architectural dialogue

I’m a mathematical and musical ignoramus. However, I do know that symmetry is rather more than what is described in Wikipedia.

The article there explains symmetry almost entirely in terms of dualistic patterning – you know: the left half is equivalent to the right half of a figure, explained more mathematically (and interestingly) in this article in terms of reflections, rotations, transformations, glides and the like. There are even comments upon quantum symmetries as well as symmetries in biology and, of course, in architecture.

Using the famous Leonardo diagram the author of this Wikipedia article almost gets there: it is “often used as a representation of symmetry in the human body and, by extension, the natural universe.”

Well, sortof … Apart from the man at the heart of this diagram, the point is his centredness with a square and circle of the same area – the circle has been ‘squared.’ A not dissimilar (though more crude) drawing from the same period might well have shown a man set half-way between earth and the sphere of heaven. What is at issue is cosmology and human life, not patterns as such …or, if you want, mankind as a patterned aspect of existence in the full, universal sense.

To ‘square the circle‘ is to transform the symbolic perfection of the circle into something earthly, to bring them into correspondence. The sphere of heaven is ‘squared’ into an earthly equivalent of the same area. They are harmonised. As Above, So Below. Geometrically, this is not straight-forward. But that’s not the essential point at issue – which is to achieve the ‘squaring’ as a living fact that characterises one’s life, not to play with a symbol.

In fact, one can’t a square a circle with compasses and straight-edge. It’s impossible (see the Wikipedia article; the fact was proven in 1882). One can be approximated to the other, but that is all. In other words, as it would be understood during the Renaissance, what is sensible cannot be exactly mapped to what is intelligible (which was disturbing). Nevertheless, mathematically and abstractly, such a correspondence of areas is inherently symmetrical in the most fundamental terms (1:1).

As Heron of Alexandria put it: “We call symmetrical the quantities that are measurable with a common standard of measure.” Symmetry is the first and most important principle of the created universe. And, as Hallyn notes (in The Poetic Structure of the World, when discussing Copernicus’s work): “the etymological meaning [is] the fusion of syn and metria signifying ‘the act of measuring together’ or ‘the common measure.’ From that perspective, Copernicus’ preoccupation with symmetry is related to his speculations on cosmic harmony.” Symmetry confers unity. For example, Leonardo described the growth of a tree in terms of ever-expanding circles. On any circle at some point in time the circumference of all the branches must equal the thickness of the tree trunk.

Similarly, given a situated challenge, the classical architect is challenged to find, use and activate a commensurable that is common to all the parts of the formal whole. And that’s not easy either.

Qualitatively rather than quantitatively, the notion of symmetry is intrinsically mythical and has a darker aspect in some popular cultures. Obtain a piece of finger nail or lock of hair of one’s enemy and one has a magical power over the whole of their being; the part embodies something of the whole in that sense characterising an organism and denied to a mechanism or aggregation. Hence, also, the traditional notion that, given a fragment of a classical building one can recreate the whole – probably nonsense, of course, like folk magic, but I’m sure you get the intentional idea and the creative aspiration. To create a symmetrical work is to fabricate a which is in harmony with the universe order, or, in the Christian tradition, to be a part of a restorative healing endeavour that corrects a local aspect of the fallen state of being. Here, architecture is an instance of alchemical magic and detail (as every architect knows)is important. In high and low culture the same principle can be found at work.

The flip side of symmetry is forms of dissonant monstrosity which, as Hallyn explains it, was expressed in Mannerism as an issue of a post-Copernicus disjuncture between sensible perceptions and intelligible realities. Sense impressions can be misleading. In Mannerism “we construct the resemblance [between man and God] by reconstructing internally the laws that govern Creation.” That resemblance is a task as well as a fact.In Mannerism, “the task of the recipient consists in working back from the exterior design to the interior design, to the idea that is signified. By reversing the direction of the flow of meaning, the recipient actualises the idea that he had potentially within himself. Having discovered the interior design of the author, he becomes the active icon and signifies, as a man, having (re)discovered his humanity, his identity to the other, in the sharing of the idea …”

Another consideration is the other side of the coin of harmony and commensurability: dissonance and incommensurability i.e., when parts can be related and compared, but there is no common factor that can be expressed. The incommensurable is an irresolvable ratio or number and thus has a frightening aspect which was so disturbing to the ancient Greeks that, we are told, they took the poor character who disclosed this truth out to sea and threw him overboard.

A tradition of a similar nature is the irrational ratio in music which has come down to us via medieval superstitions as ‘the devil’s music’ and, more latterly, is embodied in the blues and is a scary aspect of some heavy metal music to many a Christian (lock up your daughter!). The devil is in the detail.

(On tritones and dissonance see this wikipedia site.)

That is the anxious viewpoint that reads dissonance as evil. On the other hand, the incommensurable added to the traditional universal elements of earth, water, air and fire can be considered as ether – what belongs to a celestial region or is the mysterious presence of an indefinable Spirit, something infinite.

But perhaps the most interesting aspect of symmetry and its disharmonious contrasts concerns conduct and ethics rather than maths, religious values or superstition.

Behaviourially, symettry is implicity on-target. It has found the Aristotlian ‘mean’ – the keynote of ineluctable aptness – between otherwise incommensurable values. It is to exercise ‘practical wisdom.’ It is a ding not the dong, but it is hardly within a comfort zone. The mean is ultimately that commensurable that is ever an approximation to an incommensurable … if you see what I mean. One ‘squares the circle’ with intellectual difficulty and ever only approximately. In fact it is, by tradition the ‘golden mean’ (1 : 1.61803398874894820458634 … etc.) which, within the neo-Platonic tradition characterising architecture as we know it from the likes of  Alberti to Le Corbusier, has been found to be beautiful rather than frightening.

Such stuff remains within the baggage a contemporary architect lumbers around with. Or, if you want, it is one of the spirits that still haunts the profession.