The following blog post is constructed from conversations between myself and my partner, Dr. Bruce Sawhill, Stanford educated theoretical physicist and mathematician.
Creativity, Combinatorics and Patterns of Possibility
There are certain concepts in mathematics that are beguilingly simple, easily accessible, and amazingly general.
One of these is combinatorics. It is the study of counting patterns. It is the answer to the question, “How many different patterns can I make with these basic elements?”
Understanding combinatorics doesn’t require understanding calculus, statistics, or other branches of advanced mathematics. Delving deep into it uses all of those subfields and more, but the price of admission is low.
All you need is the ability to count and recognize patterns. Even a very young child has an intuitive feel for this.
Have you ever seen a child’s book with pictures of several things and the question, “Which of these is different from the others?” The illustration might show three types of birds and a dog.
This is about categorizing and becoming aware of patterns.
In the previous blog post about artificial intelligence, we talked about putting things into categories.
The ability to perceive and differentiate patterns is deep in our DNA.
Like so many other things in life, this ability to recognize patterns is key to survival and therefore evolution. Noticing patterns that are beneficial to us and differentiating them from patterns that are hazardous to us and being able to do this quickly is an essential life skill.
Much of this is taught, but it takes advantage of the fertile soil of cognizance that has evolved to the task.
Combinatorics is relevant to art and creativity, because at their root they are about patterns.
We see patterns everywhere-–patterns of color, patterns of value, patterns of musical pitches and textures, patterns of words and sentences, patterns of movement in dance.
Combinatorics is a pathway into vast possibility
Enormous numbers of patterns are easily created. Astronomical numbers of patterns can be generated without the inconvenience of space travel.
Here’s an example: Imagine you are putting books on a shelf. Let’s say you start with a pile of three books. You have three possibilities of which book you choose to place first.
You’ve chosen your first book and now you have two books left and therefore two choices for which one you place on the shelf next.
After you place your second book, you only one book left.
All in all, you have 3 x 2 x 1 choices, or six ways to arrange the books on the shelf.
You can convince yourself of this if you still would like convincing by figuring out all of the unique ways to write down the numbers 1, 2, and 3 (numbers standing in for book titles) such that you have three numbers in a row with no repeats.
- 1, 2, 3
- 1, 3, 2
- 2, 1, 3
- 2, 3, 1
- 3, 1, 2
- 3, 2, 1
This product of a descending series of numbers is written N!, where N is the top number and the exclamation mark is called “factorial” or sometimes “bang.”
The result of six is mildly interesting, but it seems like nothing to write home (or write a blog post) about.
So let’s make the problem juicier by saying you have a box full of books, perhaps 50.
Now the number of patterns is equal to 50 x 49 x 48 x 47 x 46….. 3 x 2 x 1.
That’s going to be a bigger number than six but how much bigger?
Wait for it…
I’m not even know how to say that astronomical number!
Maybe you don’t care about all of the possible patterns of book titles and just want to group the books by color. This will reduce that huge number above, because, for example, a case where red book A is in position 7 and red book B is in position 8 doesn’t count as different from a situation where the books are reversed.
It’s still two red books in a row. Grouping cuts down that enormous number above, so that big number is essentially an upper limit.
So now we see that introducing not only color but symmetry reduces the number of possibilities.
What does this mean for creativity?
You brought in constraints, in this case color and symmetry, and constraints are powerful for creativity.
Why is constraint a potent force in creativity?
The answer is complicated.
It reminds me of a conversation I overheard at a cocktail party. One person, flummoxed by the carrying on of an arrogant pedant, asks his friend, “Is Kafkaesque bad or good?”
Well, giving a cogent answer requires further discussion.
Imagine two extremes of patterns: random and regular
On one side, we have visual static, so called “TV snow.”
A vast number of patterns is viewed in quick succession, but they have a quality of sameness to them.
Each pattern is different, but the difference does not convey meaning or information to us and therefore our attention wanders.
It is because the patterns are structureless and without symmetries. There is a *statistical* sameness about them.
This same reasoning can be applied to sound- white noise is endlessly different but there’s paradoxically a sameness to it. The differences, though numerous, do not register.
On the other extreme is very high symmetry- the ticking of a metronome or a perfectly spherical form.
These structures are so pure and ideal that they may be viewed with contempt as being too obvious or, like Platonic forms, too perfect and therefore static and uninteresting.
There is no surprise, no discovery involved in comprehending them. There is a kind of coldness of perfection.
For white snow, there is nothing about each individual pattern that lets you predict the next one whereas for the metronome you can predict what it’s going to be doing until the end of time.
