fractals in architecture

FRACTALS IN ARCHITECTURE - Jinu Louishidha Kitchley “A bee puts to shame many an architect in the construction of her cells….” - Karl Marx, Das Capital. Architecture, more than any art form, enjoys the privilege of spatiality and has to address human perception and the senses, if it has to be successful. Christopher Alexander in his book “Nature of Order” has underlined the deep link between building and man as “The structure of life I have described in buildings is deeply and inextricably connected with the human person and with the innermost nature of human feeling” - An excerpt from Book 4 “The Luminous Ground”, by Christopher Alexander (1963) Architects, who have responded to this instinct and who have gone beyond structural constraints and catered to the emotional needs of the user have achieved more than the creation of mere shelters. Architecture basically, involves the organization of spaces, definition of shelters and the creation of order. Order may be defined as the sequence or arrangement of things or events subject to rules or laws (Webster’s). Throughout history order in architecture has been subjected to rules of structure and stasis. As technology encroached the realms of building, we find that order has been interpreted as straight lines, symmetry and Euclidian geometry. This in due course resulted in the creation of massive buildings, oblivious of details and human connectivity. Fortunately, contemporary architects have started questioning these rules and the very nature of order and have started rethinking on the metaphysical aspects of building. 1.0

NATURE AND ARCHITECTURE For centuries Nature has been serving as an undying source for research and inspiration. Architects and Designers have probed vehemently into the nature of natural forms and organisms to identify and understand the great concepts of the master designer. Nature, with all its complex and diverse form is bounded by certain organizational and mathematical relationships that condition and control all organisms. The organizational laws of nature govern not only nature’s physical forms but also their behaviour. Human beings being a part of this biological cosmos are also subjected to such organizational and survival rules. When mathematically inharmonious, unorganized and proportion less forms are imposed against this basic instinct it triggers a negative psychological response. In order to cope up and overcome this negative feeling, settlements and buildings creatively dispense order and hierarchy. Fractals are one of the outstanding examples of such a response. The formulation of chaos theory * occurred as an eye opener to many in the fields of science and applied science. The introduction of this theory to investigate and evaluate art and design gained wide spread prominence in the 1980s. The complexities of order and geometry started gaining their importance as against earlier designs that reduced inherent complexity of nature to simple shapes and analogies. As a result, interest in the area of Fractal *

Chaos theory. (Simple deterministic systems can give rise to unpredictable behaviour)

Geometry has increased. This paper attempts to identify and demonstrate the applications of fractals in architecture. 2.0

FRACTAL GEOMETRY Fractal geometry “is the study of mathematical shapes that display a cascade of never ending, self-similar, meandering details as one observes them more closely.”i Self-similarity “is a phenomenon where repeated elements change in scale but retain a similar shape. The most common examples of fractals are the fern leaf, clouds, coastline, branching of a tree, branching of blood vessels, etc., (Fig 1)

A Fern leaf

Clouds

Coast Line

(Fig 1) The great mathematician Benoit Mandelbrot’s book “ Fractals: Form, Chance and Dimension”,(1977) marked the beginning of the probe into the field of Fractal Geometry. It was Mandelbrot who proposed the idea of a Fractal in 1975. He defined a fractal to be “any curve or surface that is independent of scale.” This property referred to as self-similarity, means that any portion of the curve if blown up in scale would appear identical to the whole curve. iiThen the transition from one scale to another can be represented as iterations of a scale process. Prior to Mandelbrot there were a few contributions done to this field by lots of other renowned mathematicians and scientists, but they remained scattered. Some of the theories are chronologically listed below to give an idea of the delighted interest mathematicians showed towards the complex nature of fractals… Cantor’s Comb (1872): b Georg Cantor (1845-1918) evolved his fractal from the theory of sets. All the real numbers in the interval [0, 1] of the real line is considered. The interval (1/3, 2/3) which constitutes the central third of the original interval is extracted, leaving the two closed (0,1/3), and (2/3. 1). This process of extracting the central third of any interval that remains is continued ad infinitum. (Fig 2) The infinite series corresponding to the length of the extracted sections form a simple geometric progression… [ 1 + (2/3) + (2/3) 2 + (2/3) 3 + …..] / 3 This shows that this sums to unity, meaning that the points remaining in the Cantor set, although infinite in number, are crammed into a total length of magnitude zero 2.1

