Aerial view of the Ba-ila settlements of southern Zambia.

Palace of the chief in Logone-Birni, Cameroon.

11

Fractals in Hinduism The Hindu temple is an attractor with a variety

interactions. A Hindu temple is a synthesis of

of them, entering a doorway, circumambulating

proliferation and amalgamation, its total meaning

of visual aspects, and wherever one engages one (ritually walking clockwise around the whole

edifice while intently gazing upon parts of it), or

approaching the inner sanctuary, or worshipping there, one is accessing an aspect of the whole.

The dynamic is like that of a complex system

with multiple feedback flows each giving access into the whole system of metaphysical vision.

The temple offers means of engagement in the

nuances of the mystery, depending on the stage of development of the pilgrim; a child arriving for the first time experiences certain aspects according to her capabilities, while an elder

returning after many previous visits experiences others.

The spaces of the Hindu temple involve a dynamic system of traffic flows and feedback loops

organized to give a variety of ritual activities,

involving the worshippers' mental and physical 12

many symbols. By their superposition, repetition, is formed ever anew. This capacity for ever-

changing recursive experiences is a vital aspect of Hindu temples, and is often ignored by observers because it is easier to think of the temple as a static structure. The many pilgrims arrive

from different backgrounds and locations, each experiencing different aspects of the whole; yet

each aspect can reflect to some extent the whole,

or lead further into it. There is a constant renewal over the generations, through the repetition of basic Hindu rituals, the rededication to ideals,

and the perpetuation of the philosophical views; thus the temple serves to inspire. It enacts the mystery that seems to be at the heart of the

Hindu, the simultaneous co-existence of the One

and the many, the many and the One. It works to help the many pilgrims envision the One in the many, the One found in each one.

The ideal form gracefully artificed suggests the infinite rising levels of existence and consciousness, expanding sizes rising toward transcendence above, and at the same time housing the sacred deep within.

This universe is like a ripe fruit appearing from the activity of the cit [consciousness]. There is a branch of a

tree bearing innumerable such fruit. There is a tree having thousands of such branches. There is a forest with

thousands of such trees. There is a mountainous territory having thousands of such forests. There is a territory

containing thousands of such territories. There is a solar system containing thousands of such territories. There is a universe containing thousands of such solar systems. And there are many such universes contained within

what is like an atom within an atom. This is what is known as cit or the subtle sun which illumines everything in the world. All the things of the world take their rise in it. Amidst all this incessant activity, the cit is ever in undisturbed repose.

William Jackson

Kandariya Mahadeva Temple in Khajuraho, Madhya Pradesh. India: Symphony in Stone. Photography

by Brian Murphy.

13

Fractals in Art

Great Wave off Kanagawa.

Rock and Water in Storm by Leonardo da Vinci.

Both Hokusai and da Vinci observed fractal geometry in waves and used it in their paintings to imitate nature.

Smaller and Smaller by M.C. Escher.

Escher used fractals to create illusions.

The Cat in the Hat Comes Back by Dr. Seuss.

The Face of War by Salvador Dali.

Dr. Seuss used fractals as a narrative to ignite imagination.

Dali used fractals to create fear.

14

Number 8, 1949 by Jackson Pollock.

There has been much debate over whether fractals

“Not only is there a striking similarity between Pollock’s

of Pollock’s pieces do portrays fractals, and

ties between Pollock’s painting process and the process

analysis could authenticate Pollock’s art. Some

some might not, but the true essence of his art, I believe, is very well summed up by Kadish, a close friend of Pollock.

“I think that one of the most important things

about Pollock’s work is that it isn’t so much what

you’re looking at but it’s what is happening to you as you’re looking at his particular work.”

Ruebin Kadish

Pollock’s pieces might not portray fractals

precisely, but it portrays the philosophy behind fractals. The drips of paint seem to be reaching beyond the frame of the picture, so there is

no clearly defined top, bottom, or sides to the

composition, which we have gotten used to seeing in our Euclidean educated mind.

patterns and those of nature, but there are also similariused by nature to build its patterns. In particular,

contrary to popular belief, Pollock didn’t merely splatter a few blobs of paint on a canvas. Instead, he developed

a cumulative painting process of returning to his canvas regularly, gradually building layer upon layer of poured paint. This is very similar to nature’s processes – for

example, the cliff face being carved out by the repeated

pounding of the waves, or of the leaves falling day after day, building up a beautiful pattern. This link between Pollock and nature is not in itself a new observation.

