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.