Helge von Koch’s Curve (1904): b The curve generated by Helge von Koch (1870 – 1924) in 1904 is one of the classical fractal objects. (Fig 3) The curve is constructed from a line segment of unit length whose central third is extracted and replaced with two lines of length 1/3 . This process is continued, with the protrusion of the replacement always on the same side of the curve, to get the Koch’s curve. 2.2

Sierpinski’s Triangle (1915) b Sierpinski considered a triangle whose mid points where joined and the triangle thus formed extracted. . (Fig 4)The same process is repeated on the resulting triangles also. When this is repeated ad infinitum we get the Sierpinski’s Triangle, which is a good example of a fractal. 2.3

Gaston Julia Sets (1917) iii Fractals generated from theories of Gaston Julia (1892 – 1978) are based on the complex plane. They are actually a kind of graph on the complex axes, where the x-axis represents the real part and the y- axis represents the imaginary part of the complex number. For each complex number in the plane, a function is performed on that number, and the absolute value of the range is checked. If the result is within a certain range, then the function is performed on it and a new result is checked in a process called iteration. (Fig 5)

2.4

Cantors comb (Fig 2)

Koch’s curve (Fig 3)

Sierpinski’s Triangle, (Fig 4)

Julia Sets (Fig 5) 3.0 FRACTALS AND ARCHITECTURE Fractals presence in architecture is not a recent phenomenon nor as popularly believed a post-modern response. It has been realized and expressed in many traditional settlements and architecture. An ancient African settlement ( Ba –lla)iv Most of the ancient African settlements exhibit fractal characteristics. The European settlers found these complicated fractal arrangements as “primitive” when compared to their Euclidean geometry. These intuitive choices of fractal geometry by the ancient settlers show the relevance of fractals with respect to habitat building. “Ba – lla” a settlement of southern Zambia is a good example of fractal building. (Fig 6)Here they practiced an extended family system, which was housed around a ring shaped livestock pen. The pen had a gate at the front and storage houses around it. The buildings became progressively bigger around the ring. The largest house (the father’s house) is found opposite to the gate. A definite status gradient is thus established. The entire settlement is also a ring that constitutes of smaller housing units as described above. The front of the settlement is the gate and opposite to the gate (at the peak of the hierarchy scale) is the largest housing unit, which is the chief’s house. 3.1

“Ba – lla” a settlement of southern Zambia, (Fig 6)

Temple Architecture of South East Asia v A fractal character can be easily identified in most South Asian Temples. For example, in India the temple towers consist generally of a group of smaller towers surrounding a tower and diminishing in scale as it moves up. This arrangement exhibits a strong fractal character. In India there is a vast difference between the North Indian and the South Indian Temple architecture. Yet the fractal structure is common to both. The North Indian Temples exhibited the Nagara style. Quoting Anthony Batchelor c “The Nagara style which developed during the fifth century is characterized by a beehive shaped tower (called a shikhara, in northern terminology) made up of layer upon layer of architectural elements such as kapotas and gavaksas, all topped by a large round cushion-like element called an amalaka”. (Fig 7)

3.2

(Fig 7),North Indian temple.

The South Indian Temples are characterized by the Dravida style. “From the seventh century the Dravida or southern style has a pyramid shaped tower consisting of progressively smaller storeys of small pavilions, a narrow throat, and a dome on the top called a shikhara (in southern terminology). The repeated storeys give a horizontal visual thrust to the southern style.” (Fig 8) Anthony Batchelor c (Fig 8) South Indian Temple. Cathedrals in Europe vi “ Renaissance and Baroque architecture, especially as expressed in cathedrals, frequently exhibited scaling over several levels.” d The Cathedrals of Europe achieved dynamism by the use of fractal concept in their massing. Their spires were surrounded by spires, which were in turn surrounded by spires. (Fig 9) Also some of the arched windows are made of two arches, each of which is made of two still smaller arches. (Fig 10).