Two of Pollock’s more famous quotes – “I am nature” and “My concerns are with the rhythms of nature” –

serve as a springboard for such ideas. Furthermore, over

the years, Pollock’s work has often been referred to as ‘organic’, suggesting that his imagery alludes to nature.But what do we mean when we say a pattern looks organic?

What do we know of nature’s pattern?” Richard Taylor 15

WHAT IS FRACTAL

“To see the world in a grain of sand, and to see heaven in a wild flower,

hold infinity in the palm of your hands, and eternity in an hour.”

William Blake 16

Fractals display self-similarity at all scales, where each fragment is a reduced-size copy of the

whole. Thereby, fractals display magnification symmetry, which means that they appear similar at different levels of magnification.

For example, when cutting a broccoli, every floret cut off is similar to the whole broccoli. Cut off another floret from the cutoff floret, and it still resembles the whole brocolli.

Fractals have simple and recursive definitions. Take the Cantor set below for example, starting with a simple line, divide the line into three equidistant lines and erase the middle line, which

results in two broken lines with equidistant space in between. Repeat the same to the resultant lines, then repeat the same to the resultant lines again. By iterating this infinitely, the result is an infinite amount of lines in extremely short yet equidistant lengths.

17

19

WHY FRACTALS

“The forecast,” said Mr Oliver, turning the pages till he found

it, “says: Variable winds; fair average temperature, rain at times.” … There was a fecklessness, a lack of symmetry and order in the clouds, as they thinned and thickened. Was it their own law, or no law, they obeyed?”

Virginia Wolf, Between the Acts 20

in architecture From Charles Jencks in England to Itsuko

superposition of different ordering systems are

architectural press of chaos, fractals, complexity

scientific ideas about complexity. These are

Hasegawa in Japan, there is discussion in the

theory, and self-organization. Architecture and design should be informed by and express the

emerging scienti¯c view that the world around

us is more chaotic and complex than previously thought. However, the architectural response

has a tendency to be fairly shallow. Twists and

used to express in architectural form the new

moves in the right direction toward connecting

architecture with contemporary cosmic concepts.

However, knowledge of the mathematics of

fractal geometry can provide a path to an even deeper expression.

Carl Bovill,

folds and waves, jumps in organizing grids, and

Fractal Geometry as Design Aid

in graphic design “For a layman it’s hard to imagine what all the

Fractals mean, in other words, a new scientifi-

Anyway, the fractal theory has opened a lit-

human’s positive attitudes toward life. As said,

world of mathematics really contains inside.

tle the door of mathematics, which so often is sealed with the code of mystical formulas. In their clearness and clarity the fractals mean

something to all of us. They raise images and

help in many ways to understand the complexi-

ty and deep and abstract nature of mathematics, the perfectness and beauty of forms and rela-

tions. They would be very hard to understand, if

they couldn’t be dressed up in real forms, if they couldn’t be illustrated. The fractals can help

us to understand even some basic truths about life, that it spontanically brings forth beauty,

harmony and order, which in its highest forms expresses itself in man’s spiritual and cultural phenomenons and progress.

cal discovery, which hopefully can affect even

power of the fractal pictures can even strength-

en the confidence on the meaningfullness of the life. And as everybody can recognise, one can

see some refers of the great order, even outside the materia, which the universe contains also in these fractals. It can even help humans to

understand the nature of the universe. Think about it, when we go to the most unpracti-

cal things, to the depths of mathematics, we

still find the same things as from nature. And

even the irregularities are always, in some way, regular!

It’s too often ignored how much order and meaningfulness there really is in life.”

eduwww.mikkeli 21

in psychology One line of evidence shows that the human

This hypothesis is supported by the finding that

specifically, analyses of brain functioning shows

that have the same (fractal) properties as natural

nervous system is governed by time fractals. More that it displays typical noise signals, commonly

referred to as ‘1/f noise’ or ‘pink’ noise [Anderson and Mandell 1996]. The fractal character of such noise can be easily appreciated, because like

spatial fractals, it shows self-similar detail when you zoom in on it. But what is the function of

these time fractals? A common answer is that the natural world – and the way it changes over time – is also characterized by pink noise, which

suggests that our fractal minds are optimized

to process the fractal characteristics of natural scenes (see, for example, [Knill et al. 1990]).

discriminating fractal contours is best for those scenes [Gilden et al. 1993]. Interestingly, these findings can (tentatively) explain the creation of fractal artwork, and fractal architecture in

particular. Such art should be understood as an exteriorization of the fractal aspects of brain

functioning [Goldberger 1996]. Or as Goldberger [1996] puts it: ‘... the artwork externalizes and

maps the internal brain-work... Conversely, the

interaction of the viewer with the artform may be taken as an act of self-recognition’.

