Proportion: Science, Philosophy, Architecture

By Richard Padovan | Go to book overview

Chapter three

UNIT AND MULTIPLIER

For measuring and counting, two things are indispensable: a unit and a system. That is to say, one needs both a static principle and a dynamic multiplication-system.1


3.1ORDER AND COMPLEXITY

Before we explore further the historical relation between science, philosophy and systems of architectural proportion, we must first try to get a clear idea of what these systems are. I shall define them as methods of ordering the relations between the measures of a building, and thereby also between the two- and three-dimensional shapes to which these measures give rise. However, it is not just a matter of reducing the number of sizes and shapes to a minimum, but of achieving unity within a multiplicity of different elements.

In his book Aesthetic Measure2 the American mathematician George D. Birkhoff (1884-1944) treats order and complexity as contraries. The gist of his theory is contained in the formula M=O/C: that is, (aesthetic) Measure equals Order divided by Complexity. The aesthetic value of an object increases, therefore, in direct proportion to its degree of order, and in inverse proportion to its complexity. And since he holds that the order and complexity of a two-dimensional figure can be exactly calculated—its order, by determining the degree of vertical or rotational symmetry, perpendicularity, etc., its complexity by counting the number of straight lines needed to delineate it—it is possible to quantify the aesthetic merit of any geometrically definable object. At the top of Birkhoff’s order of merit are the square and the square grid.

Against this, I would argue that the elimination of complexity tends towards disorder, or rather non-order. Uniformity—the repetition of a single element, for instance an uninflected grid of square panels of standard size—is, paradoxically, the antithesis of order. Where uniformity is absolute and complexity approaches zero, as happens with a

1. H.van der Laan, Het plastische getal, E.J. Brill, 1967, p. 3.

2. G.D. Birkhoff, Aesthetic Measure, Harvard University Press, 1933.

-40-

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