View
975
Download
0
Category
Preview:
DESCRIPTION
Discussion on the Historical impact of Fractals, Mathematical Background and Theory of Fractals, Natural Fractals and Artistic Forgeries of Nature, Practical Applications of Fractal Geometry, Fractals in medicine
Citation preview
Group Work Summary
Our group consisted of Tarei King, Stephen Ticharwa, Thorsten Ziller, Jason Sabbage and Josh Scorrar.
Topics were discussed in several initial meetings and divided by interest per person. There were 4 meetings scheduled over the past 4 weeks which allowed time to independently research chosen topics and communicate zindings to the group as a whole.
All members attended mentioned meetings and the workload was shared equally, for which help offered to members through discussion via email and physical mediums.
1
Abstract
The content of our report has intended to provide an insight into function of fractals as well as investigation of their relevance to applied science and humanities. It comprises a collective effort encompassing research and individual perspective on Fractal existence through discussion of subject matter such as Geology, Art History, Computing Science and Medical Biology. As such we have detected signizicant utilization of mathematic principles to facilitate exploration and development of interdisciplinary arts and thought.
In particular, the advent of technology meant computers initially gave scientists opportunity for tangible representation of the Fractal discovery. The immediate impact on all other respective investigative disciplines was now that a scientizic method could be implemented to identify and depict the pervasive nature of Fractals in human culture itself.
2
Table of Contents
Group Work Summary.............................................................................................................. 1Abstract.......................................................................................................................................... 2Table of Contents........................................................................................................................ 31. Discussion on the Historical impact of Fractals on Western Artistic Thought.......................................................................................................................................... 4
28........................................................................................................................................ 12. Mathematical Background and Theory........................................................................ 9
2.1 The history of Fractal Theory and those involved................................... 92.2 Introduction to the maths of Fractal Geometry.................................... 102.3 Julia Sets................................................................................................................. 112.4 How to generate a Julia Fractal Set.............................................................. 12
3. Natural Fractals and Artistic Forgeries of Nature................................................. 143.1 Briezly on Euclidean.......................................................................................... 143.2 Natural Fractals and dimensions................................................................. 153.3 Recognition of self‐similarity in nature..................................................... 153.3.1 On measurements of coastlines................................................................ 163.4 Before the computer......................................................................................... 173.5 Computer Art ...................................................................................................... 173.6 Using computers to forge nature................................................................. 18
4. Practical Applications of Fractal Geometry............................................................. 194.1 Fractal Image Compression............................................................................ 194.2 Fractal Use in Visualization and Simulated Terrain generation....204.3 Fractals in Hollywood....................................................................................... 214.4 Fractal Art and Fashion.................................................................................... 224.5 Emerging Fractal Based Technologies....................................................... 22
5. Fractals in medicine........................................................................................................... 235.1 How are fractals involved in the process in which certain cells form together to create organs?...................................................................................... 235.2 Technology and Maths used for understanding Fractals in Biology........................................................................................................................................... 235.3 Fractals use in Cellular structures............................................................... 255.4 Technological advances involving Fractals in Biology.........................25
Conclusion.................................................................................................................................. 27Bibliography............................................................................................................................... 28
3
1. Discussion on the Historical impact of Fractals on Western Artistic Thought
Given the general subjective nature of Western thinking plus prolizic and voluminous accounts of History of Art, I briezly attempt to identify the 'simple' fractal, as well as provide a context for how it is applicable in unfolding 'history'. The phenomenon of evolving History is impact itself, yet the fractal can perhaps pertain to, or be juxtaposed with it.
'Nothing exists except atoms and empty spaces; everything else is opinion' [Democritus, 400B.C]
A basic dezinition of a fractal is imperative if I am to discuss its inzluence or any subsequent derivation or meaning. So, what is a fractal?
‘… a fractal is a shape that, when you look at a small part of it, has a similar (but not necessarily identical) appearance to the full shape.' [https://www.fractalus.com/info/layman.html]
A geometric pattern that is repeated at ever smaller scales to produce irregular shapes and surfaces that cannot be represented by classical geometry. Fractals are used especially in computer modeling of irregular patterns and structures in nature. [http://www.answers.com/topic/fractal]
Without attributing any specizically mathematical properties we could argue that a fractal in essence is a demonstrable output, or arrangement, of tangible yet inextricable interrelationships.
'Life on Earth is an intricate network of mutually interdependent organisms held in a state of dynamic balance. The concept of life is fully meaningful only in context of the entire biosphere.' [Paul Davies, The Cosmic Blueprint]
So how does the notion of fractals inzluence the human psyche or reach levels effecting or contributing to artistic thinking?
