Sacred Number

Here is a book with really solid foundations whilst providing popular access, if you can be bothered making the minimum effort to understand, to the key ratio of Nature. I tried to provide a good introduction to phi in Matrix of Creation because too much was spread around a number of books - but now Scott Olsen, an American liberal arts professor, has used his sabbatical year most wisely to create what is likely to be a definitive work with a disarmingly cool format and low price. (Wooden Books have been developed by John Martineau of Glastonbury to provide a series of accessible text on key subjects relation to traditional arts. Many, like this one, are classics because so much work goes into selecting and illustrating the material)

One of the powers of sacred geometry is that it can deal with these irrational numbers through a geometrical construction since an irrational cannot be generated using whole numbers, effectively in one dimension, except by using modern mathematical methods.

Thus the formula for the golden mean is (√5+1)/2. However this is rearranged in Scott Olsen's book into (1/2 + √5/2) which makes for the very excellent geometrical generator for phi using a unit square as follows:

It is easy from this diagram to see why this rearrangement leads to a better geometrical solution. The mid point of any square forms an implicit 1:2 right angled triangle to the opposite corners. Since the units are 1/2 then the hypotenuse must be √5/2 and this, when dropped onto the baseline, to extend the other half unit, produces the required (1/2 + √5/2) which equals the golden mean.