Mysterious mathematics 3

Don't read this if you are sick of rants.

I wanted to do a follow-up of my previous posts on architecture and mathematics.  What better way to start than googling the two terms.  Bad mistake.  It can give you indigestion.  Not the number of hits, but the knotting stomach induced by the sheer inanity of some amateur design journalist grappling with the subject.  If you are masochistic, go to 10 Amazing Examples of Architecture Inspired by Mathematics on Flavorwire. The bloopers make you cringe.

First up, an innocent enough project for a building for an undefined Buddhist program, incorporating a mobius strip surface.  To justify it, the author manages to describe the lumpen stasis of a traditional stupa and its circumambulatory path as a 'twisty space'.  She should visit one sometime.

This introductory example of the mathematical marvels in architecture is also conspicuously a project, not an actual, built building.  I wish they would stop doing that, especially when there is a perfectly good, prize winning built example of a Klein Bottle House (the proper 3D version of the mobius strip) near Melbourne, Australia.

At least Walter Netsch's US Air Force Academy’s Cadet Chapel in Colorado Springs, Colorado has the merit of having been built.This piece of full sized origami is described as shaped like a tetrahedron,which it clearly isn't.  But what can you expect from an author who first quotes the definition of a tetrahedron as a four sided object with triangular sides, and then promptly confuses it with pyramids, which have five sides including a square base.

I can afford to gloss over the geodesic domes of the Eden Project because it doesn't have anything offensively wrong.  But I am offended by the breathy, guileless description of Foster's office tower in London known as The Gherkin.  According to the author: "the modern tower was carefully constructed with the help of parametric modeling amongst other math-savvy formulas so the architects could predict how to minimize whirlwinds around its base."  Give me a break!

Yes, parametric modelling is a key feature of the practice's characteristic approach to generating apparently complex shapes.  Which are surprisingly buildable, because of the way the parts can be scheduled for automated manufacture and reliable logistics in assembly.  This use of bespoke software is lucidly explained in Chris Abel's admirable book Architecture, Technology and Process,  and incidentally, compared at length to the way Gerhy uses the CATIA software suite to solve his complex building geometries.

OK.  I am actually grateful for the reminder of the Philips Pavilion at the 1958 World’s Fair. And this time, the author commits no bloopers, probably because she simply names the dominant geometry as the use of hyperbolic paraboloids employing steel tension cables.  Why do I miss mention of Felix Candela and his much earlier experiments with the same form in concrete?  Or the factoid that these curved forms can be generated from repeated straight lines, which made them buildable by conventional building materials.  Therein lay the fascination with their mathematics for architects like Candela, trying to build his formwork out of timber, before casting it in plastic concrete.

Mathematical allusion is getting pretty desperate when the owner of this Toronto home, named the Integral House is "a calculus professor who wrote textbooks and wanted to incorporate the mathematical sign into the home’s name and design. Undulating glass and wood walls also echo the shape of a violin."  Might be delightful, but the maths is in the room acoustics and the material choices and assemblies of the 200 seater performance venue, not in the trite simile.

 I've previously pointed out that any undergraduate architecture student could generate Barcelona’s Endesa Pavilion by trial and error in ten minutes, with the free SketchUp modelling software.  The "mathematical wizardry" is entirely that of the people who make the program run fast enough to use, not in translating the solar geometry algorithms.


I'll gloss over Cube Village, built by Dutch architect Piet Blom, because the author doesn't attempt a mathematical description of it.

And then, joy of joys, a truly worthy project.  Gaudi's Sagrada Familia cathedral in Barcelona.

The author enthuses over the virtuoso exercise in numerology, itself unremarkable in the tradition of mystical religion.  She does mention the hyperbolic paraboloids of surfaces, and even the "catenary arches".  She obviously has no idea of the true magic of Gaudi's insight, that the optimum shapes of the compressive structure in masonry could be empirically developed, by modelling them as pure tension, upside down as catenary chains, loaded with intuitive estimates of the superimposed loads.

And so we end with a mathematical whimper, a fractal gas station makeover in Los Angeles.  This might seem a bit unfair.  There is something serious about fractal geometry, and there is definitely something important about how the form generation potential of meshed surfaces has lately influenced architecture everywhere.  My angst is two-fold.

On the one hand, I refer back to the insight that hyperbolic paraboloids can be generated from straight sticks or cables to employ conventional building materials.  Employing a software package like 3D Studio Max automatically creates 'mesh' substitutes for the complex curved surfaces, and thereby turns them into forms that can be clad by combinations of flat sheets cut or folded into triangles, the joints typically filled with generous beads of mastic.  This is the main reason why we can now build all the rendered fantasies that so dominate the architectural magazines.  But it is also the reason why so many of these tortured forms end up with all those slivered triangles all over their surfaces, as they do in the example used.

My second reason for despair?  The author clearly doesn't understand any of this.
Read the original here.

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.

I hate to break it to her, but it is not necessary to use 'genius mathematical formulas'.

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.

SEE ALSO

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!