Fractal architecture across cultures and continents

Richard Taylor, University of Oregon, USA

The strikingly intricate shape of the sunken temple at Varanasi stands out from the conventional, rectangular buildings that surround it. It seems to celebrate organic shapes rather than the artificial shapes of the other buildings. But what do we mean by ‘organic’ shapes?

In the 1970s, the mathematician Benoit Mandelbrot identified a subtle form of order within the apparent disorder of nature’s scenery. Many natural objects were shown to consist of patterns that recur at increasingly fine magnifications. Mandelbrot christened this repetition as ‘fractal’ (a term derived from the Latin ‘fractus’, meaning fractured) to emphasize their more intricate appearance when compared to the smoothness of Euclidean shapes such as traingles, squares and circles. Catalogued in Mandelbrot’s famous book Fractal Geometry of Nature, a diverse range of natural objects were shown to be fractal, including mountains, clouds, rivers and trees. The sunken temple captures the essential qualities of fractals – the shape of the building repeats at smaller scales.

Fractals have appeared regularly throughout the history of art, dating back to Islamic and Celtic patterns. More recent examples include Leonardo Da Vinci’s sketch The Deluge (1500), Katsushika Hokusai’s wood-cut print The Great Wave (1846), Jackson Pollock’s poured paintings (1943-1952) and M.C. Escher’s Circle Limit III and IV (1960). Escher is particularly well-known within the art world for his mathematical dexterity and his ability to manipulate repeating patterns at different scales. This fractal repetition has been extended to the arrangement of physical objects such as rock distributions in the Ryoanji Rock Garden in Japan.

However, the idea of creating fractal buildings is more challenging due to the repetition of the construction process at different scales. Nevertheless, the sunken temple is not the only example of a fractal building. Examples span across many different cultures. The Castel del Monte, designed and built by the Holy Roman Emperor Frederick II (1194-1250), has a basic shape of a regular octagon fortified by eight smaller octagonal towers at each corner. A more recent example is Gustave Eiffel’s tower in Paris, where the repetition of a triangle generates a shape known amongst fractal geometrists as a Sierpinski Gasket. The Eiffel Tower (1889) serves as a demonstration of the practical implications of fractal architecture. If, instead of its spidery construction, the tower had been designed as a solid pyramid, it would have consumed a large amount of iron, without much added strength. Instead Eiffel exploited the structural rigidity of a triangle at many different size scales. The result is a sturdy and cost-effective design.

Gothic cathedrals also exploit fractal repetition in order to deliver maximum strength with minimum mass. The fractal character also dominates the visual aesthetics of the building. A Gothic cathedral’s repetition of different shapes (arches, windows and spires) on different scales yields an appealing combination of complexity and order. In contrast to the ‘filled-in’ appearance of the Romanesque structures that pre-dated the Gothic era, the carved out character of the Gothic buildings delivers a distinctive skeletal appearance that results in their remarkable luminosity. The visual appeal of Frank Lloyd Wright’s Palmer House in Ann Arbour (USA) of 1950-51 has been analysed in terms of Lloyd’s use of triangular shapes at different scales. More recently, Frank Gehry’s organic architecture has been discussed in terms of fractals. Furthermore, The Water Cube swimming pool arena at the Beijing Olympic Games (2008) featured a fractal bubble design in its metal framework!

image by emily geioff

What are the possible motivations for creating a building based on fractals? One reason is the structural strength mentioned above. Fractals also disperse the energy of waves very efficiently – whether they be sound waves from noise, vibrational waves from passing traffic, or shakes from earthquakes. Thus, fractal buildings are fundamentally quiet and safe. Fractal shapes also have large surface area to volume ratios. For example, trees are built from fractals in order to maximize exposure to the sunlight. Possible advantages of this large surface area for buildings therefore include solar cells on the rooftops and windows that deliver a large amount of light to the building’s interior.

However, the main reason for building fractal architecture focuses on the associated aesthetics and the hope of mimicking a natural ‘organic’ shape. The study of aesthetic judgement of fractal patterns constitutes a relatively new research field. Nevertheless, recent studies have shown the people find fractals to be aesthetically pleasing and that they can reduce the observer’s stress-levels.

Although we think of fractal architecture as a concept for future cities, examples such as the sunken temple show that our ancestors have been exploiting the many positive qualities of fractals for many centuries.

Richard Taylor,
Professor of Physics, Psychology and Art,
University of Oregon, USA