In mathematics

The Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics.

The Golden Ratio is a complex and abstruse mathematical concept. So complex and so abstruse that some of the greatest mathematical minds in the history of the world, from ancient times right up until today, have spent (and continue to spend) countless hours ruminating over it and its properties. It would be hubris for us to presume to understand it well enough to explain it to you, so we are not going to attempt that. We are simply going to tell you what we do know and the Golden Ratio’s place in this world—which is actually quite interesting and, in many cases, some fun information to have and in others, very practical information, particularly for those involved in the visual arts.

The first record of the Golden Ratio is in Euclid’s Elements, c. 300 BCE. Euclid, the “Father of Geometry,” was a Greek mathematician living in Egypt. The Elements, a 13-volume compilation of all the known mathematics of Euclid’s time, is the most important mathematical textbook ever written. It is still used today, and it is second only to the Bible in most printed and most studied books.

Euclid’s was the first definition of the Golden Ratio.

The Golden ratio is a special number found by dividing a line into two parts so that the longer part divided by the smaller part is also equal to the whole length divided by the longer part.

a/b = (a+b)/a = 1.6180339887498948420 . . . .

We simplify that as 1.618 (Phi) same as we do 3.14 (Pi).

In the human body

Maybe it’ll be easier if we see the Golden Ratio in the flesh—literally. Examples of it are all over our bodies.

The length of our fingers, for example: Each section from the tip of the base to the wrist is larger than the preceding one by roughly the ratio of phi.

If the length of the hand has the value of 1, then the combined length of hand + forearm has the approximate value of 1.618. The proportion of upper arm to hand + forearm is in the same ratio of 1:618. That is, the length of the hand is in perfect proportion to the length of the arm.

Another example of the Golden Ratio is the measurement of the navel to the floor and the top of the head to the navel.

It’s all over our faces. The head forms a Golden Rectangle with the eyes at its midpoint. The placement of the mouth and nose and the distance between the eyes and chin—all based on the Golden Ratio, which appear to be the proportions that are most pleasing. Painters and sculptors, as far back as the ancient world, used the Golden Ratio to create works of the ideal human figure. Plastic surgeons and dental surgeons use it today to restructure the human face.

Motif name: Pyramids of Gizeh

In architecture

However, the Golden Ratio is more associated with architecture and art than the dimensions of our bodies. Some examples:

The Egyptian Pyramids

The pyramids were built thousands of years before Euclid. So, was it a coincidence that the pyramids contain the Golden Ratio? Most likely not. The Egyptians were an extremely intelligent civilization and very well could have discovered the Golden Ratio themselves and built the pyramids accordingly.

The Parthenon

One hundred years before Euclid, Phidias, the Greek sculptor, was what we would call today the “artistic director” of the building of the Parthenon. He used the Golden Ratio in his sculptures of the Greek gods, but most notably in the sculptural decorations on the exterior.
Note: The numeric value of the Golden Ratio, Phi, was named after him.

The Golden Ratio continues to appear in architecture, such as the National Gallery in London and the Sydney Opera House.

In art

Sistine Chapel, Michelangelo

More than two dozen examples of the Golden Ratio appears in Michelangelo’s masterpiece created in the 16th century. Did the Ratio underlie the whole painting? It’s possible. The most profound and affecting example of it is where God and Adam touch in “The Creation of Adam,” exactly at at the Golden Ratio point.

Another example is in the depiction of Noah being found by his sons naked and drunk. The two sons point to him, but their fingers are point directly to the Golden Ratio points from the painting’s sides. One scholar posited that Michelangelo did that on purpose, as if to say, “Look, it’s right here.”.Depends on how much sense of humor he had as he spent the better part of four years painting the ceiling when he would have preferred to be sculpting. And as his eyesight was permanently damaged from the strain.

In nature

The Golden Ratio has been present in nature for all time—long before the pyramids were built. It is expressed in the arrangement of branches along the stems of plants and of the veins in leaves (and in the branching of veins and nerves in humans and animals.).

It is there in the majestic galaxies: The Milky Way has a number of spiral arms, each with a shape identical to the Golden Spiral. Look at seashells or snail shells, flowers or a hurricane. A wonderful aspect of the Golden Ratio is that the more you see it in nature, the more you will find.