"The structure of the Universe is complex...Science and Art both attempt to explore this in order to understand, and then make use of it... Maths and science seek to analyze experience while art attempts to synthesize experience." (M.Fowler, 1996)
According to the above authors, they further suggest Architecture and Sculpture, as art forms, were inevitably, attributed sacredness during connection through (essentially institutionalized) knowledge and ritual. Places became literal 'sanctizied' environments ‐ containing these select perceivable qualities e.g. in land mass, the elements, existing vegetation and animal life. Resultant depiction ranged from echoing form, to contrasting it and not only its contents, but (historical) Architecture itself, notably became a mediating device for referential transition (as human) between earth and the celestial.
"...The artist is extremely aware of the limits of space. Felt as a physical presence with physical properties, it is through this medium that all of the arts are expressed. Dance,
4
music, theatre, architecture, and sculpture, all manipulate the space/time experience…. Intelligible space enters realms of mass and boundary… Sculpture is active volume while architecture is active container…. but both require more than mere presence of form to be discernable arts... clear concept and experience of emotion predicate sensitive use of materials in organization." (M.Fowler, 1996)
Adding to this example art historian Albert Elsen (as scrutinizer) decidedly remarks…
"… Up until abstraction evolved in the twentieth century we know that shapes such as triangles, circles, and squares historically symbolized concepts and values. All three of these shapes, for example, have stood for God.... the circle has stood for eternity, resurrection… the Pythagorus triangle referred to human knowledge. In Christianity triangular shape stood for the Trinity, and has signitied hieratic social systems... for many painters change to the face of the square (similar to changes seen in depictions of Christ) appears through the history of art... often personal history and artists intentions deliver new dimensions of meaning..." (Elsen, 1981)
As such, perhaps the (early) introduction of applied geometrical principles in two dimensional space allows for 'strengthening', 'reinforcement' or 'solidizication' of specizic variables or qualities pertinent to the 'artist', whose demonstrative role may predominantly serve to visually represent a desire for the designated relationships of these values, to also be held ‐ or at least 'received' ‐ by the viewer.
(detinition of) Euclidean Geometry Study of points, lines, angles, surfaces, and solids based on Euclid's axioms. This axiomatic method has been the model for many systems of rational thought, even outside mathematics, for over 2,000 years… This work was long held to constitute an accurate description of the physical world and to provide a sufticient basis for understanding it. [http://www.answers.com/topic/euclideangeometry1]
Observing the inclusion of Pythagorus and also Euclidean geometry, this invokes reexamination of interrelationship with the fractal.
"The Euclidean plane is an abstraction used to examine geometrical relationships in two dimensions. In Nature, a perfectly tlat, rigid plane with no thickness does not exist... In actual fact the reality is that... surfaces are twisted, convoluted, rough and irregular. In short they are fractal." (M.Fowler, 1996)
In the case of Pythagorus, he posits that ‐ if the length of a right triangle's sides are designated a, b, and c (where c is opposite 90°) then a squared + b squared = c squared. So if squares (as shapes) are placed on the triangle's sides the total number on side a and b would = total number on side c, (the hypotenuse). This is considerably important because basically, in all of art history, delegated visual space is (pronouncedly) dezined or contained by the proxies of shape and form ‐ and thus appear, inevitable relationships of intersecting line ‐ which as they straighten, comprise a natural abundance of squares, rectangles and triangular (general) convergence.
By terrestrial standards, certainly for those of the human eye ‐ 90° as a 'pull' ‐ for 'frame of reference', would be an unsuitable host if it were not to 'lead' the party. That means, that there is a tendency for the viewer to extrapolate from sometimes
5
multifaceted and intricate space ‐ some sense (literally) of 'containment' ‐ through detecting fulcrums of continuity. Enter 90°. (Complete with Pythagarol little black dress).
".... can a shape that is detined by a simple equation or a simple rule of construction be perceived by people other than geometers as having aesthetic value.... When the geometric shape is a fractal, the answer is yes. Even when the fractals are taken 'raw', they are attractive..." (Emmer, 1993)
With divisibility in mind, the Golden Ratio as well warrants mention. A ratio ultimately becomes an inzinite repeating decimal e.g. 1.666.... the 'Golden Ratio' is sometimes referred as Phi (∅) approximately equal to 1.1.618.
"... it is extracted from the ratio of lengths of two line segments, where one line is divided (using a compass) at a particular point (the 'Golden Cut') .... and the smaller segment is in relation to the larger which is relation to the whole line (namely)...
[ http://www.cosarch.com]
"Our perception of the complexity of a contiguration is altered when we can tind an algorithm and a set of modules to generate that contiguration" (Emmer, 1993)
"An entire fabric as a warp… (comprised of individual strands) its weft’s horizontal elements interlace with stronger vertical ones becoming more stable ...Symmetry is the working structure, while elements, such as the Golden ratio, are the weft threads…Depending on perception, pattern of symmetry may be hidden within the layers, suggesting only asymmetry, or absence of order, when in truth, they both coexist..." (M.Fowler, 1996)
Inzinitely presented with the logistics of characterizing space, both artist and viewer are confronted with the human 'oeuil's natural gravitation for compositing visual elements aesthetically, often identizied as symmetry. If it is suggested that symmetry is a concept that allows description of relationships in space and that the visual artist is able to explore its limits therein, (given rise and/or opportunity for aesthetic 'presence') we could dezine basic symmetry as characteristic repetition of arrangement with equality, and asymmetry as being arrangement which lacks equal quality. Therefore natural symmetrical arrangement as opposed to applied symmetrical arrangement (or movement) could be described as inherent symmetry.
