Le Golden Ratio


This is the numerical value of the golden ratio, a proportion often found in nature, art, and architecture. Represented by the Greek letter Φ (“phi”), it comes from a manner of dividing a line so that the longer section divided by the shorter section is equal to the entire length divided by the longer section. In numerical form, it would look like this:

(where “a” is the longer section of the line and “b” is the shorter section:)

10.7.14 GM FORMULA

If the line is cut so that the two fractions are equal, both fractions will divide to equal approximately 1.61803398875- which is where the golden ratio came from.

The golden ratio isn’t exactly a new concept, either; even the ancient Egyptians and Greeks have been to discovered to have used the ratio in building some of the more monumental structures left behind—including the Great Pyramids and the Parthenon.

The ratio, along with the Pythagorean equation, can be used to create a ‘golden triangle’, a right triangle with a hypotenuse of length Φ, and sides 1 and √Φ, respectively.

If one where to draw two of these with the side √Φ touching –an isosceles triangle with a height of √Φ and a base of 2- the resulting triangle would have a base/height ratio of 0.63600982475. The Great Pyramid, with a base of 230.4 meters and a height of 146.5 meters, would also have a base/height ratio of 0.63600982475.

Now, unless we assume the Egyptians were magic and just coincidentally built their sacred pyramids the with same ratio as one that’s considered the epitome of beauty, we can only assume that the golden ratio had been discovered and taken into consideration, even that long ago.

When talking about the golden ratio, one would also usually dabble into the ‘golden spiral.’ The golden spiral is an aesthetically pleasing figure typically found in both art and nature alike. To create one, draw a square. Divide the square in half vertically to create two identical rectangles. In the right rectangle, draw a line from the bottom left to top right corners, and rotate this line at the bottom corner of the rectangle until it lies flat along the square’s bottom line. From this, draw another rectangle with the new line and original side as two sides. The entire figure, including all three rectangles, is our base rectangle.

10.7.14 GM1

From this rectangle, create a square using the longest side. Rinse and repeat. Do make sure, however, that the squares rotate in a constant direction- clockwise or counterclockwise. In this case, it’s counterclockwise.

10.7.14 GM2

One could create an infinitely large rectangle in this manner; however, I will stop at 5. The rest is simply drawing the curve.

10.7.14 GM3

Placing the elements of the composition along the spiral’s path tends to create a more natural-looking and aesthetically pleasing piece. Along with that, if the canvas itself has the proportions of a golden rectangle, the widths of the squares can be used for choosing sizes of the subjects of the piece.

And, as the golden ratio is said to be so ‘beautiful’, aspects of its use can be found in more than a few famous art pieces. Leonardo da Vinci in particular, being a mathematician along with being an artist, incorporated the ratio rather often. The Mona Lisa’s face lines up with the tiny curls of the spiral. The Last Supper uses golden proportions for the windows, the table, and even the arrangement of the people.

And so, the golden ratio has more than a few footholds in not only art, but also architecture, history, and mathematics. It’s everywhere.


About barberne

an animation student at alfred state from rochester
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