Numerology

… is alive and well, and living somewhere between Chartres cathedral and Wall Street. I am one of the few remaining humans not to have read The Da Vinci Code, but it seems clear from what I have been told that it has given our collective system another potent dose of the Fibonacci virus. This is something that, as the author of a forthcoming book about Chartres, I knew I’d have to grapple with at some point. That point came last week, in the course of a fascinating day in York exploring aspects of Gothic in a summer school for architects. I’m too polite to mention names, but a talk in the evening on ‘sacred geometry’ was, for this naïve physicist, and eye-opener about the pull that numerology continues to exercise. There is plenty of room for healthy arguments about the degree to which the Gothic cathedrals were or weren’t designed according to Platonic geometry, and there will surely be no end to the time-honoured practice of drawing more or less fanciful geometric schemes on the ground plan of Chartres using thick pencil to reveal the builders’ ‘hidden code’. John James is a part master of this art, while Nigel Hiscock is one of the few to make a restrained and well argued case for it.

But the speaker last week was determined to go well beyond Chartres, by revealing divine geometry in a teleological universe. Most strikingly, he suggested that many of the planetary orbits, when suitably ‘corrected’ to get rid of the inconvenient eccentricity, become circles that can be inscribed or circumscribed on a variety of geometric figures with uncanny accuracy. One such, relating the orbits of the Earth and Venus, is shown in Figure a. The claim was that these ratios are extraordinarily precise. For example, in this case the orbits fit the construction to greater than 99% precision, it was asserted.

This seemed to me like a wonderful exercise to set A level students: how might one assess such claims? I decided to do that for myself. The Earth/Venus case is a relatively easy one to test: the ratio of the two ‘circular’ orbits should be equal to the square root of 2, which is approximately 1.414. Now, there is a question of exactly what is the right way to ‘circularize’ an elliptical orbit, but it seems to me that the most reasonable way is to use the average distances of the two planets from the sun – the mean of the major and minor axes of the ellipses. This apparently gives 149,476,000 km for Earth (to 6 s.f.) and 108,209,000 km for Venus. That gives us a ratio of 1.381. Not within 99% of root 2, then – but not bad, only out by about 2.4%. (I’m sure ‘sacred geometers’ will insist there is a ‘better’ way to circularize the orbits, but I think it would be hard to find one that is as neutral as this.)

How do we know if this near-coincidence is mere chance or not? Well, a relatively simple test is to consider all the geometric figures of the type shown by the speaker, and see how much of numerical space is spanned by the corresponding ratios – given a leeway of, say, 3% either way, to allow for the fact that (as was explained to me) the real world lacks pure Platonic perfection. So I did this, considering just the inscribed and circumscribed circles for the perfect polygons up to the hexagon, along with a couple of others in which these polygons are adorned with equilateral triangles on each side (see Figure b). (I know the latter look a little contrived, but one of them was used in this context in the talk.) I’m sure one can come up with several other ‘geometric’ figures of this kind, but this seemed like a reasonable minimal set. The ratios concerned then cover the space between 1 and 2. With the exception of Mars/Jupiter, all of the planetary orbits produce a ratio within this range when we consider each planet in turn and the next one beyond it.

Now, at the low end of the range (close to 1), one can get more or less any number by using a sufficiently many-sided polygon. For hexagons, the two circles produce a ratio range of 1.12 to 1.19, allowing for 3% variation each way. And in any event, while I don’t know the exact number, it seems highly likely from the dynamics of solar-system formation that one can’t get two orbits too close together – I suspect a lower limit on the radius ratio of something like 1.1.

OK, so adding up all the ranges covered by these figures leaves us with just 32% or so of the range between 1 and 2 not included. In other words, draw two circles at random with a radius ratio of between 1 and 2, and there is a two in three chance that you can fit this ratio to one of these geometric figures with 3% precision. With seven pairs to choose from in the solar system, we’d expect roughly 4-5 of them to ‘fit’.

It took me less than an hour to figure this out using school maths. The speaker last week was hardly lacking in arithmetical skills, but it seems not to have occurred to him to test his ‘coincidences’ in this way. I can only understand that in one way: these numerological arguments are ones he desperately wants to believe, to a degree that blinds him to reason.

That was born out by other statements about the ‘foolishness’ of science that would have been disproved with the most minimal of checking (such as that scientists discovered in 1987 that crystals with five-fold symmetry are possible, while Islamic artists had known that for centuries). The only explanation I can see is that of judgement being clouded by an anti-science agenda: we rarely question ‘facts’ that fit our preconceptions. I have to confess that I do find it troubling that educated people develop such an antipathy to science, and such a desperation to believe in some cosmic plan that generally turns out to be remarkably banal (whereby God fills nature with cheap number tricks), that they abandon all their critical faculties to embrace it – and to serve it up in a seemingly authoritative manner to people who don’t necessarily have the resources to assess it. I’d like to recommend to them the words of Adelard of Bath, who suggests that the intellectuals of the twelfth century had a rather more astute grip on these matters than we sometimes do today: “I do not detract from God. Everything that is, is from him, and because of him. But [nature] is not confused and without system, and so far as human knowledge has progressed it should be given a hearing. Only when it fails utterly should there be recourse to God.”