"Picasso… admired more the asymmetrical constructions of symmetrical human
6
features and objects in Cézanne’s art. … Picasso also received the idea of creating continuities in art where in nature there were discontinuities, and vice versa Cézanne’s reduction of myriad shapes to multiples of each other found comprehension in Picasso's conjugation of ovoid forms...and multiplicity of arcs... (Elsen, 1981)
"... Mathematics provides a rich array of materials from which to select, limit and control, and therefore compose. Mathematics not only supplies materials but can also be applied to the problem of visual consonance, consequently contributing to both the form and content of the work... " (Emmer, 1993)
"...Mathematical language uses the word "fold" for describing a rotation...The pentagon is tilled with Golden Ratios and by extension the patterns that arise through 5turn symmetry is associated with organic forms..." (M.Fowler, 1996)
If we expand on the principle of rotation and apply it to three dimensions in space ‐ this posits freedom of movement without constraint in a singular plane. Fractals dezinitively through their recapitulation relay an emancipated, interdependent synergetic process that substantiates vitality and braces infrastructure for metamorphosis.
'The crystal may be regarded as one of the basic form patterns… atoms or molecules are so spatially arranged that the unit can withstand the stresses of the environment, or be able to accommodate itself either to free space or to the pressures of continement.'David Drabkin, Biochemist
"From the distribution of foliage on a tree, to the complex neural network of our nervous system… these can be better described with the help of Fractal Geometry. In the human body, the fractal design of its components allow our organism, to greatly extend its contact surfaces to carry out the innumerable and complex functions of interchange that make life possible. This optimal structure must surely be motivated by evolutionary reasons." (Salingaros, 2000)
In my opinion, we can thus far acknowledge that fractals possess intrinsic properties that display striking resemblances to mathematic principles utilized throughout history. Discourse unsurprisingly continues over the respective roles and impetus of Maths, Art and Science in a seemingly, sometimes misled debate of collective, versus individual cultural 'endowment' .
But the necessity for 'clout' is thankfully diminished by the wisdom of those immersed in disciplines, both historical and present day, where 'artistic' thinking and subsequent impact is an interdisciplinary method or approach ‐ if not pathway ‐ best described perhaps; as an unfettered explorative journey. Here, the essence of the fractal prevails: complete reciprocity is perpetually summative.
"...I believe that the collective/individual mind, tries to protect itself from the ‘unknown’. Simultaneously tlexible, it eventually evolves by incorporating new content… a possible explanation… why people spontaneously build structures that have fractal properties. We can also connect the ‘fractal ‘ universe with the mind structurally… the impact of our experiences may account for why our mind is fractal; because so is our environment… surrounded by fractal structures for millions of years…. a great deal of
7
our mind's structure stems from this ancient relationship...." (Salingaros, 2000)
“Faced with pure algorithmic art (of the fractal) no experienced person will fail to realize eventually that the work is neither a painting nor a photograph… it certainly does not raise the issue of 'artiticial creativity'. Yet, these are often perceived as beautiful on their own terms. Therefore, creativity and beauty and the production and consumption of art must be viewed separately." (Emmer, 1993)
The fractal then, it seems, literally conjures magic. Enchanting, spellbinding, fascinatingly hypnotic ‐ at times polemic ‐ perhaps 'conjurer' deservedly and by inference becomes it. Yet most telling and astonishing of all, I believe, is its ubiquitous nature and the potency of consistency in all past, current and imminent, pending permutations and/or manifestations. Essentially, it permeates culture. By default, this includes the realm of the artistic mind.
The mind is inextricable from History as phenomenon. As homo sapiens, perhaps relentless examination of the Fractal and desire for further understanding of quantiziable or identiziable complexities belies its absolute value.
Quite possibly, the unique experience of witnessing the fractal is rezlection in a mirrored surface of interconnectedness ‐ a source for contemplative choice ‐ to be enveloped by inevitable change while seeking awareness of its profound momentum.
What do you see in the Fractal? Or does the Fractal see something in you?
"The ancient idea of the Eastern philosophers on the interconnection between the microcosm and macrocosm is acquiring rigor in our times with the scientitic understanding of the universal laws of nature. The discovery of fractal geometry and its role in the description of a great variety of natural phenomena plays an important part in this process."
"... Recently I read a very interesting book: "The Evolution of Consciousness" by R. Ornstein, a leading neurophysiologist, on the subject of evolutionary psychology. The author mentions a study comparing students from Western cities, which contain many horizontal and vertical outlines in their designs, but few oblique ones, to a group of Cree Indians, whose houses contained lines in all orientations. The conclusion of the experiment is that the urban students had less acuity for oblique lines than the Indians did, which shows that the level of complexity of the urban environment has a corresponding impact in the level of complexity of the human mind… "
– Benoit B. Mandelbrot.
