phi

 romanesque.jpg  maths.surrey.ac.uk

a line divided unevenly…

Differentiation in Nature is necessitated by inequalities – the original fabric of the universe required an uneven (non-uniformly smooth) structure to develop future star systems and galaxies as clumping masses evolved into larger ones by gravitation.

According to classical Greek belief, the “perfect proportion” is found by dividing a line unevenly so that the ratio of the larger part to the smaller part is equal to the whole line divided by the larger part.  This happens to be the same proportion found in humans:  any person’s height, no matter how tall or short they are, divided by the distance from the ground to the navel, is always the same, approximately 1.6.  Thus, the perfect division of a line is associated with an ideal harmony, reflecting Man and Nature – the geometry can therefore be replicated in art and architecture.

The ratio of the larger piece, a, to the smaller, b, is equal to the ratio of the entire length, a+b, to the larger piece, a.

quadratic solution of phi

geometry of the golden ratio…

To geometrically derive the Golden Ratio, start with a square, each side = a, then find the midpoint of the base, creating two equal sections of  a/2  each.  From the midpoint, draw a line connected to the top right vertex.  Use this length to add an additional section to the base – this is the Golden Section, b.

geometry of the golden section

use pythagorean theorem to solve for a right triangle

The first 100 digits of phi:

1.6180339887 4989484820 4586834365 6381177203 0917980576 2862135448 6227052604 6281890244 9707207204 1893911374…

phi-scratches                                                                                                                                          

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