### 2.2.2 Characteristics - A Fractal is Self-Similar

#### 2.2.2.a Self-Similar and Self-Affine

Fractals are always self-similar, at least in some general sense - what does that mean? That means that on analysis of a certain structure will bring up the same basic elements on different scales. For example, details of a certain coastline look like larger parts of the whole curve; the characteristic - the irregularity - of this natural form remains the same from scale to scale. In this way fractals can also be described in terms of a hierarchy of self-similar components - e.g. trees and branches or town-, district- and local-centers.

Any structure is self-similar if it has under-gone a transformation in which the proportions of the structure have all been modified by the same scaling factor. The new shape may be smaller, larger, rotated, and/or translated, but its shape remains similar, which means that the relative proportions of the shapes’ sides and the internal angles remain the same[01] - these transformations can be produced by a reduction-copy machine as shown in picture 02. Fractals produced by self-similar transformations are “true” fractals, the underlying algorithm is the same from scale to scale - zooming into such a fractal shows an object, which is the same as the whole. These "perfect" fractals may produce objects which look similar to nature, but there is mostly something missing - the factor of random. Nevertheless they can be used as a first approach to nature instead of Euclidean objects- remember: “... trees are no spheres”.

picture 02: The reduction ...

The reduction of a paper in a copy machine is a similarity transformation.

The picture shows a grey sheet of paper that is transformed in a reduction-copy machine with three lenses. One lens reduces the grey square by 1/3 and places the copied square on the left upper side - colored red. One lens reduces the grey square by 1/3 and places the copied square on the left lower side - colored yellow. Finally one lens reduces the grey square by 1/3 and places the copied square in the middle of the right side - colored blue. Then the same transformation is repeated on each of the new squares - this is called recursive transformation.

If a transformation reduces an object unequally in one or the other way, then the transformation is referred to as a self-affine transformation. In a self-affine transformation the internal angles of the shape and/or the relative proportions of the shape's sides might not remain the same - these curves are not exactly self-similar. For example parts of a “natural” fern or a snowflake are not exactly a copy of the whole as it is true for the computed examples of fractals, but nevertheless the parts look very much like the whole - this is called a statistical self-similarity because on average parts look equal[02] .

Some examples of statistically self-similar structures in nature are the large and small branching structures of a tree, the bays and inlets of a coastline, weather fluctuations through time. But also man-made structures can be statistically self-similar such as the average of the Dow Jones, which shows similarities between hourly, monthly and yearly fluctuations[03] .

#### 2.2.2.b Coastlines

Coastlines in reality look similarly rugged no matter from which distance we observe them. That was also one important circumstance that Benoit Mandelbrot found out when he analyzed the coastline of Great Britain: A certain part of the curve always looked similar to the whole curve no matter on which scale he searched[04] . For example Norway has a lot of fjords, which means that its coastline is very rugged - zooming in on the fjords will offer a similar roughness.

That the coastline of Norway is very rugged means that we will more quickly find a greater number of details from scale to scale and the length increases more quickly than in the case of a smooth coastline. The quickness the length approaches “infinity” is a characteristic for the examined coastline.

On looking at the border of a Euclidean object such as a circle, something different happens. As we did with the coastline we zoom in infinitely on the curve, but instead of finding similar sections the zoomed part turns into a straight line. In contrast to that we would never find a straight line on the coastline but always a rugged part of the curve.

#### 2.2.2.c Self-Similar Structures in Architecture

Castel del Monte: This is an early architectural attempt at self-similarity. The building has been erected over an octagon, with octagonal towers on its edges. The same form can be produced by putting the octagon in the copy machine of picture 02 with the input: "Reduce the octagon by a certain factor 1 to X and put it on the edges of the previous octagon". This could be repeated more often. In fact, what we see here is the first iteration of this expression - see picture 03.

picture 03: Castel del Monte

“The hunting seat of the Staufer emperor Friedrich II is an early attempt of self–similarity in architecture”. Castel del Monte was the hunting seat of the Hohenstaufer Emperor Friedrich II in Apulia (1240 - 1250).