3.3

European Cathedrals, (Fig 9) ,(Fig 10)

4.0 4.1

CONTEMPORARY USAGE Fractals and Design Fractals with its inherent complexity and rhythmic characteristics have also inspired many contemporary architectural design processes. 4.1.1 Peter Eisenmann vii Peter Eisenman exhibited his House 11a for the first time in 1978. He adopted a philosophical process of fractal scaling constituting of three destabilizing concepts of: “discontinuity, which confronts the metaphysics of presence, recursivity, which confronts origin; and self similarity, which confronts representation and the aesthetic object. viii

“House 11a, a composition of Eisenman’s then signature “L”, combines forms in complex rotational and vertical symmetries. The three dimensional variation is a cubic octant removed from a cubic whole, rendering the "L" in three dimensions. Each "L" according to Eisenman represents an inherently unstable geometry; a form that oscillates between more stable, or whole, geometric figures. The eroded holes of two primal "L"s collide in House 11a to produce a deliberately scale-less object, which could be generated at whatever size was desired.ix (Fig 11) House 11a,(Fig 11) 4.1.2. Zvi Hecker x Hecker, born in Poland, moved to Israel in 1950. Zvi Hecker’s work comes out of brutalism of the fifties. He stresses the idea of a building and its integration with surroundings. As a metaphor for his buildings Zvi Hecker uses the crystalline geometry of nature. Fractal geometry serves as a key diagramming concept in his many designs. In his Extension for the University of Applied Arts, Hamburg (Fig 12) he has achieved an architectural ensemble of great complexity, unified by the scale and the formal interrelations of its parts. University of Applied Arts Hamburg –Zvi Hecker, (Fig 12) Further in his Ramat Hasharon, City Centre (Tel Aviv, 1986-1995) project, Hecker attempts to provide a small city with identity. The fractal concept of a sunflower was incorporated in the planning of the cluster of building courtyards and pathways and thus identity was achieved at every level. (Fig 13)

The Sunflower of Ramat Hasharon( Tel Aviv), City Centre – Zvi Hecker, (Fig 13) 4.1.3. Ashton Ragatt McDougall xi The group ARM a Melbourne office creates symbolic programmes with fractal expressions. Storey Hall in Melbourne, which is a part of the urbanized campus of RMIT, stands apart from its neighbours by the use of green and purple fractals (with yellow and silver highlights).(Fig 14). The façade has a concrete surface with a huge bright green and purple doorway. The concrete surface has evident Penrose pattern of bronze fractals dancing over and it. The bronze fractal panels are articulated further by a green linear meander and the surface squiggles like folds in a rock or skin. Inside the building also the theme is elaborated – and the metaphor of an undulating skin is maintained. Storey Hall , Melbourne, (Fig 14) The Penrose tiling pattern, which connects façade, floor, walls and ceiling into a single ornamental system, has a five-fold symmetry. (Fig 15) It consists of a fat and thin rhombus which when arranged never results in a cyclic pattern. This unusual self-organizing order is called Quasicrystal and is found to exist in Nature. Quasicrystals, with an inherent fractal self similarity give a higher degree of complexity to repetition and rhythm. c (Fig 15) Penrose Tiling Pattern 4.1.4. Ushida Findlay xii Ushida Findlay Partnership was founded in 1987 by Eisaku Ushida, a Japanese graduate of the University of Tokyo (1976) and Scottish Kathryn Findlay. They were both former collaborators of Arata Isozaki (between 1976 and 1982). Their architecture explores components like symbolic systems, psychoanalysis, and “psycho-geography" and is based on pure scientific and

geometrical research of the form. For Ushida & Findlay, this turning of fundamental sciences, to the side of the geometry of chaos and non-linear mathematics, is the means of finding a real autonomy of the architectural object. Eisaku Ushida and Kathiryn Findlay in their S project (a transterminal at Tokyo) have used fractals in a different plane. (Fig 16) They seek a fractal urban order. c They superimposed the psychological geography of each of the user on to another, to form an incrementally enormous power, which is the source motivation in the shaping of the city. Ushida and Findlay’s S project (Fig 16) 5.

Frank.O.Gehry xiii Frank. O. Gehry’s Guggenheim Museum at Bilbao has a strong organizing metaphor and a diversity of form and colour. Smooth continuous forms in steel and limestone flow towards a center point where it erupts into a robust flower with its petals blowing in the wind. This 26 self-similar petals, which are dressed in steel bends upwards with, curved glass.(Fig 17)This composition exhibits a seamless continuity, organic flexibility and a fractal fluidity that captures the landscape with great power. c

