This is the second in a series of posts exploring the ideas of the mathematician and design theorist Nikos Salingaros, and by extension his collaborator, the architect Christopher Alexander.

The previous post in this series examined the idea of a universal scaling hierarchy and how it could assist designers in deciding how many scales should be employed in buildings, and what the ratios between them should be. This post and the next will consider the concept of universal distribution, as a way of answering another important question: how ‘full' should each scale be? or how many elements should each scale in the hierarchy contain?

Imagine you set out to design a tree. While universal scaling is concerned with how big the 1st order branches should be in relation to the trunk, how big the 2nd order branches should be in relation to the 1st order branches, and so on, universal distribution is concerned with how many branches should be on the tree, and how many twigs, and how many leaves.

A good place to begin exploring universal distribution is by looking at the structure of fractals. For our purposes, a fractal is any pattern that is generated recursively, and has the property of scaling symmetry: it is self-similar at various scales, meaning that you can zoom in on any part of the pattern and it will look identical or near-identical to any other level of magnification, and will contain the same amount of detail. Fractals also have the interesting property of fractal dimensionality, but this property is less relevant to our purposes.

One of the simplest fractals is the Koch snowflake. Starting with an equilateral triangle, add to it three triangles with sides 1/3 the length of the original (meaning the scaling factor is 3); to the sides of these triangles add nine triangles with sides 1/3 their length; and so on.

Another simple triangle-based fractal is the Sierpinski gasket. In a sense it is the inverse of the Koch snowflake: it is subtractive (‘perforated') where the snowflake is additive (‘accretive'), and' ‘ingrown' where the snowflake is ‘outgrown.' Also, the scaling factor here is 2, not 3. Neither of these factors are very close to 2.72, which was proposed in the last post as a good approximation of the universal scaling hierarchy, but this doesn’t matter here: we are using fractals not to prove a point about universal scaling (we could easily create a fractal with a scaling factor of 2.72 if we wanted), but to introduce the concept of universal distribution.

The distribution of elements in the Sierpinski gasket is as follows:

0th order scale: 1 element

1st order scale: 3 elements

2nd order scale: 9 elements

3rd order scale: 27 elements

4th order scale: 81 elements

The distribution factor in this case is 3, i.e. each scale contains three times the number of elements of the previous scale.

As with the Fibonacci sequence (also recursive), fractals are everywhere in nature: trees and river systems are familiar examples. People find these recursive, scale-symmetrical structures inherently pleasing, because they generate just the right amount of information compression in the brain: they are neither monotonous, like a grid of triangles all of the same size, nor chaotic, like a field of triangles of random size, position, and rotation. The former possesses the necessary quality of order, but has no sense of life; it doesn't contain enough complexity to hold the mind's attention. The latter produces a sense of anxiety, because the mind cannot derive any rules or patterns from it to reduce the computational load. The objects and environments that give us the greatest satisfaction occupy a ‘Goldilocks zone' between these two extremes. They exhibit scaling coherence: they have structures at different scales, a scaling hierarchy, and a high degree of self-similarity at different ‘magnifications.' Neither simplistic nor chaotic, they stimulate the mind without overwhelming it.

How does this apply to buildings? Consider the baroque facade of Santa Maria in Vallicella, Rome.

There are bare areas devoid of detail, and within these areas there are centres of focus where the detail is concentrated, particularly around the parts of the facade where one's attention is naturally directed, such as doors and windows (incidentally, but not coincidentally, the human face displays the same kind of detail distribution: bare areas such as the cheeks and forehead, and smaller areas of concentrated, expressive detail, such as the mouth and eyes).

The facade of the gothic/romanesque Siena Cathedral shows a much denser distribution of detail typical of much gothic architecture - its distribution factor is higher than that of Santa Maria in Vallicella - but even here there is a hierarchical arrangement of blank areas and areas of greater detail.

Now let’s look at some pathological examples of universal distribution, or rather its lack. Take a textbook example of ‘high modernism,' the Villa Savoye by Le Corbusier.

The facade contains only a few large scales; it is almost completely devoid of small-scale details, and the mid-range scales are absent. The result is flat, barren, and unnatural; the building lacks the characteristics of universal scaling and distribution found in nature.

The lifeless character of ‘white box modernism’ was recognised as a problem, at least implicitly, by the post-modernists and deconstructivists, but in rejecting high modernism without understanding the root cause of its shortcomings, they only fell into another set of problems. The facades below are representative. They contain only one scale, or perhaps not even that. Are they all detail, or all blank? There is no hierarchy, only monotony. In the end, the effect is every bit as dead as the modernist deadness these contemporary architects were presumably seeking to avoid.