8
2. Mathematical Background and Theory
“Although computer memory is no longer expensive, there's always a tinite size buffer somewhere. When a big piece of news arrives, everybody sends a message to everybody
else, and the buffer tills.” [ Benoît Mandelbrot.]
2.1 The history of Fractal Theory and those involved
When we talk about fractals we come across the Mandelbrot set which is the most famous example of fractals. It is named by the mathematician Benoît Mandelbrot.
In 1979 Mandelbrot zirst started to study fractals called the Fatou sets and the Julia sets at Harvard University. His knowledge was based on the previous work of Pierre Fatou and Gaston Julia from France who started experimenting with fractal formulas in 1917 in Paris.
Fig 2.1 Benoît Mandelbrot inspired by Gaston Julia and Pierre Fatou
The french mathematicians were working in the zield of complex numbers. Pierre Fatou started to focus and specialize in applying a function repeatedly by using the output from one number as the input to the next which is known as mathematic iteration. He zigured that iterations of a simple function can produce complex outputs in graphical representations.
At the same time at the age of 25, Gaston Julia wrote an article on iteration about rational functions which gained immense popularity among mathematicians. As a result of that he received the Grand Prix award. Despite his fame, his works were all forgotten until the day Benoît Mandelbrot mentioned them in his works.
Benoît Mandelbrot’s main focus was to come up with the most simplest possible transformation of a fractal. His main advantage to any other mathematician from the past was to use modern computers to calculate iteration functions. For the zirst time in history with the use of computer technologies he could calculate and plot a tremendous amount of images of Julia sets with the formula zn+1 = zn2 + μ. While investigating the topology of the Julia set in depth he changes the formula into zn+1 = zn2 + c and called it the Mandelbrot set. To have a better understanding of the scientizic discovery Benoît Mandelbrot made in the 1980s we have to look into more detail.
9
2.2 Introduction to the maths of Fractal Geometry
The Mandelbrot set is one of the most implemented fractals in plotting programs now a days. It is produced by the formula: [1]
zn1 zn2 c
The variables z and c are complex numbers such as 3 + 2i. The formula is iterated until |zn| is greater than or equal to the bailout value 2. Then the pixel that c corresponds to is colored according to the number of iterations that occurred before the process bailed out. The uninteresting black area of the image is the actual Mandelbrot Set. It consists of all the values for c where |zn| never got larger than 2. Of course this area is impossible to change accurately, so the program decides to color black all pixels for which |zn| never gets larger than 2 for a given number of iterations. [2]
It is astonish that such a simple formula can produce such an interesting picture. If you zoom into the image you will zind sections that look the same as portions of the image at other zooms. This inzinite level of detail is common to all fractals and makes them so fascinating.
Fig 2.2 “The best know fractal and one of the most complex and beautiful mathematical objects known.” - Adrien Douady.
10
2.3 Julia Sets
Fig 2.3: Original Mandelbrot Fractals. (Mandelbrot B. 1983)
Julia sets are created with the same formula as the Mandelbrot set, but they use the formula differently. For the Mandelbrot set, c is the point being tested, on the complex plane. For the Julia set, c remains constant. From this dezinition, you can see that there is an inzinite number of Julia sets. One for each value of c. In fact, there is a Julia set that corresponds to each point on the complex plane. There is an interesting relationship between the Mandelbrot set and the Julia sets. In a way, you can think of the Mandelbrot set as an index for the Julia sets. For values of c that are inside the Mandelbrot set, you will get connected Julia sets. That is all the black regions are connected. Conversely, for those values of c outside the Mandelbrot set, you get unconnected sets.
Fig 2.4: Discetion of the Mandelbrot Fractal. (Mandelbrot B. 1983)
11
2.4 How to generate a Julia Fractal Set
Fig 2.5 Examples of a simple and a more complex Julia set. (cite needed for picture)
We put in a complex number z, then multiply it by itself , and then add another complex number c to the result. The c is a complex constant, that is, a number that does not change throughout the entire fractal calculation. [4]
Let z = 2 + 3i and c = 4 + 5i.f(z) = z2 + c =[(2 + 3i) · (2 + 3i)] + (4 + 5i) =[(2 · 2) + (2 · 3i) + (3i · 2) + (3i · 3i)] + (4 + 5i) =[4 + 6i + 6i + (-9)] + (4 + 5i) =(-5 + 12i) + (4 + 5i) =-1 + 17iSo f(z) = f(2 + 3i) = (2 + 3i)2 + (4 + 5i) = -1 + 17i.
The next thing we need to do is to zind out the size or norm of a complex number.
The norm of a real number is just the number itself. For example the norm of 7 is 7.
