[“Time is the magnitude of motion when the size of its earlier and later are brought together in the mind.”]

This Japanese Zen calligraphic drawing beautifully shows ‘creation’ through the simple progression from the absolute unity of the circle, through the triangle (with three points forming a qualitative transition from the pure, abstract elements of point and line to the tangible, measurable state called a surface) symbolizing the passage between the transcendent and the manifest realms, to the manifest form of the square (representing materialization).

[Unity and Multiplicity]

“From both the metaphysical and natural points of view it is false to say that in order to arrive at two, you take two ones and put them together. One only need look at the way in which a living cell becomes two. For One by definition is singular, it is Unity, therefore all-inclusive. There cannot be two Ones. Unity, as the perfect symbol for God, divides itself from within itself, thus creating Two: the creator unity and the created multiplicity.

Unity creates by dividing itself, and this can be symbolized geometrically in several different ways, depending upon how the original Unity is graphically represented. Unity can be appropriately represented as a circle, but the very incommensurability of the circle indicates that this figure belongs to a level of symbols beyond reasoning and measure. Unity can be restated as the Square, which, with its perfect symmetry, also represents wholeness, and yields to comprehensible measure. In geometrical philosophy the circle is the symbol of unmanifest Unity, while the square represents Unity poised, as it were, for manifestation. The square represents the four primary orientations, north, south, east and west, which make space comprehensible, and it is formed by two pairs of perfectly equal yet oppositional linear elements, thus graphically fulfilling the description of universal Nature found in Taoist and other ancient philosophies.

The square (above) represents the earth held in fourfold embrace by the circular vault of the sky and hence subject to the ever-flowing wheel of time. When the incessant movement of the universe, depicted by the circle, yields to comprehensible order, one finds the square. The square then presupposes the circle and results from it. The relationship of form and movement, space and time, is evoked by the mandala.”

Source: Robert Lawlor, Sacred Geometry: Philosophy and Practice, pp. 16, 23


“… 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