Fractal Aesthetics in Architecture

Journal paper, in Applied Mathematics & Information Sciences. (article)

Wolfgang E. Lorenz, Jan Andres und Georg Franck. This paper deals with fractal aesthetics and proposes a new fractal analysis method for the perceptual study of architecture. The authors believe in the universality of formulas and aim to complement the architectural description in terms of proportion. ...

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Example: Sunbeams

an example for three.js

SimAUD 2017

A Building Database for Simulations Requiring Schemata. (book)

Gabriel Wurzer, Jelena Djordjic, Wolfgang E. Lorenz und Vahid Poursaeed.
Obtaining spatial representations of existing buildings for use in simulation is challenging: To begin with, getting permission to access submitted construction plans can take a long time.. ...

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Steuerberater Kanzlei

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VI Fine Arts and City Planning

Whether fine arts and city planning offer fractal geometry or not can once more be proved by finding characteristics of fractals in the form of paintings and cities, but also by calculating the fractal dimension with the aid of the box counting method or the scaling range analysis. After the section about fine arts and an analytical part about cities, a section of some possible applications of fractal geometry in city planning as a helping tool will be mentioned.

6.1 Fine Arts

Artists have always used geometry in the one or other way for their works, although they may not have been conscious of that. First the decision has to be made if something should be formed in three or two dimensions. Second there are many rules for composition and proportion found in design and art that are based on dimensions. Thinking of one of the most important proportion-rules, namely the "golden section", this turns out to be, as mentioned before, a fractal sequence. Beside that art can be interpreted as a way for finding the basics of beauty and harmony that are found in the laws of nature. In this way chaos and fractal geometry may help to explain and prove the "rules" of beauty.

6.1.1 Description of Two Paintings

 The basic difference between Euclidean and fractal geometry can be shown at two paintings of the last century which are compared in a purely formal way in the following section. That is "Composition with Black, Red, Grey, Yellow and Blue" by Piet Mondrian of the year 1921 and "number 8" painted by Jackson Pollock in 1949.

6.1.1.a Piet Mondrian

"Composition with Black, Red, Grey, Yellow and Blue" of picture 62 represents a painting that belongs to a style of abstract-geometric forms which is very close to Euclidean geometry, like the modern architecture of the 1920ies which asked for Euclidean shapes. In this sense Mondrian’s painting can be described by straight lines and simple, smooth rectangles. E.g. one may start on the right upper side with the red rectangle of a certain proportion and relate all parts of the painting to it, defining width, height and color of each single object.

picture 62:Piet Mondrian

1912, "Flowering Trees" (left); 1913, "Composition No. VII" (below left); 1921, "Composition with Black, Red, Grey, Yellow and Blue" right)

The painting is located in the development of the style and message of Piet Mondrian. Out of the organizing principle of the synthetic cubism he developed the arrangement of areas with certain colors, at which first the motive of the tree turned into an abstract set of lines. Later on, his paintings lost any reference to real objects.

Nevertheless the process of painting is not accompanied by a straight mathematical scheme, but instead of that Mondrian worked intuitively in front of the canvas. The aim was to show pure reality through pure arrangement. There should not be any influence of subjective feelings and ideas. Mondrian tried to achieve universal harmony and surmounting of individualism, which meant excluding sensuous perception and visible reality by a reduction to straight lines, rectangles and a few colors[01].

Later on Piet Mondrian wanted to express through his paintings the complex metaphor for the encounters, the misadventures and the mutual accomplishments of which life is made up[02]. By that he turned away from the abstract Euclidean geometry to an abstract "Fractal Rhythm", which displays the cascade of interest that can be also found in nature. An example for this rhythm is offered by the painting "Victory Boogie Woogie" from 1944. For describing the series of colors of one strip of the painting - and by that the Fractal Rhythm -, Carl Bovill simplified the color use to red, blue, yellow, black and , which lead to the following sequence of colors for this certain strip:  

1, 5, 1, 3, 5, 3, 4, 2, 3, 3, 4, 2, 3, 4, 3, 1, 3, 4, 3, 4 (with 1=red, 2=blue, 3=yellow, 4=black, 5=grey).