Frank. O. Gehry’s Guggenheim Museum, Bilbao (Fig 17) The contemporary usage of fractals in architecture has resulted due to a range of varied concerns. One of the concerns is the organic metaphors of design as used by Zvi Hecker and Frank Gehry. Another is an attempt to get closer to nature and its fractal language as ARM and Ushida Findlay. 4.2 Fractals as Analytical tools Fractals are very popular in the researches conducted in fields, having characteristics of rhythm, scale and progression of details (like Music, Physics etc.,). a Architecture and Design, also concerned with the control of rhythm and scale can benefit from the use of this relatively new mathematical tool. Architectural composition is concerned with the progression of interesting forms from the distant view of the façade to the intimate details. This progression is necessary to maintain interest. This is a fractal concept. 1 Carl Bovil in his book “Fractal Geometry in Architecture and Design” observes that in both architecture criticism and design, fractal geometry provides a quantifiable calibration tool for the mixture of order and surprise.

He has identified two ways through which the fractal concepts may be incorporated in architecture and design – Fractal Dimension and Fractal Distribution. Fractal Dimension is taken up for discussion in this paper. 4.2.1 Fractal Dimension Consider a point. It has no dimension, because it has no length, no width and no height. A line has one dimension, because it has no width and no height, but infinite length. Similarly a plane has two dimensions and a cube has three dimensions. But Fractals can have fractional dimension.(Like 1.4, 2.1, etc.) Fractal Dimensions may be better understood with the help of mathematical equations. Consider a line. If we multiply the length of a line by 2 we get 2 lines. But when we multiply the length and width of a square by 2, we get 4 copies of the original. Similarly in a cube, we will get 8 copies. (Fig 18) It is apparent from this that when we double the sides and get a similar figure, we can write the number of copies as a power of 2 and the exponent will be the dimension. i.e., if dimension is d then the number of copies or the magnification factor n = 2 d. Here the number of self-similar pieces is 2. Therefore we can say that Dimension Magnification factor = (Number of Self similar Pieces)

Dimensions (Fig 18) xiv

To understand a fractal or non-integer Dimension, consider a Koch curve, which is a continuous one-dimensional line, (Fig 19) divided into three equal segments .The middle segment is replaced by the two sides of an equilateral triangle of the same length as the segment being removed. This is repeated for each line segment. It will be seen that the one-dimensional line is filling a 2 dimensional space. This means that the fractal dimension of such a line must lie in between 1 and 2.

(Fig 19)xv log( number of self-similar pieces) Fractal Dimension = A Koch curve.

log ( magnification factor)

- (1)

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The first iteration for the Koch curve consists of taking four copies of the original line segment, each scaled by r = 1/3.Therefore, log (4) = 1.262(which is a non-integer) - (2) Fractal Dimension = log (3)

Therefore, it is clear that a fractal dimension of any form gives us the measure of the progression of details. As the fractal dimension increases there is a higher degree of detail in the form. xvii There are various methods by which fractal dimensions may be measured. But the most graphical method for the approximate calculation of the fractal dimension is the box (grid) counting method. “The box counting method may be less suited to the task of hugging the more intricate details of the base curve but, because of its low computer processing requirements; it is recommended as a method suitable for yielding a first approximation of the fractal dimension. This method of calculation of Fractal Dimension of buildings, is popular and easy to calculate the fractal dimension of buildings and involves the following steps: xvii Step 1: Take the elevation of the building to be considered and place a grid (of grid size S1) over the top of it. Step 2: Count the number (C1) of occupied grids i.e., the grids that have a line in them. Step 3: Now double the grid size (S2) and count the number of occupied grids (C2). Repeat the process and tabulate the results. Slight variation in the grid can result in different values for C. Grid Size S1 S2 S3 -

Box Count C1 C2 C3

Step 4: The log-log plot (resolution scale Vs number of occupied boxes) is used to determine the fractal dimension D across scales 2 to 1. D (2 to 1) = (log C2- log C1) / (log S2 – log S1) xvii This is demonstrated with an example below…

State Bank, Egmore, Chennai

c

Grid Size, S1 = 21,Box Count, C1 = 122

State Bank, Egmore, Chennai c Grid Size, S2 = 42, Box Count, C2 = 418 Fractal Dimension, D (1 to 2) = (log C2- log C1) / (log S2 – log S1) = (log 418- log 122) / (log 42 – log 21) = 1.78

