I. Introduction

... Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth ... [01]

This quotation by Mandelbrot shows that the Euclidean geometry - the perfect “clinical” shapes of cones, pyramids, cubes and spheres - is not the best way to describe natural objects. Clouds, mountains, coastlines and bark are all in contrast to Euclidean figures not smooth but rugged and they offer the same irregularity in smaller scales, which are some important characteristics of fractals - see chapter “2.2 Characteristics”. As the following pages indicate, fractal geometry, in opposition to Euclidean geometry, offers better methods for description or for producing similar natural-like objects respectively. The language in which it is expressed is called “algorithms”, by which complex objects like a fern or a cloud can be reduced to simpler formulas or transformation rules respectively. Fractals can be found everywhere from coastlines, border-lines and other natural rough lines to clouds, mountains, trees, plants, ... and maybe also in architecture. The following chapters explain what a fractal is in general and how fractals can be used for architectural analysis and in the stage of planning. Fractals are used as a helping tool for explanation in many fields ranging from medicine to economy. From this point of view fractals should not be excluded from architecture.

1.1 Mandelbrot - Fractals - Theories and Explanations

... Fractals will make you see everything differently. ... You risk the loss of your childhood vision of clouds, forests, galaxies, leaves, flowers, rocks, mountains, torrents of water, carpets, bricks, and much else besides [02]

The computer-scientist Benoit Mandelbrot introduced the word "fractal" [03] in the year 1975 to describe irregular, not smooth, curves. Fractal geometry in general has become more and more popular since Benoit Mandelbrot’s book “The Fractal Geometry of Nature ” was published for the first time in 1977 and the “Mandelbrot set” was defined in the year 1980. Since that time many of those computer-pictures and "monster curves", as Benoit Mandelbrot called curves with unusual characteristics such as the Koch curve, have been published - some of them will be introduced in chapter “3 Different Fractals”. They were mostly created with a mathematical background, but they can also be found in art because of their beautiful interesting appearance [04]. In the eighties of the 20th century the “Mandelbrot set” could be found in many scientific journals last but not least because of its beauty - this kind of visualizing was, however, only made possible by the quick development of the computer. Subsequently today it is no problem for any of us to produce classical and natural looking fractals quickly and easily on the computer at home. Of course it is dangerous to think that we know enough about fractal geometry and the mathematical background so that we can start to use this "new" geometry in every field of our life. But, are we not now - with all the background of our society - in a position where we can deal with all the questions arising from the use of these new scientific theories and is this not more important than putting all the priority on finding the final result at the moment. To use new theories first means to define how they can help in a specific field and then to find applications in the next step. These days we can already look into other possible fields of application of fractal geometry: natural science, medicine, market analysis, manufacturing, ecology - and architecture? In any case architects and city-planners already use concepts and phrases for analysis which come from other fields, such as the biological metaphor: the city as an organism with the “heart" for the central business district, the “lung" for the green space, the “arteries and veins” for the hierarchy of communications routes [05] - in this connection fractal geometry might provide a new vocabulary which explains the complexity of a city in a better way.

1.2 An Overview

We can describe mountains, clouds, trees and flowers by models consisting of simpler geometric forms based on Euclidean geometry, for example using net models in CAD, but are they exactly what nature is [06]

The first two chapters below give an introduction to fractals and fractal geometry in a more general way, listing characteristics and explaining some examples. Then one chapter follows about the differences between Euclidean and fractal geometry and their expressions in the Euclidean and fractal dimension, introducing and explaining some measuring-methods of the fractal dimension. Until a short time ago scientists described nature through so called “smooth” continuous mathematics, which is the mathematics of smooth forms such as lines, curves and planes and which is expressed in the language of Euclidean geometry. The “new” science of complexity [07]  does not try to simulate any more the rugged character of nature through smooth forms but it deals with the raggedness of the structure itself - this field of mathematics is expressed in the language of fractal geometry:

“The whole is more than its parts”. The fractal new geometric art shows surprising kinship to Grand Master paintings or Beaux Arts architecture. An obvious reason is that classical visual arts, like fractals, involve very many scales of length and favor self-similarity [08].

Chapter “5 Fractals and Architecture” to chapter “6 Fine Arts and City Planning” deal with the question if fractal geometry can be used in architecture and for the analysis of cities or at least with the question in which ways further research can be done - it lists the possible applications of fractal-theory in fine-art, city-planning and architecture, with regard to analytical, formal and practical aspects. Fractal geometry may help us to understand and analyze complexity that can be found in towns of the Middle Ages but also in cathedrals and other man-made objects up to these days. It may also help us to transfer this complexity, which also arises from the development over time, to newly planned cities and buildings - cities and buildings may then also be reduced to simpler algorithms: the automatic architect [09] ? Among other themes one part of chapter “5 Fractals and Architecture” deals with the so called “organic” architecture. In this context, natural structures, explained through fractal geometry, could serve as easy-to-use models for a new “organic”, natural-like architecture. The underlying structure, the character of a building will then follow the principles of fractal geometry that can be found in natural elements. This results in a connection between architecture and nature not only in a formal respect - e.g. with Antoni Gaudí [10]  - but also in the complexity of forms and in its character of self-similarity, irregularity and roughness - e.g. with Bruce Goff. Another section of this chapter deals with the question of the quality of buildings, which means asking why one building is regarded as "good" architecture while the other is regarded as "bad" architecture. It is clear that everyone has his or her personal access to architecture and evaluates buildings in a different manner. But there are buildings, which are more interesting in general than others, or there are buildings, which are more noticeable to most people than others - e.g. the acceptance of Gothic, which is related to fractal geometry, in contrast to the resentment towards modern buildings, which are related to Euclidean geometry. Subsequently the idea arises that maybe it is complexity that modern buildings lack. Chapter “7 Problems with Measuring” deals with measuring problems arising from the use of different computer-programs for calculating the box-counting dimension introduced in chapter “4 Dimensions”. Finally, this box-counting dimension is used in chapter “9 Statistics” to get a classification of elevations and ground plans of buildings and cities. In this context the fractal dimension is applied as an indication for roughness.

