This is the first in a series of posts exploring the ideas of the mathematician and design theorist Nikos Salingaros, and by extension those of his mentor and collaborator, the architect Christopher Alexander.
introduction
Buildings, like biological organisms, are organized at a number of different scales, from the largest (the overall dimensions of the building) to the smallest (the texture of the sand in a render coat). This fact presents the architect or designer with a choice: he may either explicitly address it, and attempt to answer the questions and challenges it raises in the design process; or he may choose to ignore it and evade its challenges entirely; either way, his choice will be evident in the results.
How many scales should a building contain? What should the ratios between them be? How many elements should each scale contain? And what should the ratios between the number of elements in each scale be? All of these questions point to a deeper issue: why are some buildings so beautiful and seem so healthy, while others are so ugly and pathological?
This and subsequent posts will try to answer the above questions via an exploration of the ideas of Nikos Salingaros, with the aim of outlining a simple and practical methodology that can might be useful to anyone whose interest is in designing buildings that are beautiful and healthy, rather than ugly and sick.
the universal scaling sequence
You are probably familiar with the Fibonacci sequence, where each number in the series is the sum of the previous two numbers, beginning with 0 and 1:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377 . . .
From this we can derive another sequence, more relevant to our purposes, called the universal scaling sequence, which is obtained by removing alternate terms from the Fibonacci sequence:
1, 3, 8, 21, 55, 144, 377 . . .
The universal scaling sequence can be applied to architectural design in the following way: assign the arbitrary size of 1 to the largest scale in a building, then to the next scale down in size (in the same linear dimension) assign the value 1/3, then 1/8, then 1/21, and so on. Or, start at the other end and assign to the smallest scale in a building the arbitrary size of 1, then the next scale up in size should be 3, then 8, then 21, and so on.
To see how this works, try the following: start with a blank building facade with an overall height of 10m, which is your ‘first-order’ or ‘1’ scale. From this, the sequence might suggest a floor-to-floor height of 10m/3 = 3.33m, a window height of 10m/8 = 1.25m, an ornamental cornice height of 10m/21 = 475mm, a sill or window frame height of 10m/55 = 180mm, and finer ornamental details at 10m/144 = 70mm and 10m/377 = 25mm. Now do the same for the horizontal dimensions. Say the facade is 6m wide: 6m/3 = 2m, 6m/8 = 750mm, 6m/21 = 285mm, 6m/55 = 110mm, and 6m/144 = 40mm. Using these figures, I drew the facade below in about 15 minutes, letting the scales determine the design without much input from my ‘individual creativity.’
It’s nothing special, but that’s the point- designing in this way is highly forgiving.
Interestingly, if you take any term from the Fibonacci sequence and divide it by the previous term (e.g. 233/144), the number obtained approaches the famous golden mean, 1.618, as the two terms get larger. Likewise, if you divide any term from the universal scaling sequence by the previous term, the answer approaches 2.618, which is the square of 1.618. However, because the Fibonacci sequence is not a true geometric sequence (where each term is the nth power of the previous term), it is impractical to use it in most design situations. Nikos Salingaros proposes the natural logarithm e, 2.718, as an acceptable geometric substitute.
The Fibonacci sequence in nature
A logarithmic spiral in nature
These constants are not just arbitrary abstractions: they are found throughout the natural world (hence the name universal), from spiral galaxies to molluscs to the number of petals on flowers. Their great aesthetic and mathematical appeal sometimes tempts architects into designing golden rectangles into their buildings, citing the (possibly apocryphal) example of the Parthenon, or designing buildings that look like seashells and other organic forms. These kinds of applications are over-literal and fundamentally misconstrued. The real significance of the universal scaling sequence is that it provides a useful tool for checking that a building's various scales (the dimensions of building elements as measured along the same axis) are a reasonable approximation to the ‘natural' hierarchy of the universal scaling sequence, i.e.:
1. Few scales of the sequence are missing;
2. There are no significant scales that fall between the terms of the sequence; and
3. The ratios of adjoining scales are close enough to 2.618, or 2.718.
Of course, real-world considerations mean that the scales in actual buildings will rarely conform to the mathematical ideal. In practice, the ratios are rounded off into rules of thumb, like the vernacular builder's ‘rule of three’: each scale in a building should be roughly three times the size of the next smallest scale, and 1/3 the size of the next largest. At any rate, the important thing is not strict adherence to the numbers, but understanding the concept of the universal scaling hierarchy as an ideal to aim for.
Adoption of the universal scaling hierarchy has several benefits: it imposes non-arbitrary limitations on design (limitations are good); it guides the designer in making more effective design decisions; and it aids in the diagnosis of design flaws. If a building or facade feels too busy, for example, it may be because it contains too many scales that fall between those on the universal scaling sequence. Conversely, omitting scales from the sequence results in a collapse of the scaling hierarchy and a barren, lifeless appearance.
Inappropriate spacing of scales and collapse of the hierarchy: only a few large and seemingly random scales are present in this facade.
This classical facade presents a full hierarchy of scales, from large (the ‘bay’ or distance between columns) to small (the width of the dentils in the cornice).
Consider another example: a door and its architrave. If the door is a standard 820mm wide door, universal scaling would suggest an architrave with a width of 820 / 2.718 = 300mm or so. This might sound excessive, but if the architrave itself is further subdivided (by mouldings, painted or carved patterns, or other ornament) into successively smaller scales in the hierarchy (say 110mm, 40mm, and 15mm), the result is a door with great presence. Economic realities generally meant that such opulent doors were reserved for classical or civic architecture; in humbler vernacular buildings, architraves were typically around 100mm wide, which skips a scale but is still far more effective in expressing the idea of ‘doorness’ than a modern ‘architectural’ door, which might have an ‘architrave’ as thin as 10mm. In this case there are three scales missing between the scale of the door width and the scale of the frame width, which is a bit like a tree consisting of a single, massive trunk covered in tiny twigs: our brains cannot ‘span the gap' between the two scales to form a coherent connection between them, and the hierarchy collapses.
Classical door with ornamented surround
Modern door with "frame"
Given that the human perceptual system evolved in the natural world, where these scaling sequences and ratios are the literal rule, it should not be controversial to suggest that people have an instinctive affinity for correct scaling ratios, and that buildings designed around a universal scaling hierarchy hold an innate aesthetic and emotional appeal for us, as evidenced by the fact that such buildings are found across all ages and cultures, in both classical architecture and vernacular building traditions. In fact, as Salingaros points out, there are only two significant exceptions to this universality: one being the ‘death architecture’ of Egyptian Pyramids and defensive fortifications, both of which are deliberately designed to be repellent; and the other being modern architecture.