There’s no mystery to 'Sacred Geometry'

The reasons premodern designers used it are mostly obvious and banal

I think everyone has seen an image like this one. The ubiquity of ‘sacred geometry’ in the great architecture of history, as demonstrated by drawing shapes on top of a photo.

The use of a certain set of irrational ratios in the geometry of pre-modern building is a real phenomenon, and as I’ve mentioned previously on the podcast, I’m actually broadly a sacred geometry believer, in the sense that I think that compositions based on these sorts of ratios often have real aesthetic value.

There is obviously a low, as well as a high, form of this sort of analysis. The low stuff is often pure mystification — cryptic figuration, greek letters and scary-looking decimalised irrationals. But even the more elevated diagrams often serve to mask a process which is actually quite natural and straightforward behind a hieroglyph.

I can’t tell if this one is meant to be a joke? I love the application of this practice to William Wilkin’s National Gallery — a very weirdly proportioned building, by an scholarly architect with no real aesthetic sense at all. Link

From a modern, arithmetic-brained point of view, Golden Ratio-style figures and relations can seem mysterious, and their use thus suggestive a kind of higher, primal ‘forgotten’ wisdom. This basically their popular reputation. Their appearance in both nature and ‘Great Art’ lends them a sense of contact with fundamental or even gnostic truth.

But actually, at least in design, the reasons for their appearance and use are basically quite banal. They arise in the process of drawing itself. That they appear, in hindsight, as mysterious or forgotten at all is really about the displacement of a primarily geometric understanding of design by one grounded in new technologies of setting out, arithmetic and calculation.

Paestum — from Tons Brunès The Secrets of Ancient Geometry

In fact, if you really consider the generation of irrational rectangles within premodern practices of drawing, rather than being misled by the spirals, greek letters and infinite decimals, they are not merely simple and straightforward to create, but a natural, obvious, reliable choice, and even, in some sense, an easy one, a path of least resistance within the design process.

The reason that irrational figures like the Golden Ratio appear a lot in pre-modern design geometry, is that they are closely adjacent to the process of producing any kind of geometry at all.

For most of history, all precise geometric construction relied on a straight edge ruler (with few, if any, markings for distance) and a compass. This technology is incredibly old. It was clearly well-established when Euclid wrote his Elements c. 300 BC. Figures drawn with a compass have been found in a 14th c. BC tomb in Egypt.¹ Pretty much all early architectural treatises from Vitruvius (1st c. BC) onward stress the centrality of ruler and compass technique for proper architecture over naive building.

In ruler and compass construction, the geometry is all relational. With a modern CAD programme you can easily create precise geometry whose relationships are wholly arbitrary. You can draw a wall, and draw a window anywhere on it. In ruler and compass construction this is actively difficult and requires a certain amount of ingenuity, because drawing any geometry requires you to plot a series of interrelations, and naturally proceeds from the existing geometry already on the page.

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I want to show what I mean by demonstrating a bit of ruler and compass drawing.

Let’s start by doing a square —

i. First we draw a straight line with a point on it;

ii. Then we need to generate a perpendicular line — this is a three stage process. First we create two equidistant points on the base line;

iii. Then we use these points to create two intersecting radii above or below the original point;

iv. – vi. Now we have two perpendicular lines — we create the two corners on these lines and repeat the perpendicular line process at least once;

vii. – viii. We reproduce the side using the compass and join the points together, and now we have a square.

From this point, if we want to make a root-2 rectangle,² a classic piece of ‘sacred’ (i.e. irrational) geometry, it goes something like this (shown both by extension and insetting).

To point out something rather obvious — the process of creating the ‘sacred’ rectangle is actually lot less involved than generating the original ‘banal’ square. Even generating a root-2 rectangle out of nothing is no more involved than creating a square.

It’s actually a lot more work to generate a figure of arbitrary and rational dimensions (say 5:2 or 7:3) than an irrational one (e.g. 1:√2) — because in the latter case the geometry is already in the drawing to start with.

If we want to set out a Golden Rectangle it’s a little more involved, although the extra overhead is mostly in finding the centre line of one of the sides (see below).

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When you draw with a ruler and compass, the only way to create geometry is out of geometry that already exists on the page. Thus the construction of any drawing is a process of cultivating relations, ideally in as direct and succinct a way as possible.

The classic characteristics of ‘sacred’ geometry are this exactly the kinds of complex interrelations that you would expect from a process with this sort of nature. Complex ramification and involution of irrational rectangles and other simple figures is just what naturally results from a process in which the geometry is generated from the geometry that is already there. It’s not magic.

Parthenon — from Tons Brunès — The Secrets of Ancient Geometry

Naturally, none of this should be taken to undermine either the beauty or ingenuity of these constructions. Designing the Parthenon is still a considerable feat.

But it’s one that should be understood as arising naturally out of a certain modality, a design process. In a sense, the real mystery is how this process ever became mysterious. Somewhere in the history of design, perhaps around the introduction of the parallel motion and scale ruler, the arbitrary dimension became naturalised and the ‘sacred’ proportion became mysterious.

1. Arguably whenever you find precise geometric setting out there is some prima facie evidence of ruler and compass use. The Pyramid Khufu is 4600 years old — QED.

2. An explanation of what this is —

Magic Rectangles