To zind out the norm of a complex number is a bit differently:
z x yi (x 2 y 2)
As an example we have:
3 4i (32 ( 4)2 (916) (25) 5
With this knowledge in mind we can understand now how a computer program handles a function with complex numbers to output a Julia set in a cartesian coordinate system:
1.) We have to determine the size of the cartesian coordinate system on the computer screen. For example: Resolution: 900 x 900 = 81’000 Pixels
2.) We set up the computer to use the mandelbrot formula f(z) = z2 + c. Then we
12
input a complex constant value for c. For example: c = 4 + 5i
3.) The computer starts calculating in the upper left corner of the coordinate system. In this example it would be (‐900,900). The computer program is set up to interpret these values as (‐900,‐900). These two values are then be used as x and y in the function.
4.) The computer calculates the function and uses it’s result to calculate its norm.
5.) If this norm is smaller then the parameter of the coordinate system, then the resulting x and y values of the function become the new input value for the same function. This iteration will be repeated until the norm is bigger then the parameter of the coordinate system. There is also an inbuilt limitation of 256 iterations otherwise some functions run in a loop forever.
6.) Once the norm is bigger than the parameter of the coordinate system the calculation stops and records the amount of iterations that occurred. Then the computer program goes to the next pixel in the cartesian coordinate system and returns to step 4.).
7.) The number of iterations tells the computer what color this very zirst pixel has. For example the iteration would be 55 which could mean that the color of this pixel is red. The next pixel has an iteration of 54 which could mean that it has a slightly lighter shade of red. etc. [5]
8.) Step 3.) to 7.) are repeated 810‘000 times for all the pixel on this particular coordinate system. If we multiply the amount of maximum iteration that could occur to the amount of pixel then we can see how it can take up to 900 x 900 x 256 = 207‘360‘000 calculations for each fractal image. Also take into consideration that some of the complex numbers are much more complicated to calculate than the one shown above.
13
3. Natural Fractals and Artistic Forgeries of Nature
“the more things change, the more they stay the same”
‐Glenn Elert, Mathmatics in the Digital age, 1995‐1998.
Fractal based visualisation was not realised in a complex iteration prior to the invention of the computer.
Benoit B. Mandelbrot (who worked for IBM) in the 1970’s argued against Euclid geometric laws and theories, which ignore “formless” shapes such as coastlines, mountains and clouds. As with any progression of knowledge and the increasing contribution to the ‘global intelligence’, Mandelbrot’s ideas separated from the existing Euclid theory – Euclid of course, separating from Newton theory before that and herein lies a great irony which is; progression of ideas can be seen to have a selfsimilar nature and are in fact, fractal in nature.
Fractals in nature have a different story in the context of this report. The continued discussion will visit a brief history of why fractals are a more accurate measure for natural occurrences (such as coastlines, leaf perimeters and snowzlakes), provide examples of the theory and describe the social context and continued outcomes of the discovery of fractals in nature.
3.1 Briefly on Euclidean
Euclidean geometry described all natural shapes as “cones, sphere or cylinders”. Any object with did not fulzil such a dezinition was coined as “noisy Euclidean geometry” and such were parasitic forms.
When one observes natural shapes such as coastlines, mountains and even snowtlakes it becomes obvious that nature cannot be included with the Euclidean dezinition.
At this point, Fractal geometry takes centre stage in dezining and exploring such shapes. Giving them dezinition, theoretical duplication and allowing for such visuals as “the forgery of nature” (Mandelbrot, 1983)
14
3.2 Natural Fractals and dimensions
The Koch snowzlake illustrated the true nature of fractals and could also present one of the early representations of a “forgery of nature” (Mandelbrot, 1983)in the form of a snowzlake.
Fig 3.1 Four iterations of a Koch curve to create the Koch Snowtlake (GNU License).
Representation on the Lindenmayer System is derived from the following:
F F F F FF = forward one. –F = turn 60 left. +F = turn 60 right.
3.3 Recognition of self-similarity in nature
Mandelbrot discovered that coastlines viewed from aerial photos say 1 km in radius, presented similar shapes when a level of magnizication was applied to the same image – lets say 0.5 km in scale.
Applying another level of magnizication, the same result was evident and proving a level of self‐similarity was present in coastlines.
15
3.3.1 On measurements of coastlines
In a basic form, to measure a coastline one would start with two or more point at two heads of the coastline and join with a straight line. We would then take a smaller scale and repeat the process – therefore gathering a more accurate depiction, and repeat the process again.
Fig 3.2: Example of scaling measurements of the university(Yale University , 2009)It should be quickly deduced that this method yields two fundamental issues:
1. The reduction of scale can occur a theoretical inzinite amount of times – making the length of the coastline inzinite in theory.
2. Measuring at such scales in practically impossible.
3.3.2 On Rivers and mountains
Fractals are evident within satellite images of rivers and channels of a continent or land mass. Magnizication on the whole will reveal another self‐similarity within rivers themselves as displayed below.
