Golden Rectangle

GOLDEN SECTION, GROWTH AND FIBONACCI SERIES

The golden rectangle, also known as the golden rectangle or golden ratio, is a particular ratio between the lengths of the sides of a rectangle. A rectangle is said to be golden if the ratio of the length of the longest side to that of the shortest side is approximately equal to the ratio of the sum of the two lengths to the length of the longest side. Mathematically, this ratio is approximately equal to 1,6180339887…and is often represented with the Greek letter φ (phi). This ratio was deemed aesthetically pleasing and was used in many works of art and architecture.

Ø (fi), was described by Johannes Kepler as one of the “two great treasures of geometry” (the other being the Pythagorean theorem).
The golden rectangle, of dimensions  Ø and 1, can be easily constructed with a ruler and compass according to the technique shown in the image below. Click on the image to download the dwg.

Starting from the golden rectangle, we can construct the golden spiral.

It seems that these ratios were known since the time of the Egyptians, as they are found as particular harmonic ratios in the study of the dimensions of the pyramid of Cheops. Similar proportions are also found on the Parthenon in Athens.

If we want to construct a sequence of ratios in which the golden ratio between the new quantities is constantly maintained, we can proceed as follows:

(1-x):x=x:1=1:(1+x)=(1+x):(2+x)=(2+x):(3+2x)=(3+2x):(5+3x)=(5+3x):(8+5x)=
(8+5x):(13+8x)=(13+8x):(21+13x) ……

The numerical sequences that we have thus obtained indicate the close correlation with the in the development of the Fibonacci sequence. As the values ​​increase, this increasingly approaches a geometric progression of reason. Ø.