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**Mathematics and Mysticism**

*by Andrew May*

First published in British Mensa's *Aquarian* newsletter, February 2006

In philosophy, there is a fundamental dualism between the material and spiritual worlds -- between the physical and metaphysical, between science and religion, between logic and intuition, between rational and irrational. The word "rational" comes from a Latin root meaning to reason or calculate. Its opposite, irrational, is often used as a pejorative but can also mean "ineffable" or "beyond mortal ken".

In mathematics, rational and irrational have a different but curiously parallel meaning. A rational number is one that can be expressed as a ratio of two integers (for example "three-quarters" is the ratio of integer 3 to integer 4) while irrational numbers cannot. In this mathematical sense, the word "rational" is related to "ratio", but ultimately its etymology is exactly the same as the mainstream usage. An irrational number is one that cannot be calculated according to the rules of arithmetic. "Pi", the ratio of the circumference of a circle to its diameter, is irrational, as is the square root of two, or the square root of three.

So where do irrational numbers come from? Are they simply a theoretical abstraction invented by a nerdy little man with thick glasses and a propeller beanie? Far from it -- irrational numbers are actually MORE natural than rational numbers, not less. Arithmetic, that bastion of rationality, is an artificial man-made construct -- but one that is so well-established, and taught from such an early age, that few people realize it. Nature’s own mathematics is geometry, and geometry is all about irrational numbers.

The longest side of a right-angled triangle is equal to the square root of the sum of the squares of the other two sides. This is Pythagoras’ theorem, but Pythagoras wasn’t a mathematician or scientist -- he was the foremost mystical philosopher of ancient Greece. He was a contemporary of the Buddha, and the last mainstream Western philosopher to believe in reincarnation and the transmigration of souls. Pythagoras realized there was an intimate connection between geometry and true, underlying reality, and this is what motivated him to create his theorem -- a theorem that is inseparable from irrationality. Imagine a unit square, and use Pythagoras’ theorem to calculate the length of the diagonal. The two shorter sides are both one, so their sum is two, and the square root (which is the number we’re looking for) is the square root of two. An irrational number!

The connection between irrational numbers and mysticism can be seen most clearly in mediaeval Gothic architecture. This was a style of architecture, characterized by pointed arches, that flourished in France, Britain and neighbouring countries from the 12th to 15th centuries. All the great cathedrals, abbeys, parish churches and chapels of this period were built in the Gothic style, from Chartres Cathedral in France to Rosslyn Chapel in Scotland. Yet viewed in a broader perspective, Gothic was something of an anomaly in the history of Christian ecclesiastical architecture. Earlier churches, later churches, churches further south, churches further east -- all have conventional "Classical" styling with round arches. There’s an indefinable quality about these Classical-styled churches that makes them somehow less sacred, more secular than their Gothic counterparts. Perhaps this isn’t surprising, because they generally belong to less pious, more humanistic times -- St Paul’s Cathedral is a wonderful building, but it’s a monument to Man, not God.

Geometrically, Classical architecture is based on the circle, straight line and square. To draw a circle, on an architectural scale, you simply need a piece of string. You anchor one end of the string, and sweep the other end around to mark out the circle (or semicircular arch). You can draw a straight line with a piece of string, too, by stretching it out tight. But you can’t draw a square with just a piece of string. There’s no problem drawing four sides of equal length, but you also need a perfect right-angle between them. For that, you need a separate tool -- a set-square. That’s a lot more complicated than a piece of string, because the set-square has to be constructed and calibrated properly in the first place.

Gothic architecture doesn’t need a set-square -- it only needs a piece of string. While it’s difficult to draw a perfect square with string, there’s another shape that’s much easier -- a regular hexagon. This is illustrated in Figure 1. First you draw a circle, then you use the same length of string (i.e. radius of the circle) to mark points A, B, C, D, E , F around the circumference. Finally you join these six points together. The result, after just a few seconds, is a perfect hexagon.

(Figure 1)

Unfortunately, it’s not a hexagon we were after but a square. The hexagon’s no use because all the angles are 120 degrees, and we want right-angles (90 degrees). However, it’s still possible to extract right-angles from the hexagon, as shown in Figure 2. First draw in the horizontal diameter AD, then the two vertical lines BF and CE. These produce right-angles at points G and H, where they intersect AD. Finally, you can draw a second horizontal line FE, parallel to GH.

(Figure 2)

Now we’ve got something that looks like a square! GHEF -- that’s a square, isn’t it? Well, no it isn’t. The width (GH = FE) is equal to the radius of the original circle, which we will call 1 unit. But the height, (GF = HE) is smaller than one unit. Mathematically, it can be shown that it’s half the square root of three, which is an irrational number. My computer says it’s 0.866025403784 (followed by an infinite number of non-recurring digits), but the mediaeval Gothic masons didn’t know that. They just knew it wasn’t a rational fraction -- and that was enough to convince them it was God’s number and not Man’s. There are no squares in Chartres Cathedral -- just rectangles with sides in the ratio of 1 to 0.866025403784.

If you go back to the original circle (Figure 1) and move the anchor point of the string to point A on the circumference, you can draw another circle which intersects the first. This is shown in Figure 3. By highlighting the upper half of the intersection you can see that we’ve created a perfect Gothic pointed arch! What’s more, the ratio of the width of the arch to its height is once again the irrational ratio 1 to 0.866025403784. Not all Gothic arches have this precise ratio, but it’s always an irrational number. That’s one of the reasons it appealed to the Gothic masons in preference to the Classical semicircular arch. Another reason is that it pointed upwards, to heaven, rather than curving back down to Earth. In their minds, that meant the same thing -- Classical architecture is about Man, while Gothic architecture is about God.

(Figure 3)

Reference: *"Chartres: Sacred Geometry, Sacred Space"* by Gordon Strachan (Floris Books, 2003).

Copyright © 2006 Andrew May

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