Fig 3.3 self similarity of a river bed, magnitied to show self (Yale University , 2009)
16
3.3.3 Visual examples of other fractals found in nature
Fig 3.4 Other fractals found in nature (Yale University , 2009)
3.4 Before the computer
Although simple fractals where found historically throughout many ages in art, religion and nature their visual conception circa 1970‐1980 in a complicated way extending past say 10+ iterations. To appreciate this statement, one would only have to consider drawing zigure 1. – the Koch snowzlake up to 10 iterations are realise the time involved and the level of intricacy extends past that of even the more patient human.
Throughout the report Mandelbrot and his fractal set has been mentioned. Some 256 iterations were required to create the complex set, of which would be the life work of one person had this been undertaken by hand.
However, simple fractals are witnessed throughout history in different places and it would be prudent to mention these places by visual reference only.
Fig 3.5 Castel del Monte was the hunting seat of the Hohenstaufer Emperor Friedrich II in Apulia 1240 1250 (SALA, 2000).3.5 Computer Art
The modern computer overcame the difziculty with iteration and with powerful graphics engines, more and more interesting forms of fractals were able to take place.
17
Artists, users and mathematicians were able to now visually realise the inzinite pleasure and beauty of self‐similarity and self‐iteration on a large scale.
Like a well‐known mathematical equation, viewing a fractal shares an elegant and innately understood beauty. As well as being beautiful, they represented the visual state of inzinity.
Left Fig 3.6: “The Mandelbrot Set” – B. Mandelbrot, 1987.Right Fig 3.7 : “Composition with Black, Red, Grey, Yellow and Blue” – Piet Mondrian.
3.6 Using computers to forge nature
Evidence of fractals in nature should be clearly illustrated up to this point and paint a clear picture of their applications and more importantly the notion of intinite dimensions.
To recursively end on the zirst point made in this section, fractals are natural and nature exists within fractals.
Computers were initially used to visualise the vast history of fractal art and compute it at a higher iteration rate to ultimately create something, which exists in nature.
As nature itself started the discover and research into fractals, it seems only fair to illustrate how well the theory of Mandelbrot and his predecessors were by the following images created through computation means, but still fractal based.
Fig 3.8: 3D renditions of fractal geometry produced by wiki username: Solkoll.
18
4. Practical Applications of Fractal Geometry
Fig 4.1 ‐ Sydney Harris, Fractal Cartoon.
This part of the report attempts to show the impact of fractals and the practical applications that have emerged as a direct or indirect result of fractal geometry.
Nowadays barriers among math, science, art, and culture are increasingly being diminished due to advances in fractal geometry.
4.1 Fractal Image Compression
The use of fractals in compressing images originates from the basic notion that in certain images parts of the images resemble other parts of the same image. Fractal algorithms convert these parts, into geometric shapes into mathematical data called "fractal codes" which are used to recreate the encoded image. In the book the fractal geometry on Nature Benoit Mandelbrot put forward a theory that traditional geometry with its straight lines and smooth surfaces does not resemble the geometry of trees and clouds and mountains .It is no coincidence that Fractal Image Compression is best suited for nature images.
In 1987 Michael Barnsley author of the book Fractal Everywhere and Alan Sloan formed a
19
company Iterated Systems which went on to receive patents on Fractal Image Compressing.
In the 1997 Arthur Clarke documentary “Fractals- The Color of Infinity ” Barnsley describes the notion of fractal image compression to have come about from what he called the collage theorem.
The basic idea of the theorem was to cover an arbitrary picture with tiles of smaller copies of its self to form a collage, then tell the computer to look at the picture and automatically find the fractal formula of the picture then you could turn it into an entity of infinite resolution.
“If you took a fern and covered it with little ferns then you would have created a formula for a fern” . Michael BarnsleyThe fractals used in the image compression system are iterated functions .
In commercial application Microsoft has used fractal image compression in the Encarta Mutimedia encyclopedia.
As this is heavily patented technology one would argue that perhaps this has hindered the development of this technology and the use and development by other commercial enterprises.
4.2 Fractal Use in Visualization and Simulated Terrain generation
Using geometric primitives such as lines, curves ,rectangles and polygons CAD software succeeded in creating graphic representation and illustrations of man made structures and objects . This success however could not be replicated when it came to representing natural objects such as mountains, clouds, rivers and coastline.
In 1975 Richard F Voss created a computer generated graphic illustration of realistic looking mountains using fractal iterations. This was a breakthrough in the world of computer graphics.
Fig 4.2 Voss succeeded imitating the roughness of a mountain landscape. ‐ Richard F. Voss. Mountains Illustration.
With advancements in computer technology bringing the development of super computers like the IBM Blue Gene capable of reaching speeds of petaFlops, computer
20
graphics have become more and more realistic. User requirements have driven developments in the zield of simulated computer graphics, Military, Film Industry, Gaming Industry.