Looking at the middle picture at the bottom reveals another characteristic of fractals, the surface of the building is rough and nowhere smooth - the structure comes from the material itself: stone. Zooming closer means finding more and more details of the weather-beaten rough surface - the smaller parts of the material.

Bruce Goff: Some of Bruce Goffs buildings contain characteristics of fractal geometry, for example the Eugene Bavinger house, near Norman, Oklahoma, built in 1950. The floor plan describes a curve that shows a form like the self-similar Cephalopode Nautilus - the units of the Nautilus follow the structure of a logarithmic spiral curve. This curve is called self-similar because the angles of the tangents are equal in all points - see picture 04.

picture 04: Bruce Goff

Some pictures of the following two examples, the Eugene Bavinger house and Joe Price Studio, both by Bruce Goff, are taken from “The Architecture of the Jumping Universe” by Charles Jencks, pages 42 to 45.

For Charles Jencks the occupation with new mathematical theories is also an attempt to find the answer to a central question which has been dealt with in architecture since the 19th century: “in which style should we build”.[Jencks Charles, The Architecture of the Jumping Universe (1996), Academy Editions an imprint of Academy Group Ltd, ISBN 1 85490 406 X, p.43.]
Joe Price Studio by Bruce Goff, Bartlesville, Oklahoma, 1956, floor plan. The central theme, motif, is the equilateral triangle which can be found on different scales. Many variations, multiplications and subdivisions of the 60-degree angle in triangles and hexagons of many different sizes can be located. This is a fractal-concept: the angles are similar from scale to scale.
It is true that Bruce Goff did not know anything about the definition of fractals, which came up later with Mandelbrot in the seventies, but nevertheless some of their characteristics can be found in his buildings. This may be due to his reference to organic architecture, to nature. Many plants and natural shapes as mountains and clouds have self-similar patterns, which he transferred to his architecture by unity with variety like the 60-degree angle of the Joe Price Studio.

Another example of fractal architecture by Bruce Goff is the Joe Price Studio in Bartlesville, Oklahoma from 1956. All shapes include equivalent triangles on different scales; the angles are similar from scale to scale - 60 degrees or a multiple of it. The "meeting-area" is e.g. hexagonal, the walls of the music-room are three-cornered and the ceiling of this room consists of a three-cornered decoration for better sound quality through reflection - see picture 04[05] .

Facade: Looking at a building from some distance, we will get a first impression of it. Coming closer, it depends on the respective building itself if some more details of interest appear. As an example the view of Robie house by Frank Lloyd Wright will offer more and more details on a lower scale with a repetition or variation of forms respectively. The building “remains” interesting - and in addition to that the repetition helps us with regard to orientation and classification. In contrast to Frank Lloyd Wright, the smooth modern buildings of the twenties and sixties often lack detail on lower scales - as we will see later on they also have a lower fractal dimension.

Architecture: For more examples of self-similarity from scale to scale and other fractal characteristics in architecture - from ground plans to facades - see picture 05.

picture 05: Some more examples for self-similarity

One example for an early attempt of using self-similarity in architecture is the floor plan of the Tadsch Mahal in Agra/India. The middle octagon that is repeated in the four diagonals of the outer square forms the basic element.

The Hindu Temple “Rajarani” in Bhuvanesvar also shows some characteristics of fractals, which can be found at many Hindu temples of India. First, some of the architectural elements have undergone a transformation by which size and position have been changed. For example the whole parabolic form of the temple can be found in a modified way in the smaller parts on the surface, which is rendered prominent by different colors. Second, the surface of the temple is similarly rough no matter which scale is used.

In the picture of the Pokrov cathedral in Moscow the prominent element turns out to be the bulb-shaped dome. This element is transformed in size and position from one stage to the next, which is rendered prominent through the colors yellow, orange and red. But in addition to that there are also other forms that are repeated on different scales.

