of
White’s Dimensional-Shift Operator %
by
Douglass A. White, Ph.D.
Unit Circle and Unit Square Diagonals and Ratios
The constant ratio of the circumference of a circle to its diameter is called Pi and is a
pure algebraic irrational number that approximately equals 3.14159.... If a circle’s
diameter is 1, the circumference is Pi. The diagonal of a unit square (each side has
length = 1) is the square root of 2, another famous algebraic irrational number.
Phi is the ratio of the radius of a circle to the side of a regular decagon inscribed in it.
There are many other ways to derive Phi. It is the ratio of the diagonal of a double
unit square rectangle plus its height to its length. It is also the ratio obtained when a
line segment is divided such that the ratio of its long part to its short part equals the
ratio of the whole line to its long part. This is called the Golden Section. If (a) is
the short part and (b) is the long part, b / a = (a+b) / b. This means that a^2 + ab –
b^2 = 0. Also, a golden rectangle generates a fractal spiral of itself at increasing and
decreasing scales. (For a diagram of this see Observer Physics Chapter 17, p. 25.)