Yannick Joye,

Fractal Architecture could be Good for You.

in cinematography This fabric of perceived reality seems to be

it relayed back to us again… Through the lens

level were subtly stirring up the ones around

my imagination into a desire to understand

produced from within itself… As if each

it, so as to create in each other a unique, yet

interrelated, turbulent, writhing flow… Each level emanates from the rest… Yet the rest

emanate from it… Feeding back into and out

of the others… Undulating across all levels of creation, perception of the universe seems to

stem from a fractal-like feedback loop… From the smallest to the largest and all the way

back again… Here, within this sea of chaos, arises our world… And what better way to

become more familiar with it, than to watch 22

of this magnificent film… One which stirred

my position within the natural order of things better…

Here, the abstract patterns of creation that gave rise to the atomic matter that our bodies – and

the material universe – seem to built from, now unfold in ever more complex ways… Ways that we seem to take for granted in an everyday, presumed, regular – almost clock work – reality…

Reggio Godfrey,

Koyaanisqatsi: Life out of Balance

QUESTIONING EUCLIDEAN GEOMETRY AROUND US

Philosophy is written in this vast book – I mean the universe - ‌ in the language of mathematics, and its

characters are triangles, circles and other geometric figures, without which is humanly impossible to understand a single word of it; without which, one wanders in vain in a dark labyrinth.

Galileo Galilei

Galileo was merely expressing the sentiment that had reigned supreme in the days of the ancient Greeks, in whose world truth and beauty were sought in terms of composites of perfect Euclidean solids. It is a

ceaseless wonder that classical mathematics, which is based on such perfect and abstract forms, has the impact it does on human civilization. It reduced geometry into straight lines, planes, circles, squares,

triangles, octahedron, decahedron etc. It is only well suited to study the things we have created, things we built using that classical mathematics.

The patterns of nature, plants, mountains, the weather system, these were outside of mathematics, at least until Mandelbrot introduced fractals in the late 1970s.

23

On Order versus Chaos Our current idea of order and organisation is so constrained by the Euclidean geometry that anything

outside that is often considered chaos. By re-arranging mundane objects in our daily lives, Ursus Wehrli challenges the “chaos” around us in “The Art of Clean Up”.

"Ursus Wehrli demonstrates how much creative potential there is in the act

of cleaning up and how good it feels when everyday chaos is resolved into its components and reorganized."

Guenther Reinhardt, Stuttgarter Nachrichten

However, there is a field in botany, phyllotaxis which studies the arrangement of plant leaves on a stem; Fibonacci’s sequence, golden angle, fermat’s spiral for example. The science behind much of beauty and structure in the natural world is in fact an intrinsic part of the laws of physics. Amazingly, it turns out

that the mathematics of chaos can explain how and why the universe creates exquisite order and pattern.

Spiral, or astichous phyllotaxis, leaf arrangement numbered according to their successive apearance.

24

Die Kunst, aufzur채umen by Ursus Wehrli.

25

On Manufacturing

euclidean

chocolate jelly candy noodles sausages french fries

fractal

vegetables fruits meat bread popcorn cotton candy

“Food is the ultimate embodiment of our continuing attempts to tame, transform, and reinterpret nature.�

Massimo Montanari, Food is Culture

Manufacturing method seems to be key in determining the geometry of the outcome, even in objects as

susceptible as food. In manufacturing terms, chocolate, jelly and candy are molded, sausages, french fries and noodles are extruded, cookies are stamped, lasagna is compress rolled.

Bread, allowed to rise, forms apollonion gasket with air bubbles produced by yeast, a type of selforganised packing.

A slice of bread.

Soap bubbles.

Microscopic structure. Apollonion gasket.

Often deemed complicated and irregular, fractal geometry has been shied upon in favour of regular ones, which could be easily tamed and manufactured. However, the 3D printing has made manufacturing of fractals possible.

