The Golden Section

The Golden Section is a proportion realized by the ancient Greeks in which the smaller term is to the larger term in the same way that the larger term is to the sum of the smaller and the larger [a:b=b:a+b]. In other words, it is the only proportion defined by one irrational number, and is reproducable into an infinite series. That quantity is [Maple Math] or [Maple Math]

> evalf(1+sqrt(5))/2;

[Maple Math]

> evalf(1-sqrt(5))/2;

[Maple Math]

If we evaluate the reciprocles of [Maple Math] , then we obtain very expected values. This illustrates how [Maple Math] works in an infinitely decreasing and increasing manner.

> evalf(2/(1+sqrt(5)));

[Maple Math]

> evalf(2/(1-sqrt(5)));

[Maple Math]

Proof: [Maple Math]

[Maple Math]

[Maple Math]

[Maple Math]

[Maple Math]

Greek Use of the Golden Section

The Pythagoreans made an observation by the subdivision of pentagons: diagonals of the pentagon made a smaller pentagon in the center. Furthurmore, these diagonals were divided unequally by other diagonals; the ratio of the smaller part to the larger part of the diagonal is as the larger part is to the whole of the diagonal. This is how the Golden Section was discovered.

> pentagon:=plot([seq([cos(Pi/2-2*Pi/5+2*i*Pi/5),sin(Pi/2-2*Pi/5+2*i*Pi/5)],i=0..5)],scaling=constrained,color=green,thickness=3):

> pentagon;

[Maple Plot]

> diagonals:=plot({seq([[cos(Pi/2-i*2*Pi/5),sin(Pi/2-i*2*Pi/5)],[cos(Pi/2-(i+2)*2*Pi/5),sin(Pi/2-(i+2)*2*Pi/5)]],i=0..4)},color=yellow,thickness=3):

> plots[display]([pentagon,diagonals]);

[Maple Plot]

Though the proportion is irrational the Greeks found it useful because of its relationship to nature. The human body is divided by the Golden Mean: the navel divides the body into two unequal portions equal to the Golden Section.

The Vitruvian Man/ Arm Sketch

The Golden Section and the Fibonacci Series

Leonardo of Pisa (ca. 1180-1250) has been credited with the discovery of the Fibonacci series. The Fibonacci series is a sequence of numbers from the sum of the previous two numbers in the set (i.e.: 1, 2, 3, 5, 8, 13...). The Fibonacci series is used in relation to nature; the branching of trees, the distribution of leaves, and the scaling of fingers and hands are all proven examples. It has been observed that as the Fibonacci series progresses it comes closer and closer to the actual value of [Maple Math] .

The Golden Section in Architecture

Le Corbusier deciphered a system of measurement for architectural design based from the human body and the golden section. He called it the modular . The measurements are similar to a Fibonacci series: 6'-0", 3'-8 1/2", 2'-3 1/2", 1'-5", 0'-10 1/2", 0'-6 1/2", 0'-4", 0'-2 1/2", 0'-1 1/2".

The Modular Man (measured in mm)