Architecture x Mathematics

Architectural design has dramatically changed in the past century. How? Why? There is no simple answer to these questions but one could definitely argue that the introduction of computer aided design softwares has opened a new field of possibilities to designers. In fact the constant evolution of our knowledge in the field of computing and mathematics has become available and usable by anyone and has generated a new set of boundaries. In this essay an attempt will be made to understand how recent discoveries in the field of mathematics have influenced today’s architecture.

Mathematics and architecture have always been intrinsically linked. In 1st century B.C Vitruvius had already described how arithmetical methods was at the heart of the architectural conception of his time. Later artist like Albrecht Durer and Leonardo Da Vinci reinforced the link between art and science by introducing perspective, proportion, chemistry and design in their masterpieces. In the beginning of the 20th century, Le Corbusier and Ozenfant developed purism a branch of cubism that focused on classical forms rather than decorative features. Purism was founded on mathematical rules such as the golden ratio, human measurements and Fibonacci numbers and strict artistic features (pilotis, free façade, roof garden…) Le Corbusier developed the “Modulor” system based on the golden number to develop future architectural designs.

Modulor by Le Corbusier

In 1946, Art and Science literally collided when two of the greatest minds of the past century, Le Corbusier and Albert Einstein, meet in Princeton. After explaining the concept of the “Modulor”, Einstein wrote to Le Corbusier: “It is a scale of proportions which makes the bad difficult and the good easy.”

Le Corbusier & Einstein in Princeton, 1946

In our days a new relation between architecture and mathematics exists. It is no longer solely based on perfect perspective, proportions or shapes. In fact, the advances made in the field of mathematics have opened the door to a new way of viewing our world and how to build around it.  The increase in the complexity of the mathematical tools (due to the evolutions made in computational science mostly) has rather surprisingly entirely changed the way architects go about their work. In fact, the introduction of easy to use architecture and design softwares have on one hand generated an almost limitless array of architectural possibilities but on the other changed entirely how architects communicated. In fact architects had always communicated through 2D plans, length and angles. These softwares have allowed the designers to communicate directly in 3D. Therefore this has generated a large decay in the usage of numbers. The architects have now access to a part of mathematics that was only mastered by few scientists. It includes the chaos theory and non-Euclidean geometry. This has enabled architects to, as well as pushing the boundaries of their designs, push their own boundaries, discovering new knowledge amongst the field of mathematics. Computer-aided design has initiated a transition from the usage of classical mathematics to the usage of many different specific fields of fundamental mathematics. Some architects decided to focus on curved surfaces and series to express their art others, in the footsteps of great mathematician Benoit Mandelbrot, focused on the iteration of shapes at different scales similar to fractals and others based their art on optimizing space and minimizing structures.

MONUMENTA is an exhibition held once a year in the “nef” of the Grand Palais in Paris. In the spring of 2011 Anish Kapoor proposed the LEVIATHAN. At first the size and shape of the structure and the red monochrome are overwhelming. Also the contrast between the structure itself and its location is extremely interesting; the Beaux-art architecture of the palace was surprisingly harmonious with the organic shape of Anish Kapoor’s masterpiece. The organic shape of the structure and the iteration remind the viewer of a mathematical function. In my opinion, the purity of the curved surface is accentuated by the monochrome. 

To me, this work of art is an incredible example of how much mathematics and science can influence architectural design. Anish Kapoor, one of the most prominent sculptors of our time, has also made the choice to make curved surfaces and series the heart of his work with Marsyas, Tate Modern London, 2002 and Cloud Gate, Chicago, 2004.

Marsyas - Anish Kapoor

Cloud Gate - Anish Kapoor

So one could ask why such a fascination for surfaces and series? In my opinion, this is linked to the nature itself of surface (in a mathematical sense). In fact surfaces in mathematics have zero thickness but still work as separation between two spaces. In other words it is “nothing” that separates two environment. This is an extremely abstract property but also a great challenge for the artist to overcome. Other examples exist where surfaces and series are at the heart of the design challenge: the Walt Disney concert hall in Los Angeles and the Abu Dhabi Airport. 

Walt Disney Concert Hall in LA

Abu Dhabi Airport

Surfaces with single or double curvature were used in both examples. Curvature is inversely proportional to the radius hence the smaller the radius is, the higher the curvature will be. Designing these shapes is fairly simple on a computer but the architect must take into account the feasibility of the physical structure and the behaviour it will have in its environment (wind, gravity…). The architect must also be aware of the purpose of the structure. For example, acoustics were of a premium importance in the Walt Disney hall. The sinuous surfaces of the exterior created an incredible challenge for the engineers. In fact acoustics are best in a room with relatively flat walls (infinite curvature). Similarly at the Abu Dhabi airport terminal the purpose is to pack as many planes /gates as possible over the smallest surface of land. From the Geometry and the imagination by David Hilbert, one of the properties of a sphere is “Of all the shapes having a given length of perimeter, the circle is the one with the largest area”. This property can be extended to sinusoidal waves that comprise larger areas than saw tooth waves for same length of perimeter. In other words, curved surfaces allow higher interior area of smaller distances than flat surface. This feature was used in a very clever and beautiful way at the Abu Dhabi airport.