26

Proliferation of Additive Manufacturing

“Additive manufacturing is changing not only how things are made, but what is made.” Vance, A. (September 13, 2010) 3-D Printing Spurs a Manufacturing Revolution. New York Times.

“But as 3-D printing machines have improved and fallen in cost along with the materials used to

make products, new businesses have cropped up.” (December 10, 2011) The shape of things to come. The Economist.

Australian Vogue Living Sept/Oct 2011.

27

On Modularity collective noun

noun Grammar that appears singular in formal shape but denotes a group of persons or objects. animals

inanimate object

a swarm of a flock of a flight of a herd of a school of a murder of an ambush of a pride of a fling of a den of

a box of a stack of a can of a ream of a pack of a pair of a bundle of a pile of a string of a roll of

The semantically rich collective nouns for animals, evocative and poetic, illustrate the dynamism of

modularity in nature; while inanimate objects have mundane and expected modularity. Perhaps a large

part of this is due to the shape which we impose on objects; mostly of planes, rectangles, and cylinders.

The packing of florets of the Romaneso broccoli, the dendritic branching of electrical discharge; these

fractals are not only known to be highly efficient in terms of packing, but also ingenious in the way they are arranged.

What if we could count and describe the quantity of objects we have with different collective noun

due to its nature of packing? A bud of tissue paper for example; what if tissue paper is packed like how nature packs flower petals?

28

MAPPING FRACTALS PROPERTIES TO DESIGN

The Coastline Paradox While studying the causes of war between

that the coastline of a landmass does not have a

for a relation between the probability of two

of the coastline depends on the method used to

two countries, Richardson decided to search countries going to war and the length of their common border. While collecting data, he

realised that there was considerable variation

in the various gazetted lengths of international borders. For example, that between Spain

and Portugal was variously quoted as 987 or

1214 km while that between the Netherlands

and Belgium as 380 or 449 km. The coastline paradox is the counter intuitive observation

well-defined length. More concretely, the length measure it. Since a landmass has features at all

scales, from hundreds of kilometres in size to tiny fractions of a millimetre and below, there is no

obvious limit to the size of the smallest feature

that should not be measured around, and hence

no single well-defined perimeter to the landmass.

Infinite repetition of a pattern at infinite scale, the

coastline is statistically self-similar at a wide range

of scale, thus it is an example of stochastic fractals.

In Euclidean geometry, the perimeter and the area of a shape are directly proportional to each other;

take a square as example, when the perimeter increases, the area increases as well, and vice versa. For fractal geometry however, an infinite perimeter encloses converging surface area. When brought one dimension further, an infinite surface area encloses converging volume.

29

Fractional Dimension In Euclidean geometry, a line is one dimensional, a square is two dimensional, and a cube three

dimensional. When a fractal curve is introduced, and the curve so dense that it almost fills up an area, is it still one dimensional?

In the coastline paradox, coastline portrays infinite perimeter and converging surface; perimeter is one dimensional, and surface is two dimensional; the paradox is suggesting a dimension in between one

and two. Fractals portray fractional dimension, meaning that instead of integers, dimensions can be fractions.

* See appendix for the mathematical proof.

Expanding on Hilbert curve, the fractional dimension of fractal is translated into a prototype for

packing wires. Using Hilbert curve, a long line is packed into a pocket size square, like retractable wire to solve the issue of wires tangling up due to their lack of structure.

30

31

Deconstructing Fractal Tessellation Self-organisation

Fermat spiral.

Fermat spiral.

Fermat spiral.

Equidistant spiral.

Golden angle.

Archimedean spiral.

Archimedean spiral.

Figure on opposit page. The hexagon is divided and smaller versions of the pieces used to build a tessellation with a border-limit.

32

33

concentric waves by pacemakers

spiral waves after hydrodynamic breaking of a concentric wave 34

Fractals as Cognitive.

In the midst of whiteness, how do we differentiate the clouds from the snow, the snow from the mountains.

Though we cannot describe the shape of clouds, snow or mountains, we could still tell them apart, almost intuitively. It might not be coincidental then, that all of them are examples of fractals in nature.

35

Did you notice that the above picture is the same as the previous page? Only that now it is not upside down.