4.3 Fractals in Hollywood
When Loren Carpenter ‐A Graphic Image researcher presented a two minute animated zilm in 1980 showing a complicated landscape called Von Libre Hollywood immediately realized the potential of the technic ,he had used fractal geometry to create the intricate mountain ranges and valleys in the animation. Carpenter was called to Lucas Films to take a leading role in the preparation of Star Trek 2 – The Wrath of Khan. “The tilm is credited as having the tirst feature tilm sequence created entirely with computer graphics” Loren Carpenter went on to be the cofounder of Pixar Animation Studios.
Fractals graphic images of simulated terrain and landscapes and worlds appear in most Hollywood blockbusters utilizing computer graphics and special effects as well as entire productions created and completed on the computer .
The technology has revolutionized the zilm as well as the gaming industries. More and more games are being released with realistic graphics of natural objects like mountains clouds and rivers.
21
4.4 Fractal Art and Fashion.
The beauty of nature has inspired many fashion designers and artists. It is not uncommon to zind fractal inspired materials on todays fashion runways.
Fig 4.3: Mandelbrot Fractal Art Shoe by Aquavel
4.5 Emerging Fractal Based Technologies
Questions are being asked on the possibilities of creating new materials based on fractal geometry. Countless aesthetic products are inspired by nature's properties such as roughness ,Since roughness is dezined by fractal geometry, Questions are asked wether there are possibilities for new manufacturing or assembling techniques using fractals.
“To optimize the structure and properties of alloys, it is necessary to take into account the effect of the selforganization of a dissipative structure with fractal properties at load. This requires the development of selforganizing technologies for material production. Fractal material science takes into account the relation between the
parameters of fractal structures and the dissipative properties of alloys. It also takes into account the base properties of highly nonequilibrium systems and the self
organizing process of the fractal structure in bifurcation points.”
V.S. Ivanova, I.J. Bunin, and V.I. Nosenko, Fractal material science: A new direction in materials science.
22
5. Fractals in medicine
Fractals recently become a large area of research in the zield of biology and medicine. The realization of fractal patterns inside the liver and lungs has resulted in the ability to reproduce certain organs like a rats liver. By adapting the use of 3D printing to print out cell structures offers the human race an extended form of life.
5.1 How are fractals involved in the process in which certain cells form together to create organs?
The term used to describe a number of cells (primordium) that becomes a complete functional organ is called organogenesis. This requires a precisely timed sequence of events to allow the successful generation of a working organ. Each organ has its own mechanism (method of construction) which is applied to different individuals of the same species. Organogenesis initially consists of rapid expansion of immature functional parts (parenchyma) of an organ, differentiating into the expansion of functional mature tissue (stroma) (Castro, 2006). A good example in which fractal patterns appear in an organ is the bronchi in our lungs. These fractals follow similar principles to a trees trunk ‐ branch ‐ twig pattern in order to cover a large surface area to absorb more oxygen into the blood stream. The bronchial tree generates an optimal oxygen reservoir at minimal energy dissipation.
5.2 Technology and Maths used for understanding Fractals in Biology
As technology has adapted so has our understanding of our biological engineering. Studies using a light microscopy had measured the internal surface area of an adult human lung at 60‐80 m2 whereas using a electron microscope the results showed the estimate to be around 130 m2 (Weibel, 2005). The human lung is a fractal structure that contains 23 generations of branches (this is a very high number of branching generations because most deciduous trees have only 7 or 8). The self similar replication of the diameter of the lungs airway vessels takes on a parent daughter design relationship of bifurcation (d0, d1, and d2). The equation reads as follows d0x = d1x + d 2 x exhibiting the property of self organization. The bifurcation exponent, x, is said to be a specizic design because its value rezlects energy dissipation during transport. (Bennett).
23
www.stat.rice.edu/~riedi/UCDavisHemoglobin/fractal3.pdf
It is believed that this process is genetically specizied. But, it is hard to grasp that the one hundred trillion cells in a human can be coded in about 100,000 genes. All these cells can then be broken down to the trillions of trillions of fractals that make up cellular membranes. It is argued that during organogenesis that not all the cells might require a method of construction or that this method only applies at certain stages of development (Weibel, 2005). But, this contradicts how fractals self replicate. Fractals are always changing in either scale, size, rate or rotation. This is represented all throughout nature e.g. fern leaves get smaller in scale as they near the tip of the branch.