Footnotes

[01] Comparison of natural objects with Euclidean geometry by the mathematician Benoit Mandelbrot in "The Fractal Geometry of Nature". Bovill Carl, Fractal Geometry in Architecture and Design (1996), Birkhäuser Bosten, ISBN 3-7643-3795-8, p.4, Mandelbrot Benoit B., Dr. Zähle Ulrich (editor of the german edition), Die fraktale Geometrie der Natur (1991) einmalige Sonderausgabe, Birkhäuser Verlag Berlin, ISBN 3-7643-2646-8, p.13.
[02] Warning of Michael Barnsley in the introduction to his book "Fractals Everywhere" (1988), Academic Press. Bovill Carl, Fractal Geometry in Architecture and Design (1996), Birkhäuser Bosten, ISBN 3-7643-3795-8, p.3.
[03] The Latin adjective "fractus" - from which the term "fractals" is derived - was used by Benoit Mandelbrot to denote a phenomenon with a new word. The Latin verb 'frangere' means "break into irregular fragments", Mandelbrot Benoit B., Dr. Zähle Ulrich (editor of the german edition), Die fraktale Geometrie der Natur (1991) einmalige Sonderausgabe, Birkhäuser Verlag Berlin, ISBN 3-7643-2646-8, p.27.
[04] Maybe it is this beautiful interesting appearance that brings a better access to mathematics: the theory of fractals and chaos is not only an abstract theory but it can also be visualised - mostly through the computer - which surely is helpful for better understanding of the underlying mathematics.
[05] Batty and Longley, Fractal Cities (1994), Academic Press Inc., ISBN 0-12-4555-70-5, p.31.
[06] That is what architects may be interested in: forms - their mathematical explanation and their application to architecture. Today we use the mathematical explanation of complex geometries, fractals, in animated cartoons and scienc fiction movies to generate trees, ferns, mountains, unknown worlds and whatever is found in nature. The fractal geometry reduces the complex forms to much simpler mathematical expressions which make computer files smaller and, reduced to a few iterations, above all quicker to compute - which means to produce. So one field of application may be in CAD where complex forms of nature such as clouds and trees may be constructed without using strange complicated wire models or 2-dimensional pictures.
[07] Peak David and Frame Michael, Komplexität - das gezähmte Chaos (1995), Birkhäuser Berlin, ISBN 3-7643-5132-2, p.19. Complexity in architecture means that elements of e.g. a facade - columns, capitals, base, architraves etc - can be grouped in a way that there is more than one equivalent interpretation presented to the observer. This is true for Michelangelos’ facade for the funerary chapel of San Lorenzo in Florence (1516-1534). Complexity does not mean complicated forms but the interconnection of elements that produce the whole. Von Meiss Pierre, Elements of architecture, from form to place (reprinted 1998), E & FN Spon, ISBN 0 419 15940 1, p.45.
[08] Comparison by Benoit Mandelbrot in "The Fractal Geometry of Nature" - Die fraktale Geometrie der Natur (1991) einmalige Sonderausgabe, Birkhäuser Verlag Berlin, ISBN 3-7643-2646-8, p.35. Bovill Carl, Fractal Geometry in Architecture and Design (1996), Birkhäuser Bosten, ISBN 3-7643-3795-8, p.115.
[09] Of course this will not lead to an automatic architect: we know the limits of artificial intelligence. There exists, for example, a computer program which produces ground plans of villas by Palladio through algorithms. The grammar of buildings by Palladio was analyzed and brought into rules - for example the repeating symmetric axes and the continuous main-room. With the set of rules and the repeating recursive application of them, a defined grid and additional rules for the porticos, windows and doors under the control of Palladios rules of forms, the computer works out a couple of results that look like the villas built by Palladio - the underlying structure, the character is reproduced well. But there are some limits shown, for example in the third dimension, where too many rules for computing would be needed. Beside that the machine has a lack of knowing the meaning of functions - the relation to the neighborhood is missing. The program is neither able to sort out "wrong" ground floors which are not fitting that well, nor is there a limit in using the program for more complex forms.
[10] The form of the roof of the Casa Batlló by Antoni Gaudí looks like the back of a dinosaur, the columns of the ground floor seem to be the feet of elephants and the balconies look like faces of animals. Jürgens Tietz: Geschichte der Architektur des 20. Jahrhunderts (1998), Könemann, ISBN 3-8290-0512-1, p12.