“… an analogy by which this universal and archetypal sense of Number can be understood. A revolving sphere presents us with the notion of an axis. We think of this axis as an ideal or imaginary line through the sphere. It has no objective existence, yet we cannot help but be convinced of its reality; and to determine anything about the sphere, such as its inclination or its speed of rotation we must refer to this imaginary axis. Number in the enumerative sense corresponds to the measures and movements of the outer surface of the sphere, while the universal aspect of Number is analogous to the immobile, unmanifest, functional principle of its axis.
Let us shift our analogy to the two-dimensional plane. If we take a circle and a square and give the value 1 to the diameter of the circle and also to the side of the square, then the diagonal of the square will always be (and this is an invariable law) an ‘incommensurable’, ‘irrational’ number. It is said that such a number can be carried out to an infinite number of decimal places without ever arriving at a resolution. In the case of the diagonal of the square, this decimal is 1.4142 . . . and is called the square root of 2, or √2. With the circle, if we give the diameter the value 1, the circumference will also always be of the incommensurable type, 3.14159 . . . which we know by the Greek symbol π, or pi.
The principle remains the same in the inversion: if we give the fixed, rational value of 1 to the diagonal of the square and to the circumference of the circle, then the side of the square and the radius of the circle will become of the incommensurable ‘irrational’ type: 1/√2 and 1/ π.
It is exactly at this point that quantified mathematics and geometry go their separate ways, because numerically we can never know exactly the diagonal of the square nor the circumference of the circle. Yes, we can round-off after a certain number of decimal places, and treat these cut off numbers like any other number, but we can never reduce them to a quantity. In geometry, however, the diagonal and the circumference, when considered in the context of formal relationship (diagonal to side; circumference to diameter), are absolutely knowable, self-evident realities: 1: √2 and 1: π. Number is considered as a formal relationship, and this type of numerical relationship is called a function. The square root of 2 is the functional number of a square. Pi is the functional number of a circle. Philosophic geometry—and consequently sacred art and architecture—is very much concerned with these ‘irrational’ functions, for the simple reason that they demonstrate graphically a level of experience which is universal and invariable.
The irrational functions (which we will consider rather as supra-rational) are a key opening a door to a higher reality of Number. They demonstrate that Number is above all a relationship; and no matter what quantities are applied to the side and to the diameter the relationship will remain invariable, for in essence this functional aspect of Number is neither large nor small, neither infinite nor finite: it is universal. Thus within the concept of Number there is a definite, finite, particularizing power and also a universal synthesizing power. One may be called the exoteric or external aspect of number, the other the esoteric or inner, functional aspect.”
Source: Robert Lawlor, Sacred Geometry: Philosophy and Practice, pp. 10-12