At art school, students would casually refer to the Golden Mean or Golden Ratio in critiques, and it seemed that I was the only person who had no idea what they were talking about. I’m sure it had been explained to me, maybe even by an art teacher, but because it was math-related I made little effort to try to understand it. I’m tired of being in the dark. So I recently read The Golden Section: Nature’s Greatest Secret by Scott Olsen from the Wooden Books “Small Books, Big Ideas” series. Here’s what I learned.
The Golden Mean is a ratio. Plato describes the ratio visually, as a line segment divided into two unequal parts. The ratio between the shorter portion to the larger portion is the same as the ratio between the larger portion to the whole. The greater portion is called “Fye” and the smaller portion is called “Fee.” Their whole is termed “Unity.”
Two examples of this ratio are 1:2 :: 2:4 :: 4:8 and 1:3 :: 3:9 :: 9:27. Both sequences repeat the same invariant ratio (either 1:2 or 1:3, respectively). Plato refers to these ratio patterns as “extended continuous geometric proportions.” This is most significantly found in the Fibonacci Sequence, where each number is the sum of the previous two. 0, 1, 1, 2, 3, 4, 5, 13, 21, 34, 55, 89, 144, 233, 377… Each Fibonacci number is the approximate golden mean of it’s two adjacent numbers.
The second most important number sequence is the Lucas Numbers: 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199… This sequence is like the Fibonacci Sequence, as each number is the sum of the previous two. Lucas numbers are formed by alternately adding and subtracting the golden powers of Fye and 1/Fye.
Now here’s where it gets interesting: these two sequences, The Fibonacci Sequence and The Lucas Numbers, which both hold the Golden Ratio, are everywhere. Here are a few spooky examples.
- The Natural World
“Despite its seemingly endless variety and diversity, nature employs only three basic ways to arrange leaves along a stem: disticious, like corn, decussate or whorled, such as mint, and the most common, spiral phyllotaxis (for about 80% of the 250,000 different species of higher plants), where the rotation angles are Fibonacci approximations to the golden angle, 137.5. This pattern aids photosynthesis, each leaf receiving maximum sunlight and rain, efficiently spirals moisture to roots, and gives best exposure for insect pollination” (Olsen, 14).
This spiraling phyllotaxis pattern, determined by Fibonacci numbers, also builds the cell structure of the stalk and arranges scales and petals. Take a close look at the black face of a sunflower- the seeds are arranged in a golden spiral phyllotaxis.
- The Human Body
The three bones of each of finger are in a golden ratio. As are the lengths of the hand, forearm, and upper arm. A baby’s midpoint is at it’s navel, and the golden section is at it’s genitals. In adulthood, these reverse. An adult’s midpoint is at it’s genitals, and golden section is at the navel. The face and skull are treasure troves of golden ratios.
- Ancient Art and Architecture
The simple ratios and grids used in Egyptian architecture include the √3 bisection of an equilateral triangle, and the √5 based golden section. Golden proportions and Fibonacci approximations are used in the foundational structure of pyramids (both Egyptian and Mayan), Greek pottery designs, the Parthenon, Gothic cathedrals, and mandala designs.
- Compositions of Master Paintings
Da Vinci, Van Gogh, Dali, and Jean Colombe, to name a few.
- Music Composition
“The structure of both rhythm and harmony is based upon ratio. The most simple and pleasing musical intervals, the octave (2:1) and the fifth (3:2), are the first Fibonacci approximations to the golden section. The series continues with the major and minor sixths (5:3 and 8:5)… Russian musicologist Sabaneev discovered in 1925 that the golden section particularly appears in compositions by Beethoven (97% of his works), Haydn (97%), Arensky (95%), Chopin (92%), Schubert (91%), Mozart (91%), and Scriabin (90%).” (Olsen, 38).
- That Coke can you’re holding.
The golden ratio is designed into most things lying around your house, such as cigarette packs, coke cans, playing cards, coffee pots, and bicycles. Credit cards measure 86x54mm. That’s an 8:5 rectangle, and one of the most common Fibonacci approximations to the golden rectangle.
I realize this is a very long post, but to cut any of these facts would surely detract from fully describing how amazing The Golden Mean is.