Fractals for architecture and urban planning

Recently two major works have left their mark on architecture by introducing the fractal approach. These are Peter Eisenman's Bio Center in Frankfurt and Jean Nouvel's Qatar Museum. In 1987, as part of a program that included research spaces in biology, biotechnology and biochemistry, Eisenman used to guide the design of the project fractal figures corresponding to the iterations of DNA molecules.
Then using a fractal technique to duplicate these figures at different scales and in oblique orientations, these shapes aligned along their common faces generated cantilevered spaces for the cafeteria, library and meeting areas, arranged the along a central linear circulation trunk.
View of the construction of the Eisenman Biocenter.

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Inspired by fractals, the new Qatar Museum (2019) by Jean Nouvel is an interweaving of discs inspired by the desert and interlocking and intersecting with each other to form a gigantic rose of the sands. All these discs with variable and random geometry, made of Ductal concrete, are not only placed on the structure. Ductal concrete (invented by Lafarge) is a high-performance fiber-reinforced concrete that reduces steel consumption and above all greatly reduces the porosity of concrete.
They are the ones who make up the building, inside and out, over an area of 40,000 square meters. The exhibition spaces are organized around a central courtyard, the “caravanserai”. With its large curved discs, intersections, cantilevered elements, the sand rose, inspired by the natural desert, would have been impossible without computers and engineers. In addition, the protective shadows of the discs limit the number of windows and the building is energy efficient.

The sand rose inspired Jean Nouvel.

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What do we mean by fractal in the field of planning and architecture?
A fractal figure is a mathematical object that has a similar structure at all scales. These figures had long been known as curiosities. The shape remains similar, the length increases to infinity but the occupation of space has a limit.
It was B. Mandelbrot who in the mid-1975s theorized the notion of fractal. From there, one can imagine spaces made up of self-similar architectural or urban surfaces.
These surfaces would obey fractal laws, therefore regular, while drawing lines of non-integer dimensions.
The Euclidean approach that had prevailed until then is called into question. Fractals would therefore no longer be, as it was once said, "monsters", but natural forms: collagen proteins or rose des sables above.
What are the effects of abandoning Euclidean geometry in town planning and architecture?
In classical geometry (Euclidean, named after the Greek mathematician Euclid), the dimension of a line is 1, that of a surface is 2, etc. This is how architectural objects are composed, from lines (dimension 1), surfaces (dimension 2), volumes (dimension 3). Mandelbrot imagines new figures, constituted by elevation to the same power, therefore in a regular way, not whole, for example dimension 4/3. They are therefore no longer lines, nor surfaces, of Euclidean geometry, but “fractal” figures. The properties of these figures differ greatly from the properties of Euclidean figures. In particular, “self-similarity” means that at all scales a fractal figure resembles the one that gave it birth.
The results differ according to the domains and the countries. For town planning, the interest was limited. The very broad vision of B. Mandelbrot had difficulty infusing. Nevertheless, France has experienced a certain breakthrough in the field of fractals applied to urban planning, as evidenced by the work of P. Frankhauser and C. Tannier.
How to replace a compact set of buildings with a subdivision of airy houses by dividing the same space according to a fractal principle (P.Frankhauser).
For architecture, the situation is contrasted. Foreigners from different countries seem to have a great interest in the use of the fractal tool in architecture, while France remains on the sidelines.
Due to the lack of integrated training, the rarity of combined courses (architect/engineer), the weakness of research in this field and despite the commendable efforts of a handful of teachers (including Floriane Deléglise in Lyon), the compartmentalization of knowledge prevents a greater interest in fractals in architecture in France. There is a lack of a “middle” that would allow a pooling and the passage of a critical threshold.
The importance of all this production, which can be quantified at around fifty practitioners and around thirty "theoreticians", shows that the question of fractality in architecture is taken seriously. Architectural practice, at least for major works, knows how to draw inspiration from fractals, as we have seen for Eisenman and Nouvel.
According to critics of the use of fractals in architecture and urban planning, the application is necessarily limited by the size of the achievements and does not always have the desired regularity. It also depends on the perception of users. Others regret that we do not find a general approach for fractality, the use of fractals being useful only for certain aspects of the composition. It should be noted, however, that the "conciliatory" critics are dominant. Finally, the use of fractals by physicists falls under the same criticism, which has not prevented them from benefiting from the contribution of fractals to explain, for example, the behavior of flames.

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In the more limited framework that we have just drawn, fractality therefore finds its transcription in important aspects of architectural creation (surfaces, facades, ceilings, decorations, etc.). Biomimicry is making a comeback. Achievements of the past or exotic, famous recent works (Wright and Le Corbusier). Why not apply these principles to today's world? J. Nouvel, P. Eisenman show what can be done for major works. It is therefore a biomimicry with spectacular achievements that currently seems to be democratizing for architecture and urban planning.
The era of brilliant intuitions has passed. In a world increasingly dominated by minimalist achievements of ordinary architecture and town planning (therefore Euclidean), many call for the need to rediscover other constructive principles. Designing suburban environments of acceptable density and structure, as P. Frankhauser's team does, is a good path for urban planning today.
Finding in architecture, like Wright or Le Corbusier, decorations taking into account the existence of new materials but with modern instruments, is undoubtedly also a way of the future. However, to draw inspiration from nature today and therefore practice biomimicry, you must first build a model of nature and then try to apply it to the concrete case you are trying to solve. These are two extremely complex operations that the fractal principle simplifies by simultaneously providing the natural model and its architectural or urban application.
Should we condemn this trend in the name of the reservations of certain purists? Should we encourage it? Should France stay away from the international concert? The movement is underway and French reluctance will not stop it. But two possibilities are open. First of all, to ensure that in France, from places like Strasbourg or Lyon, combining training, with the help of engineering or architectural firms, we are more interested in these questions in a context that is now international. Then, maintain the science of fractals as French urban planning has done. We must take into account the above criticisms and increase exchanges with foreign journals. The use of fractals must lead to a "perceived realism". If the perspective of an overall fractal vision for the built urban space seems out of reach, the easiest use is in the direction of fractals “by pieces”. Biomimicry should not, however, distance us from the scientific principles laid down by Mandelbrot.

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