So we have a spectrum that is problematic at both ends.
Novelty in the form of richness of pattern is not enough by itself, and the strictures of structure are equally dissatisfying if pursued single-mindedly.
This brings to mind one of my favorite quotes from Aristotle.
Any virtue taken to an extreme
becomes a vice.
Even a good thing, taken to the extreme, can become problematic.
A tale from architecture
Bruce has long been fascinated by architecture, and even studied it for a year at Stanford before the major was discontinued.
Right about then, he encountered a quote by Goethe stating that,
Architecture is frozen music.
Bruce decided that he wanted the fresh rather than the frozen kind of music and eventually became a music major along with physics, but with a connection to architecture in that his chosen instrument, the pipe organ, is an architectural instrument.
It is said that the most important stop on an organ is the building.
This tends to acoustically favor buildings built with a million tons of stone, like European cathedrals.
The convenience of not having to carry your instrument around like a trombonist or cellist is counterbalanced by having to fly 5,000 miles to play under ideal circumstances.
But Bruce remained an armchair architect with a scientific bent of mind.
Why were some spaces architecturally interesting and others not? Like many before him, he became fascinated with proportion.
I started to notice some mathematical properties of spaces, starting with a cafe on the coast of northern California about ten miles up the road. It had a sun-splashed south-facing dining room with a long row of clerestory windows that was preternaturally pleasant, even above and beyond the good breakfast food.
After looking at it enough times to make my way through the entire menu, it occurred to me that the length, width, and height of the room had a special property.
The proportions were roughly 11:5:3, which I confirmed by pacing it off, much to the amusement of the dining staff. I had to estimate the vertical part.
The interesting thing about those three numbers was that no one of them divided evenly into any other.
I believe that my mind stayed fascinated with the space because it could not simplify it by breaking it in halves or thirds. I started to notice this in other spaces as well.
The dining room I observed had no natural break points, so it had to be perceived holistically.
I believe human minds are built to comprehend by cognitive dividing and conquering. If they succeed in doing this, they internally mark the concept as “understood” and move on.
Deep art resists simple understanding.
Bruce Sawhill, PhD
When you buy a chocolate bar, it is scored so as to break into convenient pieces. This may be great for chocolate but it’s bad for art.
In art, we want surprise, not predictability or repetition.
Painters will tell you that making significant marks in the exact middle of a painting (like a horizon line) or placing figures either centrally or symmetrically renders a painting visually static and less compelling.
There are certainly exceptions that make a point of symmetry and repetition, such as asymmetrical symmetry, where there is symmetry yet you add a bit of asymmetry to make it surprising and visually interesting.
It seems that the mind is fascinated by complexity and un-resolvability.
Complexity of pattern has elements of both regularity and randomness.
There’s a dialogue between the contrasts- between the spontaneous and the considered- and hopefully one of them is dominant.
If there are equal amounts of regularity and randomness, it’s not as visually exciting.
It is almost as if whatever thought process generates the pattern needs to have enough randomness to qualify as creative and original, but enough structure to let the viewer, listener, or reader remember the structural details and capitalize on the spontaneity.
It doesn’t count if you can’t remember it, and some things are more memorable than others.
Combinatorics & The Middle Ground
In terms of combinatorics, there is a middle ground between utter dominance of symmetry like the metronome with only one pattern and TV static with a maximum amount of pattern.
The number of possible combinations of a small number of things is so enormous that we can afford to sacrifice some of that huge number to create something memorable.
Symmetries distinguish patterns from each other, such as grouping books or brushstrokes by color.
There is an intuitive feeling that structures that have some regularity in them are unlikely to occur randomly- perhaps we sense this because we embody structures such as the armature of our vertebrae and the exquisite architectural intricacies of our cells and our neural networks that are not random.
There’s a sense that something more than randomness is at work, namely intent and intelligence.
We look at such a creation and say, There’s no way that could have just happened by itself, it must have been imagined and brought to life by an author, composer, artist or creator.
We recognize our kindred humanity in works of creativity and know that we are home.
With gratitude from my studio to yours,
P.S. NOW is the perfect time to create.
This is the existential moment–this is the time where we see what our life is about. We notice what is meaningful and alive for us.
You might be thinking…I’m just too blocked, too down, too scared or frozen….or even just shy….
You may be feeling that you can’t create now….
But I say to you that you’re a creator…you’re an artist and artists create.
And there are many ways to create and be creative….
Pair your explorations in your art studio with our Art of the Possible Book Series!