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One so-called 'irrational
number' that I have not yet explained, relates to what is known as "The
Divine Proportion". This contribution also comes from the Pythagorean
school. This a window that is
pleasing
to the eye and which has a relationship in which the
width to the length of the
window is 8 to 13. This
relationship is derived from the
Pythagorean theorem for a rectangle as shown
in the sketch below.
 The
construction of the Golden
Rectangle is done by first
constructing a square, and then by projecting the diagonal of half the
square, onto the base as shown in the sketch
below.
 The Golden section is obtained by projecting the side of the Golden rectangle onto the base. The following dimensions are then established: The whole ( base ) = 13 units The long section = 8 units (also referred to as the mean) The short section = 5 units The ratio of the long section to the whole is 8/13 The ratio of the short section to the long section is 5/8 The least is to the mean as the mean is to the greatest. 5/8 is as 8/13 The equation that satisfies these ratios is: 64=65 The
rectangle is composed
of 64 small rectangles.
It will be irrational to say that the the rectangle continues to diminish indefinitely. The rectangle is finite and is made up of 64 rectangles. The idea that there is an infinite number of rectangles is a fallacy. The sequence of depletion of the rectangles is as follows: 32,16,8,4,2,1,1 for a total of 64 rectangles.  Divine Proportion |