http://en.wikipedia.org/wiki/File:Thorax_Lung_3d_(2).Jpg http://en.wikipedia.org/wiki/File:H_fractal2.png
Above (left) shows the large amount of bronchi and airways that till the lung. (right) is the Koch tree model, this is very similar to the human airway tree
"Using [the] fractal concept will make it easier to mimic... nature and also to scale up our designs from one animal to another." (KaazempurMofrad.) (Patch)
"In order to make living replacements for large vital organs such as the liver and kidney, it is essential to integrate the creation of vasculature with the tissue engineering," “And the growth of these vascular networks has to be highly controlled and precise” said KaazempurMofrad. (Patch)
24
“…in 1977, morphometric studies on liver cells presented controversial results because the surface area of the endomplasmic reticulum membrane, (the sight of drug metabolism and protein synthesis had been estimated and 6m2/cm3 by Loud, whereas we (Weibel ER) had obtained a value of 11 m2/cm3” both methods used were the same but Weibels was used at 90,000 X magnitication on the electron microscope whereas Loud used a smaller magnitication. (Weibel, 2005)
5.3 Fractals use in Cellular structures
The compact design of a well functioning organism (with all its complexity) depends on a high density of internal membrane systems. The internal membrane systems are made up of small fractal rectangular shapes and play the role in which chemical reactions and transfer processes take place in a precise manner. These membrane systems constitute 'space zilling' geometries and are thus agreeable to fractal analysis. These substances must then be transferred between different organs and cells which requires a distribution network of blood vessel and airways that also has to be 'space zilling'. Fractal concepts now lead me to believe that they are everywhere and help us to understand underlying construction principles.
5.4 Technological advances involving Fractals in Biology
In 2006 a new type of technology called 'bioprinting' was developed by Gabor Forgacs at the University of Missouri in Columbia. This technique uses droplets of 'bioink' (clumps of cells a couple of micrometers in diameter). Forgacs has found this 'bioink' behaves just like a liquid. When the 'bioink' is applied the cells fuse together and form a layer. When the 'bioink' layer is set an alternate layer of 'biopaper' is applied and a structure can be formed. This can result in printing out any desired structure. e.g. For blood vessels successive rings of muscle and endolethlial cells are laid down on top of each‐other to create a tube like blood vessel. (Ruis, 2006)
25
http://www.musc.edu/bioprinting/assets/images/bioprinting02.jpg
This type of technology could provide transplant patients with new organs without worrying about their body rejecting them. This is because the printed organs are constructed out of their own stem cells which are recognized and accepted by the patients body. This would minimize the waiting list for an organ donor as well as giving the patient a less stressful operation and recovery. The discovery of fractals in biology offers us unique insights into how our cells, organs, veins, blood vessels etc. are constructed. We can now understand the process behind our biological evolution in the inzinite aspect of fractal patterns.
26
Conclusion
Our collective zindings suggest to us that the Fractal exists as an identiziable pattern and is a multiplex process. Within it, is consistent demonstration of properties pertaining to interrelationships of space, where voids cannot be separated from importance to their surrounding. This conforms to our examination of the fractal where not only division of space occurs but it also appears dezinitively recurrent.
From this we conclude that the signizicance of the appearance of such an element (whether astutely aware of it or not) is clear indication that the fractal is a transfer process enabling economy and efKiciency of space. Often this is perceived as expansion or contraction of changeable structure. This successfully allows interconnection of multiple structures, which in turn potentiates activity for (animate) growth or fundamental change of overall structure as a whole. Quite literally ‐ the Fractal makes sense.
On an evolutionary scale, it would be prudent not to consider its total absence.
27
Bibliography
Bennett, S. H. (n.d.). (Department Of Pediatrics) From http://www.stat.rice.edu/~riedi/UCDavisHemoglobin/fractal.html
Castro, L. N. (2006). Google books. (L. Taylor & Francis Group, Producer) From https://docs.google.com/Doc?docid=0AVL90cL7sKJuZGhmc2ZnOGpfMGM4NTVobmhm&hl=en
Elsen, A. E. (1981). 'PURPOSES of ART' . Harcourt Brace College Publishers.
Emmer, M. (1993). The Visual Mind Art and Mathematics. Cambridge, Mass. : MIT Press.
FRACTAL MODELS IN ARCHITECTURE: A CASE OF STUDY, CH‐ 6850 (2000).
M.Fowler, R. N. (1996). 'SPACE, STRUCTURE and FORM' . Brown & Benchmark Publishers.
Mandelbrot, B. B. (1978). The Geometry of Nature.
Mandelbrot, B. (1983). The fractal geometry of nature. New York: W. H Freeman and Company.
Patch, K. (n.d.). Fractals support growing organs. From ‐1 http://www.trnmag.com/Stories/2003/073003/Fractals_support_growing_organs_073003.html
Ruis, J. J. (2006, December 29). From http://www.fractal.org/Fractalary/Fractalary.htm
Salingaros, V. P. (2000). ECOLOGY and the FRACTAL MIND in the NEW ARCHITECTURE: a Conversation' by Victor Padrón and Nikos A. Salingaros (This conversation was published electronically by RUDI Resource for Urban Design Information . From http://zeta.math.utsa.edu/~yxk833/Ecology.html
Weibel, E. R. (2005). Fractals In Biology and Medicine Volume IV. Birkhauser Verlag, Basel ‐ Boston ‐ Berlin.
Yale University. (2009). Introduction to Fractals. Retrieved October 2009 from http://classes.yale.edu/fractals/
28
Recommended