Monday, 18 February 2013

Mysterious Mathematics

Any regular reader of this post (and I don't really have too many of those) would know that one of my pet peeves is misinformation.  My usual objection is to the limited technical performance data to be found to properly explain how buildings work, and how that holds back the growth of architectural knowledge.  What I rarely mention is some of the trite naivete that bedevils at least some writing about architecture.

Under the intriguing title "Architectural Icons Inspired by Mathematics" Kristen Avis writes in Architecturesource about.... well, the supposed role of mathematical inspiration in architecture.  I have no idea what compels her to invent it as she goes along, but she makes a brave effort to establish the claim that at least some architects are mathematical geniuses.  Unfortunately, that's just not true – with the possible exception of Christopher Wren, who really did change careers from famous mathematician to even more famous architect, responsible for much of the reconstruction of London after the Great Fire.

The reality is actually much more interesting.  The greatest strengths of architects are visualising forms and space, and using analogies, metaphors and similes for everything from planning principles to structural solutions.  You could almost say that their skills are in bypassing mathematics.  

One of the best illustrations of this is exactly the example Avis uses to headline her article: the Sydney Opera House.  According to Avis, Utzon spent over six years searching for the means to realize exactly the shells he wanted.  And if her invented quotation is to be believed, he didn't care how long it took, or how much it cost.  Had she bothered to consult any of the numerous sources, she would have known how far from the truth that is. 

For sure, Utzon's mind would have been fully engaged in the search for a solution, but the people who were trying to employ the mathematics and computational power for that purpose were in fact the nascent international firm of engineers led by the formidable Ove Arup.  Where the engineers failed with mathematical theory, Utzon succeeded by approaching the problem directly.  For him, the 'aha' moment came from the realisation that the surface of a sphere was by definition of a uniform curvature.  And that therefore thin slices of it could work as formwork for precast concrete ribs, using the simple expedient of blocking the one big steel form out at different lengths.

These were hardly esoteric mathematical investigations.  The story of the shells actually makes the opposite point.  Neither Utzon nor Saarinen (the jury member credited for plucking the evocative sketches from the refusees pile) actually understood enough about structures to know the original forms were impossible to build as shells.  But Utzon was in fact remarkably flexible in changing from the unrealisable thin, flat shells of his competition drawings, to what are in technical terms not shells at all, but a bunch of pointy arches leaning on each other.

In a similar way, Utzon's ideas for the interiors, never realised, were powerful practical observations from the natural world.  Thus the metaphor of the walnut to describe the separation of the interior halls from the exterior shell, and the attempt to develop the forms of the hall interiors from his simile for propagation of sound in the image of the stone dropped into a pond.

For the same reason as she misses the practical, experimental essence of Utzon's work, Avis invents a fanciful but incorrect interpretation of other key buildings.

Thus – notwithstanding the heady mathematical reverse engineering in Prof. Mark Burry's inspired, lifelong work with Gaudi's Sagrada Familia – the simple fact is that we have the  original physical models Gaudi used to ensure that his brickwork remained in pure compression.  He did it, possibly ignorant of any mathematical theory, by the simple expedient of building the models upside down in string and weights.  Any forms that the strings took up in simple tension could be relied on to be in simple compression when turned downside up, so to speak.

And I hate to break it to her, but the apparently complex forms of the Endesa Pavilion in Barcelona can be produced by a nine-year-old pushing and pulling on a SketchUp model. It is not necessary to use 'genius mathematical formulas'.  Arguably in this case, were it not for the necessity of being able to measure the outcome in order to construct it at full size, physical model studies using a simple sundial might well be even quicker and more efficient than relying on software.

Avis comes closest to an accurate proposition on mathematics and architecture, when she quotes Mark Burry on the loss of esoteric number systems to guide the development of proportions and patterns in buildings.  There is no doubt in my mind that those number systems still should be understood, and still could be deployed in contemporary design.   They were transmitted by artists through the experience of millennia, critically examining satisfaction with proportions and other aesthetic concerns.  But otherwise, complex mathematics will now inevitably reside in the software, and quite contrary to Avis' hopes, are likely to be less and less understood directly. As we already see with Rhino and Grasshopper, the complex digital models are most likely to be manipulated with parametric sliders, to produce mathematically optimised but not necessarily good architecture.

If architects generally are to be credited with any sort of mathematical insight at all, the best that could be said is that some of them, like the artist Escher, sometimes produce forms which the true mathematicians recognize as giving concrete expression to some of their more difficult to explain ideas.That is quite enough of a miracle.  For the rest, mysterious as it might seem that certain number systems and aesthetic satisfaction converge, we are not much helped by inaccurate and fanciful misinterpretation of the relationship between architecture and mathematics.

For anyone who doesn't trust me on this, I commend reading that wonderful passage in the middle of Umberto Eco's book Foucault's Pendulum, where in a few short sentences he demonstrates that human beings with their ten fingers and various other anatomical attributes, were destined to attribute magic to some numbers, rather than others.

And anyone who needs a quick, authoritative review of the story of Utzon's's shells, could do worse than to read  Eric Ellis in The Spectator from 2008, fortunately available online, here.


A much better story of the Opera House shells was sent in by Nino Bellantonio, and has been posted as an item in its own right at Mysterious mathematics 2: The Sydney Opera House
Well worth reading!

1 comment:

Freddie Kareh said...

Much in the same way that classical architects Michaelangelo and Garnier were not accomplished musicians, John Utzon and the architects at IaaC do not have degrees in mathematics. Even though their architecture may pay homage to these fields, they are not factors which drive the design process. Subsequently, Kirsten Avis's argument that mathematics underpins architecture stems from over analysis, and is grossly misleading as it attempts to rationalise what is essentially a creative form.

Architecture subconsciously harks back to music and mathematics purely because they are fields in which different elements can be composed to produce a harmonious outcome. Furthermore Avis's argument is akin to an article published in the Daily Mail1 on March 9, 2001, which argues that “the best footballers rely on mathematics to reach the top of their game.” Although this statement can be argued in theory, it is not applicable in reality as during a game, footballers only have a split second to calculate how much how much power or swerve they should apply to the ball.

In a similar fashion to architects, their actions are a direct outcome of hard work and natural ability, not an intricate understanding of mathematics. If Utzon sat down to resolve every detail of the Opera House to concur with mathematical principles, it would never have been built.