The Dharmaradscha Rath in Mamallapuram, India displays a transformation from one level to the next. For better orientation each level is marked by a blue rectangle. In each step one architectural element, colored green, is added.
The rough surface is produced by only a few similar or equal elements, which is a fractal characteristic: complex looking structures can be reduced to only a few transformation-rules or algorithms.

picture 05: The cathedral of Cologne

The cathedral of Cologne is a good example for self-similar patterns in Gothic architecture.

One of the most important elements of this example of the Gothic style certainly is the “pointed arch”. This element is transformed in size and position and can be found in the entrance, above windows and all over the surface - colored violet and blue respectively. But it is also an important element used indoors as the form of the arch shows. Another characteristic element is the gable above the pointed arches of the entrances and windows - colored green and yellow respectively. This architectural element can be recognized on turrets of the facade but also inside the cathedral at the pulpit. Both elements reflect the overall concept of the cathedral, the pointed rising form.

#### 2.2.2.d Self-Similar Structures and Cities

Cities: Self-similar patterns can also be found in cities in a more abstract respect. If we compare the function of rooms of a dwelling or a one-family house - living, working, walking, relaxing -, that is to say the way of dividing up space, with a quarter and the whole city we will find analogies in areas for working, living etc. Functions are repeated on different scales. But also the repetition of ways observed over years between the separate functions inside a dwelling within a day or ways in the city, e.g. to and from work, can be seen as fractal concepts.

Africa: The site plan of a village in Cameroon/Africa is based on a circle, which is then repeated on the smaller scale of the ground plans of the cottages - see picture 06.

picture 06: Cameroon

The image below shows a village of these days in Cameroon in Africa and is taken from “Die Geschichte der Stadt” by Leonardo Benevolo.

The basic figure of this village is formed by a circle, which is marked green. Along this outer curve circular huts of different sizes, rendered prominent by different colors, are arranged. These huts take in different usages or different rooms respectively, used for sleeping, men, wives, kitchen, granaries, small livestock and stocks. Inside the “kitchen-huts” the forms of the fireplaces again display the shape of the circle, but this time on an even smaller scale.
That is to say the circle, belonging to the Euclidean elements, has undergone two transformations, the variation in size and position. And though the appearance of the village itself is rugged an overall ”concept”, an order, can be found: the circle.

#### 2.2.2.e Other Disciplines

Self-similarity is also found in other disciplines - the structure of our memory for example is also fractal. A certain word conjures up a complex network of images in connection with that word. It is not the object or similar objects that come up, but events and associations, for example a certain mountain tour for the word "mountain". By that a set of memories comes up to one’s mind, people that were present at the tour and other thoughts in connection with them. The word does not bring an abstract image to one’s mind but a complex set of interrelations to events and persons. Hardly any boundary can be drawn around these interrelations, which is also true for the border of the Mandelbrot set. The network for a certain word is of course different from person to person, because of individual experiences, but this is not so important. The different networks have enough similarities for communication. For conversation and books the fact that different networks exist is getting more and more of a problem because the information increases and so the nets that are conjured up and as a consequence also the interpretation of the meaning differ more and more.

###### Footnotes

[01] Bovill Carl, Fractal Geometry in Architecture and Design (1996), Birkhäuser Bosten, ISBN 3-7643-3795-8, p.15
Jürgens Hartmut, Peitgen Heinz-Otto, Saupe Dietmar: Fraktale - eine neue Sprache für komplexe Strukturen, Spektrum der Wissenschaft (9/1989), p.54.
[02] Peak David and Frame Michael, Komplexität - das gezähmte Chaos (1995), Birkhäuser, ISBN 3-7643-5132-2, p.41.
[03] Bovill Carl, Fractal Geometry in Architecture and Design (1996), Birkhäuser Bosten, ISBN 3-7643-3795-8, p.17.
[04] Though maps of different scales differ in their details, they have equal characteristics: smaller and bigger bays or inlets are rough outlines and, besides the scale, geometrically identical.
[05] Jencks Charles, The Architecture of the Jumping Universe, (1995), academy editions, ISBN 1 85490 406 X, p.44.

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