Due to its self-similarity and tesselation unlike what our minds are taught to interpret, fractals has

36

Form exploration inspired by mountain ranges as a study on the cognitive and haptic dimension of fractals.

abstraction of fractal properties

self-similarity

self-organization packing

repetition

fractional dimension

maximizing surface area

objets which

maximizing perimeter

fermat spiral

require transmission eraser filter

sieve

heater

air freshener

humidifier

38

joint

DIRECTIONS AND REFINEMENT

“Our interest in the invisible world stems from a desire to find a form for it in the visible one, which means to prise open, to decompose, to atomize the deceptively familiar, the visible exterior appearance, before we

can deal with it again. […] We are interested in the hidden geometry of nature, in an intellectual principle, and not primarily in the appearance of nature.”

Jacques Herzog, La Geometria Oculta de la Naturalesa.

39

Eraser

40

Due to its character of sharp angles, the fractal above optimises the number of corners. With usage, the corners get worn and due to its self-similarity, there will be no big lump of eraser without corner with this eraser as compared to the ones on the left.

41

Spiral Joint

Exploration of different spirals with different number of part it could join.

42

The system of spiral joint is extended to a branching structure formed to blend in with its spiral joints.

43

Humidifier

44

A simple experiment with various materials to test the rate of spread and evaporation of water on different materials. Conclusion from the experiment is that fibrous materials such as wood, wood veneer, paper and

textile are most efficient for spreading water. While porous materials such as clay, concrete, and sponge are most suitable for evaporation of water.

45

With the conclusions from the material experimentation, I proceeded to a different approach to

designing a humidifier with fractal properties. By setting the material constraint, I played with various methods of folding to generate a form which compresses a big surface area into a small volume for

diffusion purposes. With a simple primary tessellation, I played with crinkles which create a flexible structure with paper.

46

47

The similar approach as the previous page was done with gauze instead. Gauze of different yarn density

was experimented with and simple sewing used to control the folds to create a structural form capable of holding up its shape.

48

Employing the fractional dimension of fractals, this passive humidifier would optimise the surface area for water evaporation. Theoretically, nylon is a porous material and is

capable of capillary action, which I found on hindsight to the previous two exploration.

However, whether this humidifier does work is still subject to testing with a prototype.

49

References Addison, P.S. (1997) Fractals and Chaos: An Illustrated Course, Institute of Physics Publishing, Bristol and Philadelphia.

Bak, P. (1996) How Nature Works : the Science of Self-Organized Criticality, Copernicus Springer-Verlag, New York.

Briggs, J. (1992) Fractals: The Patterns of Chaos : A New Aesthetic of Art, Science, and Nature, John Briggs, Simon and Schuster.

Dewey, T.G. & Novak, M.M. (1997) Fractal Frontiers, World Scientific Publishing, London.

Eglash, R. (1999) African Fractal: Modern Computing and Indigenous Design, Rutgers University Press, New Brunswick, New Jersey and London.

Frame, M.L. & Mandelbrot, B.B. (2002) Fractals, Graphics, and Mathematics Education, Washington D.C. Mathematical Association of America.

Gleiniger, A. & Vrachliotis, G. (2008) Simulation: Presentation Technique and Cognitive Method, Birkhauser Verlag, Basel.

Glynn, R. & Shafiei , S. (2009) Digital Architecture: Passages through Hinterlands, Digital Architecture.

Hensel, M. & Menges, A. (June 2006) Morpho-Ecologies, AA Publications, London.

Hofstadter, D.R. (1999) Godel, Escher, Bach: an Eternal Golden Braid, Basic Books.

Kappraff, J. (2002) Beyond Measure: A guided Tour through Nature, Myth, and Number, World Scientific.

50

Lindenmayer, A. & Prusinkiewicz, P. (1991) The Algorithmic Beauty of Plants, Springer-Verlag.

Olafur, E. & Weibel, P. (2001) Surroundings Surrounded: Essays on Space and Science, MIT Press, Cambridge, Massachusetts.

Padovan, R. (1999) Proportion: Science, Philosophy, Architecture, E & FN Spon, London.

Richardson L.F. (1961) The problem of contiguity: Statistic of Deadly Quarrels, General Society for the Advancement of General Systems Theory, Ann Arbor, Michigan.

Spuybroek, L. (2009) Research and Design: The Architecture of Variation, Thames and Hudson, London.

Taylor, R. (February 2006) Personal Reflections on Jackson Pollock’s Fractal Paintings, The Scientific Electronic Library Online. Retrieved from http://www.scielo.br

Yale University (2011) Fractal Geometry. Retrieved from http://classes.yale.edu Wolfram Research Company. Wolfram Alpha. Retrieved from http://www.wolframalpha.com

51

Appendix

The relationship between perimeter and area of a koch curve.

