Formulations: Architecture, Mathematics, Culture

In this book Witt describes how in the first half of the 20th century the design component of architecture was influenced by the evolution of mathematics, science, and technology, until the computer took over around 1960-1970. There was a tendency to collect, classify, and eventually unify, knowledge, which resulted in larger research projects that should bring order in the cabinets collecting curiosities from different scientific disciplines, including mathematical polyhedral models, mechanical instruments, and molecular structures. Two examples: the Principia Mathematica of Whitehead and Russell was an attempt to build up the whole mathematical construction from the bottom up and Klein's Erlanger Programm was set up to classify geometries.

 

In the 10 subsequent thematic chapters, Witt goes through different aspects of the interaction between architecture, science and mathematics that has led to different subcultures. The exposition of the book is however not mathematical. There are no formulas or popular mathematical theories, just some philosophical analysis of the epistemological aspects in architectural design as influenced by the technological and mathematical evolution. The designer draws from a network of knowledge and insights in which mathematics, although important, is one of the ingredients. Since design is mainly experienced visually, the book has many illustrations, not only of architecture, but they also show how similar ideas penetrate into mathematics, chemistry, biology, or architecture.

 

The order of the chapters is somewhat chronological, with much overlap though. Witt starts with surveying mechanical instruments that are used for drawing curves. These tools are in support of descriptive geometry as conceived by Monge at the end of the 18th century. They constitute a black box encapsulating mathematical knowledge, which liberates the mind of the designer to engage in the more creative aspects. These mechanical tools are the precursors of computer software we use today. Other tools are maquettes and models of architects (e.g. Gaudí's inverted string model for the Sagrada Familia) and different forms of art (e.g. the dodecahedron and the hypercube in Dalí's paintings) but also solid and wired models are as old as mathematics to help our imagination.

 

Triangulation is a technique intensively used in computer graphics and to solve differential equations numerically on irregular domains. Intensive use started with the triangulation to measure France (Picard, Cassini's), the Himalaya (Everest) and the earth's meridian (Delambre and Méchain) eventually leadint to our GIS systems. On 3D objects, the use of Möbius's barycentric coordinates boosted the numerical computations. The hyperdimensional geometry of Lagrange and the geometry of Riemann and Poincaré brought new challenges. For example Jouffret introduced 3D projections of 4D objects. The converse operation of stereography and stereoscopy resulted in 3D models from 2D images. These we are familiar with when watching a 3D movie in a theater with special filtering glasses, or as used in tomography applied to scanner images. Earth's rotation was shown with a pendulum drawing Lissajous figures, and this evolved in drawings with light like when registering a gait in biomechanics or the interaction of particles in bubble chambers. All this has early origins and was gradually automated, hiding away the procedures in instruments and later in software of digital black boxes.

 

The grid structure is another tool heavily used in numerical computation. It was at some time a general cultural-economic phenomenon related to standardization, modular construction, and mass production, necessary after the Wall Street crash in 1929. The Centre for Cubic Constructions (CCC) of Graatsma and Slothouber in the Netherlands, voxelizing almost anything, reached its peak of success around 1970. But rigid geometry evolved in topology and strictly square grids evolved in lattices based on more general symmetry and of course the non-Euclidean geometry. The latter is marvelously illustrated in the graphics of Escher. This also illustrates that hyperbolic geometry had a mediagenic aspect and its architectural futuristic look explains its frequent use in scifi movies. It fits the 1960's idea of freedom and throwing off the yoke of regulations and a strong believe in technology and the future. However, thin warped surfaces were first adapted by engineers who patented their design, which made it difficult to be used by the wider community of architects.

 

Witt considers two forms of topology. There is of course the flexibility of the form with knots and minimal surfaces but he also considers a dual topology which he relates to networks like connections between cities, supply hubs, or movements on the workplace. Symmetry and group theory has some origins in crystallography, but it is also an incentive to study space-filling problems which play an important role in today's nanoscience. The science of materials, chemistry, biology, and computer technology, is similar when studied at nanometer scale. Witt doesn't discuss nanotechnology, probably because it is too recent. He mentions it as the inspiration for computational models like cellular automata, of the hypercube model in parallel computation. On a macro scale, these structures were embraced by designers like Buckminster Fuller, Arthur Loeb, and many others.

 

In a short concluding chapter, Witt adds his vision on the future of the interaction. His conclusion is that the designer will more and more make use of the 'black boxes' (now in the digital age taking the form of software) in which the procedural knowledge is hidden and these can be used as modules in a larger abstract design project. This is how (scientific) progress is realized.

 

The book is a nice illustration of the fact that is well known: mathematics is everywhere, also in the 'softer skills' of a designer or an artist. This is a well-researched and broad epistemological review, illustrating in detail how mathematics has cringed in the smallest capillaries of society and how social, economical, artistic, and scientific subcultures intertwine and mutually influence each other. The backbone of the story is however the design aspects of architecture interacting with mathematics which has sinews in many scientific-economic-cultural aspects of our society.

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Adhemar Bultheel is emeritus professor at the Department of Computer Science of the KU Leuven (Belgium). He has been teaching mainly undergraduate courses in analysis, algebra, and numerical mathematics. More information can be found on his homepage https://people.cs.kuleuven.be/~adhemar.bultheel/