A modern building

Grid Size, S1 = 21,Box Count, C1 = 119

Grid Size, S2 = 42, Box Count, C2 = 270 Fractal Dimension, D (1 to 2) = (log C2- log C1) / (log S2 – log S1) = (log 270- log 119) / (log 42 – log 21) = 1.18 The fractal dimension thus calculated measures quantitatively the quality of a façade’s geometrical features and its amount of intricacy and irregularity. In the above example, the first building has a higher fractal value than the later. This is because it possesses greater details. As the amount of detailing increases, the visual experience is enhanced and the human connectivity to the building increases. It is generally noticed that a fractal dimension of 1 to 2 is exhibited in buildings. The result of this exercise might look like a complicated way of deciphering a simple, visually comprehended fact. But, what is suggested here is that a mathematical quantifiable measure may be established on the basis of a fractal dimension to a factor that is totally qualitative. This might serve two immediate purposes. One that is purely pedagogic. The other is to serve as a possible basis for a discourse on human perception, behavior and Cyber Architecture .However, this area of application of fractals is yet to be fully investigated. CONCLUSION Fractals have not gained much importance outside the academic environment. May be the limited resources of architectural practice could be a reason. Architecture has, for long, been dwelling on simple mathematics in arriving at its analytical and proportioning tools (ex. golden sections). The inherent repression towards mathematics by most architects has rarely allowed the designer to venture into the complexities of Mathematics. Fractal geometry though may seem relatively complex is based on simple geometry and comprehendible structure. With the advent of digital technology in almost all phases of designing, the complexity of fractal geometry may be negotiated easily. Architects produce a representational model of a building and not a building as such. “Architectural concepts only exist fully in their realization as discoveries through the non linear process of designing.” This situation has led

to a lot of inquiries about the process rather than that of the product. This quest has hitherto used building typology as a prime tool in design process. More benefits accrue when typology as a design method incorporates dynamic process that gives rise to infinitely variable generative suggestions. Diagram is one such dynamic process. Diagram is a generative device which unfolds/opens up new design possibilities. In this context, fractals serve as an effective and useful generative diagram. Fractals when used in a generative process “intensifies the interpretive demands on the architect by connecting the building – as- diagram to ever more complex conditions and requirements”. A wholly adaptive, flexible architectural solution is possible with such an intensive organic design process. * I thank Dr. Srivatsan for the valuable inputs on the earlier draft. Reference Websites: i http://www.inform.umd.edu/EdRes/Colleges/ARCH/Newsletter/fractal.html, Carl Bovill, Fractal ii iii iv v vi vii viii

ix x xi xii

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Geometry in Architecture. http://life.csu.edu.au/complex/tutorials/tutorial3.html http://stu.cofc.edu/~spwood/definition.html http://classes.yale.edu/math190a/Fractals/Panorama/Architecture/AfricanArch.html http://www.templenet.com/temparc.html http://classes.yale.edu/math190a/Fractals/Panorama/Architecture/EuropArch/EuropArch.html http://www.nexusjournal.com/Rossi-eng.html Michael J Ostwald ‘Fractal Architecture”, Late Twentieth Century Connections between Architecture and Fractal Geometry” prelectur.Stanford.edu/lectures/eisenman www.zvihecker.com http://aardvark.tce.rmit.edu.au/area-a/ARMSTORE/home.htm www.ushidafindlay.com

www.archilab.org/public/2000/catalog/ushida/ushidafr.htm www.salon.com/people/bc/1999/10/05/gehry/gehry2.html

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http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/workshop/Fractals.html

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http://library.thinkquest.org/26242/full/tutorial/ch4.html http://math.bu.edu/DYSYS/chaos-game/node6.html

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homepages.uel.ac.uk/1953, Mark Jeffrey.

Books : a) b) c) d) e)

Fractals Everywhere, Bansley Michael, 1988. Fractals and Chaos, Crilly A.J. Earnshaw. R. A .et al., Reference ,M.Arch., Urban Design Studio, 2001-2002, School of Architecture and Planning, Chennai “Landform Architecture, Emergent in the Nineties – Charles Jencks, Architectural Design, “New Science = New Architecture?” Vol 67, No 9/10, Sep-Oct 1997, Pgs 15-21. “After Typology: The Suffering of Diagrams” – William Brahan, Architectural Design, “Contemporary Processes in Architecture” Vol 7. No.3, June 2000, Pgs 9-11.

Note: Most sketches are downloaded from the I-net. (References given)