Abstract. Dirk Huylebrouck queries Nexus Network Journal readers on the possible relationships between architecture and higher mathematics.

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Query: Is There a Relationship Between Architecture and Higher Mathematics?

Date: Sat, 14 Jan 2006 16:27:07 +0100
From: Dirk Huylebrouck <huylebrouck@gmail.com>

I have a question for the Nexus readers: "Are there any relationships between architecture and higher mathematics?" By "higher" I mean, mathematics at the level of 1st master and up. The topology paper in the NNJ vol. 7 no. 2 by Jean-Michel Kantor goes in that direction, but the reason why I propose it is different: I recently wrote a paper (in Dutch) on "Africa and higher mathematics". "True, die-hard" mathematicians sometimes take architectural math as "baby math", and of course, related to Africa, there are some down-to-earth social (extremist) influences involved. Nevertheless, in the architecture case, the question could be seen as a modification of Mario Salvadori's query at the first Nexus conference for architecture and mathematics in 1996: "Are there any relationships between architecture and mathematics?"

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Date: Sat, 14 Jan 2006 16:27:07 +0100
From: Kim Williams <kwilliams@kimwilliamsbooks.com>

This is a question that I have given quite a lot of thought to, so I want to put some of those thoughts in black and white, and then encourage NNJ readers to respond. The kind of "baby math" that Dirk refers to is probably that which is often discussed in our pages: the proportional relationships involved in the architectural orders, for instance, based on simple ratios of whole numbers; or the ratios of architectural dimensions derived from elementary geometrical figures such as the square or golden rectangle. In this case, I agree: there is not much "higher mathematics" here. But when we look at the world of ideas, there is more going on than might meet the eye to begin with. The relationship between architecture and ideas is the inverse of the relationships between mathematics and ideas: where we often find that very simple, easy-to-understand ideas underlie some very complex mathematical structures (such as how the idea of sensitivity to initial conditions underlies the science of chaos), instead some very complex mathematical ideas underlying their more elementary architectural expression. For example, architects a few years ago were very excited about breakthroughs in chaos, and now research in topology is providing fertile ground for architects. However, architecture imposes two great limits on its practitioners. One is that we human beings have some very simple needs that must be met: we have to have horizontal surfaces on which to walk; we feel disoriented without vertical walls; we can feel downright frightened by non-vertical supporting elements; architecture must be constructed. These requirements limit architectural experimentation: we might like to hypothesize about the Möbius strip in architecture (see the paper by Vesna Petresin and Laurent-Paul Robert) but we probably couldn't live in a Möbius house. But there is no limit to the architectural imagination, and virtual architecture provides very important tools for visualizing the application of higher mathematics to architectural forms. Such visualization can then be modified to meet human requirements.

Another source of higher mathematics in architecture is to be found in mechanics. Building collapses provide particularly good, if tragic, opportunities to study structural behaviour, into which I suspect higher mathematics often enters. Ah, but you say, that is engineering and not architecture. Well, I suppose that depends on how you view architecture: in my opinion, any aspect of the built environment is included under the umbrella of architecture, and our aim at interdisciplinarity reflects that fact.

Dirk's question is related to the query proposed by James McQuillan:

Do any contemporary architects understand enough about mathematics (beyond additive planning and accidental occurrences) to apply it in their work? And even if they did, why should they do so, given the collapse of ancient mimesis and related understanding until the 18th century, which was the foundation of such applications in the past? (Click here to access this page in the NNJ.)

In spite of many architecture students' objections to learning mathematics, more than just a fundamental understanding of mathematical concepts and even a way of thinking is essential to the architectural education. This is why we publish papers about Didactics in the NNJ.

I hope that readers will think about and respond to these queries: this is an important part of keeping dialogue open within the NNJ community.

Date: Thu, 19 Jan 2006 12:17:41 -0500
From: Chandler Davis <davis@math.toronto.edu>

Modern study of collapse (meaning, since 1970) often uses modern (since 1960) deep mathematics. It is true that we don't get many architecture undergraduates interested in our mathematics courses, and even the mechanical engineering students may be sort of resistant; nevertheless there is this current of study of failure of beams, etc., and the people who do it (whether or not they are called engineers) are doing mathematics. This is important, but it is not what Dirk is mostly fishing for. He wants to know, does an advanced mathematical idea go into the architect's conception of a structure. Please let us understand the question as referring primarily to mathematical ideas correctly understood and relevant. Only in passing are we concerned with misunderstood ideas, or superficial reference to ideas, or use of exciting geometry as ornament distinct from structure(like the Moebius band sculpture on the facade of TsEMI in Moscow).