The sequence can then be translated into a graph, in which the resulting plot is similar to noise and offers a mixture of order and surprise which can finally be measured and expressed through the fractal dimension, see picture 63[03].

picture 63: Piet Mondrian - "Victory Boogie Woogie" 

left: "Victory Boogie Woogie" 1942-1944; right: "Broadway Boogie Woogie" 1942-1943 (with 1=red, 2=blue, 3=yellow, 4=black, 5=grey).

results in the sequence:
1, 3, 5, 3, 4, 2, 3, 3, 4, 2, 3, 4, 3, 1, 3, 4, 3, 4, 1, 4, 3, 1, 3, 4, 1, 3, 3, 1, 3, 4, 3, 5, 4, 1
Victory Boogie Woogie color sequence:

6.1.1.b Jackson Pollock

"Number 8" by Jackson Pollock is one of his abstract compositions in which the color is applied to the canvas by a spontaneous movement. The results of these action paintings are harder to describe in Euclidean vocabulary than the abstract-geometric formed ones by Piet Mondrian.

For his painting Pollock used the method of "drip painting" which allowed a free form of abstract painting. To get a well-balanced object he put the canvas on the ground and worked on it from each side not knowing what he was doing exactly in the moment when the colors were dropping on the canvas - the painting had its own "life". Though this method was spontaneous, he nevertheless could make corrections and ultimately it was his decision which part of the canvas he cut out and put on the frame. Highly composed, constructed compositions and working with the paintbrush confined him too much. Therefore he did not use any existing forms and motives any more.

6.1.1.c Conclusion

Is the painting "number 8" really a more "fractal" painting? Maybe this can be proved through an algorithm which is recursively applied to the canvas. A simpler algorithm, similar to the Brown motion, may run like that: put the canvas on the ground, step on the canvas and walk around dropping color on it; the way one takes and the kind of dropping is determined at random telling whether one moves to the left, to the right or even rotates, using much color or less, circular, straight or drop-shaped. Of course this does not exactly produce the same painting but the impression may be similar - we know that there are drops, curved lines, thick and thin parts on the canvas, see picture 64[04]. Anyway what this painting offers in contrast to Mondrian’s "Composition with Black, Red, Grey, Yellow and Blue" is a mixture of order, through circular and straight lines, and surprise, where no pattern is recognized.

picture 64: "Jackson Pollock - "picture number 8".

Most paintings are situated between those extreme cases. By that paintings which mirror nature in the one or other way, display a cascade of fractal detail in the same way nature does - e.g. by the rhythm of the spacing of trees in a wood.

6.2 Cities

Most cities look like complex structures to us that can hardly be grasped, both with regard to form - city-boundaries, traffic-networks - and their distribution or connection of functions - e.g. living, working, traffic, recreation and administration. This kind of irregular looking cities are often called "naturally" or "organically" grown, indicating that the geometry of such a city does not seem to be planned in the large - the form may rather be produced by many detailed and individual decisions which are coordinated in the small[05]. To tame the "chaotic", disordered looking structure of cities, planners attempted to bring simple, smooth, visual - Euclidean - order into the city[06]. This leads to the group of so-called "planned" cities, involving fast growth with planning on a large scale by one or few authorities.

Most cities are formed by a mixture of both groups, depending on the changing assumptions throughout the history of a city: e.g. a newly founded Roman military camp - "planned" -, may have been changed slowly through the Middle Ages - "naturally" grown -, ..., before the population explosion of the 19th century once more had caused fast growth - "planned".


[01] Bruckgraber Iris and more, Kunst des 20. Jahrhunderts - Museum Ludwig Köln (1996) Originalausgabe, Taschen Verlag GmbH Köln, ISBN 3-8228-8819-2, p.507.
[02] Bovill Carl, Fractal Geometry in Architecture and Design (1996), Birkhäuser Bosten, ISBN 3-7643-3795-8, p.150.
[03] Bovill Carl, Fractal Geometry in Architecture and Design (1996), Birkhäuser Bosten, ISBN 3-7643-3795-8, p.150.
[04] von Lengerke Christa, Malerei heute - von Pollock bis Warhol, in: Walther Ingo F. (editor), Museum der Malerei, Edition Atlantis, p.37.
[05] Batty and Longley, Fractal Cities (1994), Academic Press Inc., ISBN 0-12-4555-70-5, p. 28. The decisions on smaller scales may be the reason why "organically" grown cities display irregularity in form and do fit more comfortably in their environs.
[06] Batty and Longley, Fractal Cities (1994), Academic Press Inc., ISBN 0-12-4555-70-5, preface p. VI.