Assuming (for simplicity) that the length of the first line is 1 unit, iteration 0 1 2

length of each segment

number of segments

perimeter

1

1

1

1/9

16

1/3

4

1.33

64

2.37

1024

4.21 7.49

3

1/27

5

1/243

7

1/2187

16384

...

...

...

4 6 8

25

1/81

1/729

1/6561

1/847288609443

1125899906842624

1328.83

...

...

...

50

...

...

4503599627370496

... ...

The perimeter is growing exponentially towards infinity.

52

5.62

9.99

1/2541865828329

51

4096

3.16

65536

26

...

256

1.78

... ... ...

...

1771.77

1765780.96

2354374.62

...

Appendix

Assuming that each of the triangle grid is 1 unit² in area, iteration 0 1 2 3 4 5 6 7 8 9

10 ... ...

area of each

number of added

-

-

1

12

added triangle

9 1/9

1/81

1/729

1/6561

1/59049

1/531441

1/4782969

1/43046721 ... ...

triangles

3

48

192 768

3072

12288 49152

196608 786432 ... ...

area increment

total area

-

81

12

120

27

5.33

2.37

1.053

0.4682

0.2081 0.0924 0.0411 0.0182 ... ...

108

125.33 127.70

128.753

129.2212 129.4293 129.5217

129.5628 129.5810 ... ...

The area is converging towards 8/5 of its initial area.

53

Appendix

Fractional Dimension

d=1

1

2=2¹

3=3¹

d=2

1

4=2²

9=3²

d=3

1

8=2³

27=3³

d=?

1

3=2 d

9=4 d Solving 3=2 d, d=1.58496 .

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Appendix

By modifying the definition, even so slightly, would generate different

fractals. The L-system above has the same principle as those on the left.

Changing the distance of growth before further branching between the two branches, the mirror symmetry fractals can now be assymetrical.

Even without modifying the definition, a wide variety of similar yet

different outcomes could be generated if parameters were set. All the

L-system examples above have the same simple definition of branching

into two lines at the end of the line, but playing with different parameters such as the number of branches, the number of iterations, the angle of

branching, the growth length ratio, the line thickness of each growth; yields a variation of similar yet distinctly different fractals.

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Appendix

Deterministic fractal, branching L-system

tochastic fractal, branching L-system with

with exact self-similarity.

approximate self-similarity.

Deterministic Fractals versus Stochastic Fractals Exact versus Loosely Fractal

Deterministic fractals are generated through infinite recursion, therefore they portray exact self-similarity at all scales. They are created by the

imagination of scientists and mathematicians, theorized and idealized, they do not exist in the real world.

Fractals that occur naturally portrays self-similarity as well, but only

statistically, or approximately in layman’s terms. Take for example the coastline of Britain (the coastline is considered a stochastic fractal,

explained later under the coastline paradox), one cannot expect to find microscopic Britains by looking at a small section of the coast.

deterministic -n. the principle in classical mechanics that the values of dynamic variables of a system and of the forces acting on the system at a given time, completely determine the values of the variables at any later time stochastic -adj. statistics (of a random variable) having a probability distribution with finite variance

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Appendix

Terrain generating fractal.

A computer generated image of a mountain landscape with stochastic fractals.

Like deterministic fractals, stochastic fractals have simple and recursive definitions as well.

The terrain generating definition on the extreme left, for example, starts with a simple triangle.

Subdivide the two sides for the midpoint, and displace the midpoint randomly within a defined range. Repeat the same to the resultant lines, and again the same to the resultant lines. Unlike deterministic fractals, the reiteration stops after a point, as naturally occurring fractals do not portray self-similarity across an infinite scale. They do portray self-similarity over a range of

scale, but only within a limited scale. A leaf vein for example, probably has branching up to 4

iterations at most, and the structure at smaller scale would be composed of different patterns, as they are composed of different substructure for different functions.

Stochastic fractal can mimic nature so well that when used to generate natural systems, there is almost no telling that it is computer generated. The first fractal generated animation was used

to model an entire planet in the Genesis Sequence of Star Trek 2, the Wrath of Khan, back in

1980. Watch http://vimeo.com/5810737 for the principle behind the animation breakthrough.

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