The Golden Mean & Creativity
In your artistic explorations, you may have encountered the idea of the Golden Mean.
It has existential tentacles encircling philosophy, mathematics, art, and morality. It also extends back into Ancient Greek history.
In Plato’s Philebus, Socrates says,
But now we notice that the force of the good has taken up refuge in an alliance with the nature of the beautiful. For measure and proportion manifest themselves in all areas of beauty and virtue.
Art as the exaltation of beauty and mathematics in the guise of proportion are not only intertwined but connected with morality as well.
Later, Aristotle said,
Any virtue taken to an extreme becomes a vice.
Virtue is connected with moderation, and could even be construed as being a kind of early optimization, of balancing opposing influences to get the best mixture of influences to create a desirable outcome.
The Golden Mean is related to the Golden Ratio, which at its root is a number representing a proportion, such as the ratio between the width and height of a painting or an object.
This Golden Ratio is a very non-intuitive number at first glance:
Irrational & Rational Numbers
The dots at the end of the number mean that it goes on forever. It’s also called an irrational number that cannot be expressed as a simple ratio of two whole numbers like 5/8.
A number that goes on forever and doesn’t repeat, for example pi (3.14159…) is an irrational number. Rational numbers can repeat but the number that repeats is the same. For example 1/3 = 0.333333333………
Even though there are many irrational numbers (the vast majority of numbers are irrational), all of them have approximations as ratios of whole numbers.
Moving Closer & Closer To The Purity Of Pi
Our old friend Pi has a sequence of approximations: 22/7, 333/106, 355/113, and so on. Each one is closer to the unreachable Platonic purity of Pi than the last. But they never get there.
The Golden Ratio has a very unusual property. It, of all irrational numbers, is the hardest to pin down.
Successive approximations get closer and closer, but at the slowest rate of all approximations to irrational numbers!
The details of showing this are beyond the scope of this blog post, but they use something called a “continued fraction approximation.” The Golden Ratio makes you work hard to achieve it!
The Fibonacci Sequence & Creativity
The Golden Ratio has a deep connection with one of my favorite beasts in the mathematical zoo, the Fibonacci sequence. The Fibonacci sequence is an endless series of increasing numbers generated by a simple process requiring only addition.
It was first written about by Leonardo of Pisa, later known as Fibonacci, in his book Liber abaci, published in 1202. But its origins date back to 200 BCE in India, where it was developed to understand patterns in Sanskrit poetry (more connections of creativity and math!), as well as being discussed by Euclid.
If one starts with two numbers, our old favorites zero and one that we’re always wanting to go between, the beginning of the sequence looks like this: [0, 1]
To create the next number, take the two numbers at the end of the sequence (this is why we start with a list of two numbers) and add them together.
Now we have [0, 1, 1]. Try it again, and we get [0, 1, 1, 2].
Eventually we get a sequence that looks like this: [0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …]
The Fibonacci Sequence & Nature
These numbers have been used to understand a variety of natural processes, from the growth rate of rabbits (each new generation is created by the combined fertility of the two previous generations) to the shape of nautilus shells and the leaf patterns of some plants.
Here is a rendition of the Fibonacci sequence turned into a jigsaw puzzle of squares by making a left turn before the next square, again and again. The left turns allow each term in the Fibonacci sequence to nest adjacent to the previous two that created it by being added together.
Now connect those squares with a spiral curve
It looks strikingly like this cross section of a nautilus shell, even though it curves right instead of left like the above picture.
Nature reflects art and art imitates nature, with math as its guide.
Birthed by the midwife of mathematics we see art reflected in nature. Art is a process of human thought, humans are part of nature and we understand these patterns both intuitively and through mathematics.
It turns out that the *proportion* of each Fibonacci number to the one behind it is a better and better approximation to the Golden Ratio, and is closing in on it as efficiently as possible!
No pair of numbers in the sequence does a particularly poor or great job of closing in on the Golden Ratio.
Each successive ratio shaves off about the same fraction of the remaining distance.
The task is very equitably spread out.
As in life, everything matters, and there is no faster way to do it in terms of ratios of integers.
If we take the 4th Fibonacci number and divide it by the 3rd, we get 2/1 = 2. Not a very good approximation to the Golden Ratio.
The ratio of the 5th number to the 4th gives us 3/2, or 1.5. Better.
The 6th to the 5th gives 5/3, or 1.6666… Closer yet.
Like a kind of mathematical missile targeting system, the approximations alternately undershoot and overshoot while getting closer and closer to the target.
Each approximation gets better, but no single pair of numbers gets a huge improvement at the expense of some other pair getting a lousy one as happens in the case of Pi.
It’s all very equitably spread out, a very moderate kind of behavior. Aristotle would be proud.
In a previous blog post, we discussed fractals, remarkable patterns that look the same no matter how close or far you are from them, as exemplified by the snowflake (Koch) curve.
The Golden Ratio is also deeply connected with fractals.
In the Studio Journey Masterclass this spring, Bruce delighted us with a surprise talk on how mathematics can illuminate the underlying structure in the seeming chaos of Jackson Pollock’s paintings.
I’m reminded of Buttercup’s words in Gilbert & Sullivan’s H.M.S Pinafore:
Things are seldom what they seem,
Skim milk masquerades as cream;
Highlows pass as patent leathers;
Jackdaws strut in peacock’s feathers.
Watch the snippet from Bruce’s talk:
If one starts with a rectangle and then slices off one side, one can ask:
What should the proportions of the rectangle be such that the sliced off rectangle has the same proportions? (though rotated by 90 degrees)
And then one could repeat the process with the sliced off rectangle and get a slice off the slice, and so on ad infinitum.
If you do that, it looks like the Fibonacci squares above. In the picture below, we want the proportion of the long side of the rectangle (a+b) to the short side (a) to be equal to a/b.
Math & Art Explore Patterns & Relationships
Math and art don’t seem so far apart when one thinks of both of them as being about patterns and relationships, not about numbers and equations.
Saying math is about numbers and equations is like saying art is about paints and canvas. True but missing the deeper point.
In gratitude from my studio to yours,
P.S. If you enjoy this blog, you’ll love my book The Artist’s Journey: Bold Strokes To Spark Creativity