Self-similarities have always had a great importance in architecture. They have been incorporated in design before we even studied them in mathematics. This is an example of how architecture (and nature obviously) can influence research. They can be found in structures from the dome on St Peter’s square to the Eiffel tower in Paris. Mandelbrot stated in the fractal geometry of nature “My claim is that (well before Koch, Peano, and Sierpinski), the tower that Gustave Eiffel built in Paris deliberately incorporates the idea of a fractal curve full of branch points.” 

Fractals are complex mathematical shapes defined by Mandelbrot in this same book as: “a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole”. In other words the shape repeats it self indefinitely regardless of the sclale.

 Illustration of fractal following Mandelbrot set

Illustration of fractal following Mandelbrot set

Ateliers Jean Nouvel used this process in the design of the Louvre Abu Dhabi. The “floating” dome of the Louvre is formed of five rotatable layers of meshed structure with an identical pattern at different scales placed on one another. This self-similarity has a double purpose: generate a microclimate inside and develop an incredible luminosity. In my opinion, this is one of the most spectacular architectural performances of the last ten years. This design brings perfectly together tradition, functionality, beauty and complexity. 

Some architects focused their work on optimizing the use of space. Optimization and architecture are closely intertwined when the surface available to the architect is limited. Skyscrapers in busy cities are a good example. Mathematical optimization is how to find the maximum within a fixed (or imaginary) set of boundary conditions. In the architectural world this translate into finding the best combination between the design requirements and the architect’s vision. It has also found another use as stated in The New Mathematics of Architecture by Jane and Mark Burry, “Optimization has been used in architecture as a form-finding tool.” In fact architects develop several concepts and choose by establishing the pros and the cons of each one. As with material selection for a particular design, all materials show advantages and inconveniences. The secret lies with prioritising which properties are the most crucial for the design and choosing from there. The perfect model does not exist. Optimization can be divided into subgroups: use of space, environmental friendliness, use of environment (weather, noise…) and many others. The Pinnacle in London is an example of optimization. The location of the building and the nature of the soil were definitely major design requirements. Carefully organising the glass panels on the façade and generating a simple and elegant shape optimized the design of the building. In fact the geometry and the glass panels allow the structure to be more environmentally efficient.

Pinnacle Tower in London

Pinnacle Tower in London

The curiosity of the architects for mathematics has even extended to topology (deformation of objects). This area of mathematics comprises geometries like the Möbius band or the Klein bottle.

As a conclusion, I think the progress made in the field of computing has brought architecture and mathematics closer together than ever before. Of course in a certain way computer aided design are hiding the complexity of the mathematical algorithms behind the 3D model on the screen. But in another way CAD has also had an introductory role, like a steppingstone towards an area of knowledge architects would have never dared to explore before. CAD has given designers the courage to go out of their comfort zone and explore fields of fundamental mathematics like topology, fractals or non-Euclidean geometry. I have personally always viewed mathematics as a subject very similar to art. As it was written in the 16th century during the construction of the Duomo in Milan Ars sine scientia nihil est. The beauty of both subjects is that the range of possibilities is limited only by our imagination and creativity (and maybe knowledge!). So for me the link between architecture and mathematics is magical especially when it is visible. I believe that the way of reshaping our cities through science and progress in this manner is visually exciting and leads the way to further progress. In my opinion, computer aided design has revived architecture entirely. Contemporary architects are exploring, trying and experimenting in every directions taking mathematics as the foundation of their art getting rid of restraint and control and establishing new standards. I think these times we live in are extremely exciting. Times where innovation is at the heart of motivation. Times where we have yet to discover the limit of computer-aided design. Times where architects can almost be viewed as messengers between very specific fields of Mathematics and the general public through art.

References:

The main referentces were the new mathematics of architecture by Jane and Mark Burry and Mathematic and architecture in architectural design july/august 2011 . Other data was found on the following websites.

http://www.boston.com/ae/theater_arts/articles/2009/02/22/beauty/?page=full
http://www.ilet.yildiz.edu.tr/oozcan/PDF_PUB/MATHEMATIC_DESIGN.PDF
http://www.fusebox.com/2011/07/mathematics-in-design/
http://www.cairn.info/revue-d-histoire-des-sciences-2006-2-page-245.htm
http://www.persee.fr/web/revues/home/prescript/article/rhs_0151-4105_2006_num_59_2_4585