From: Doug Boldt <Dboldt@lcmarchitects.com>
Date: Thu, 19 Jan 2006 11:41:41 -0600

I dispute the paragraph from your email above! We don't need 90 degree angles to be happy and feel safe. We have been conditioned and fooled into thinking we are comfortable in our boxy prisons. Fractal Geometry is the key to the future of architecture. We are on the brink of a new Century and a new wonderful new architectural age has begun. It is right before our eyes and no one can see it -- just as in 1906 how many people could see and understand that they were living in the age of Modernism. Look in any arch magizine in 2006 and the pages are full of fractal architectural examples.

The biggest hurdle to overcome is our understanding of fractal geometry. I can't solve a logarithmic equation but I suspect I understand fractal geometry or at least its application better than most. If Fractals are the geometry of nature as Mendelbrot proposed, then fractals rarely if ever reiterate at smaller and smaller scales to infinity. Those pretty fractal computer generated pictures we are so used to seeing are artifical. In natural fractals, shaped over time by the forces of wind, water, fire, gravity, etc., reiterate 1 time or 2 or 3 times, then the molucular composition causes a new fractal to appear. The same is principle can be (and is being) applied to Architecture. Unlike the rigid and dogmatic Modern Movement, the Fractal Movement is inspired by the limitless patterns of nature in all all its aspects, infinite and ever changing.

Date: Fri, 20 Jan 2006 03:17:12 -0800
From: Lionel March <lmarch@ucla.edu>

I read your note from Dirk and thought you might like to know that George Stiny's SHAPE: Talking about Seeing and Doing is about to be published by MIT. George's work is based on extensions to Boolean algebra known as Stone
algebras, Euclidean group transformations, lattice theory -- all of which might come under the heading of 'higher' mathematics. As you probably know an area like fractals is subsumed by George's shape formalism as a special case. Incidentally, Mario Salvadori was a strong supporter of George's work and wrote a forward to Algorithmic Aesthetics.

I recently wrote an endorsement for MIT at their request:

"Stiny is to 'shape' as Chomsky was to 'word' or Wolfram to 'number.' In my view, though, Stiny may well prove to be the most radical of the three. How different a place his pictorial world is from standard textual or digital worlds: with shape there is no vocabulary, no syntax, no bits, no atoms. As Stiny draws, he talks. Shapes and shape rules bear the force of argument. These drawings are to be looked at keenly, even traced and redrawn by the reader. The supportive text illustrates what can be seen and done, providing both a personal and intellectual history. Through its drawings and maxims, Shape challenges much conventional wisdom in philosophy and education, in computer science and artificial intelligence, and in design and the visual arts."

Date: Sun, 22 Jan 2006 13:37:08 -0400
From: Emanuel Jannasch <ejannasch@hfx.eastlink.ca>

As I see it, builders can “do” math in two distinct modes. In the first, we calculate a structural form, set out a geometric pattern, or devise some discrete structure, and try to build accordingly. Our intentions in this mode may be aesthetic (creating pleasant visual proportions) they may be purely instrumental (sizing pipes), or they may be hybrid (as in shell roofs) Sometimes deep aesthetic qualities emerge from the mathematical solution of a technical/commercial problem, as in Maillart’s bridges. But whatever we achieve with builder’s math, in a world where an “applied” subject is considered inferior to its dreamier cousins, it will always be seen as lowly or in the phrase of your hard-core types, “baby”-ish.

In the second mode, mathematics is directly embodied in our work, and only later does an abstract thinker devise a symbolic description of our accomplishment. Many catenaries and funiculars were built before mathematicians had a vantage point “high” enough from which to describe them. Tesselations were treated exhaustively in practice centuries before the theory was in a podsition to concur. For millenia, boatbuilders have been bending battens and planks to arrive at complex surfaces of least local change in curvature which accord very neatly with their hydrodynamic objectives. In this respect, the bent spline on the naval architect’s drawing board is a calculator, and the bent battens enveloping a hull form under construction are computer controlled machines, and I mean this in a real, fuctional sense, not as a metaphor. In both cases, the workings of the calculator were not given abstract form until Schoenberg’s 1946 theory of mathematical splines. In this second mode, the real world is always richer and more complex than mere description, and the mathematicians’ approximation is destined to play catch-up.

The opposition between these two perspectives is ancient. Medieval philosophers kept up a heated debate between the (Platonic) “realism” of the former orientation and the “nominalism” of the latter. I suspect replies to your query will fall into one camp or the other, with a preponderance coming from the realist side, as this is the inherent tendency – I would submit – of the Nexus readership. I am reading into your query a nominalist bias more like my own and wish you all the best with your investigation. I look forward to your discoveries!


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