August 21st, 2007
An exercise in contemplative geometry from Robert Lawlor’s Sacred Geometry, 1982.
“There are a number of diagrams in the literature of Sacred Geometry all related to the single idea known as ‘Squaring of the Circle’. This is a practice which seeks, with only the usual compass and straight-edge, to construct a square which is virtually equal in perimeter to the circumference of a given circle, or which is virtually equal in area to the area of a given circle. Because the circle is an incommensurable figure based on π, it is impossible to draw a square more than approximately equal to it. Nevertheless the Squaring of the Circle is of great importance to the geometer-cosmologist because for him the circle represents pure, unmanifest spirit-space, while the square represents the manifest and comprehensible world. When a near equality is drawn between the circle and the square, the infinite is able to express its dimensions or qualities through the finite” (p74).
Here I have drawn the former arrangement (perimeter equal to circumference):
The derivation begins with an initial circle (within the square) of radius unity. Along its horizontal diameter are drawn two tangent circles, each with radius one half. Observe that the total circumference of the smaller circles equals the circumference of the initial circle, but the total area of the smaller circles is one half that of the initial circle: “One has become Two” (p73), an image of the primary duality, of yin-yang.
Next are drawn two arcs from the ends of the initial circle’s vertical diameter with radius tangent to the far sides of the smaller circles. This radius is φ, the golden ratio, dividing the vertical radius of the initial circle into the golden section of lengths 1/φ and 1/φ2. The two arcs meet to create a vesica that encloses the primary duality — the mouth of Ra, the Word, the vibrating string.
Around the initial circle is drawn a tangent square, with side 2, perimeter 8; and, finally, a large circle is drawn with diameter equal to the width of the vesica, 2√φ, giving a circumference of 2π√φ = 7.993, approximately equal to 8. The circle is squared.