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Combinatorics

 

Combinatorics is one of our favorite subjects, we return to it again and again.  It is simple counting, one of the oldest activities of human civilization.  

And, at it’s root, it is about patterns, not things, as the following joke makes clear:

A new student shows up in Mr. X’s second-grade math class. Mr. X asks the student, “If I have three oranges and Johnny gives me four more, how many oranges do I have?”

The new student responds, “I don’t know, Mr. X. In my old school we learned on apples.”

 

Abstraction is the root of complex understanding, in art and elsewhere.

The Dance Of Order & Disorder

 

We have also talked about the boundary between order and disorder, the tension between structure and chaos, between a metronome and radio static. 

Physicists have attempted to quantify this concept and make it independent of subjective perception, since one person’s order might be another person’s disorder.

In an earlier post, we used the idea of a bookshelf to convey the richness of combinatorics, that there are a lot of ways to arrange things. We rapidly got to astronomical numbers of possibilities. 

 

 

 

 

The Power Of Constraint

 

We also spoke of the power of constraint.

 

 

 

We want to reduce the number of possibilities enough to deliver a relatable coherent message.

Since art is about communication between the creator and the audience, the message and its content matters.

Meaning is found in the context.

 

 

 

Back To The Bookshelf

 

In the bookshelf example, we could tame the “combinatorial catastrophe” by arranging the books alphabetically.

This is a strong constraint, as there is only one way to do this! (Assuming there are no copies) It might be too strong of a constraint, as it leaves nothing to the imagination. We would say that this configuration of books is very ordered and disorder is at a minimum.

We’re also using information in the form of book titles to reduce the disorder. So there’s some kind of relation between information and order/disorder. We’ll return to this in the future.

A slightly less imposing constraint might be to arrange the books by color. Since all the books of any one color can be in any order, we have some more possibilities, but not too many.  The result is clearly not random, artificial as in “art,” but still interesting because creative choices were made. 

 

Scale

 

One of the great mysteries of physics is how physical laws seem different at microscopic scales than they do at human scales.

For instance, if you were to make a movie of two elementary particles colliding with each other and then make a second movie which was the first one played backwards, it would be impossible for an outsider to tell which one was the original.

This goes by the mouthful moniker of “Time reversal symmetry.”

But certain things are not symmetric in our daily physical experience.

My best friend from childhood was a sloppy eater as a small child, as reported by his father after the fact.

The father told him at one point, “How about you sit on the floor and eat, and if anything falls up, we’ll pass it back down to you.”

The father was a Harvard-trained English professor. Clearly a safe bet on his part, requiring no detailed understanding of physics.

 

 

 

 

If food falls off a table and lands on the floor, something is lost that prevents the easy spontaneous reversal of this process. A mixture of structure and energy is lost or dissipated.

This is not an easily reversible action.

So physical laws might be reversible in the microscopic world, but not in our world. This, ultimately, has to do with combinatorics.

 

 

 

 

Pencil Experiment

 

Imagine the following experiment:  Start with ten pencils in your hand and drop them to the floor from chest height. They’ll usually point in all different directions on the floor after they stop moving. There’s a very, very small chance they’ll all line up. 

What if you start with a bundle of them in your hand with them all pointing the same direction? They’ll still end up pointing in all different directions, even though there’s no physical law forbidding them from lining up. 

If you introduce a constraint, such as a V-shaped trough on the ground, you can readily make them all line up, though perhaps with a mixture of 180-degree opposing orientations.

The upshot of the pencil experiment is that there are *a lot more ways* for them to point all different directions than the one way that there is for them to line up.

From a probabilistic point of view, it is extremely unlikely to get all the pencils in line. As one considers systems with more and more elements, “extremely unlikely” becomes “essentially impossible.” 

 

There are a lot more ways to be disordered than ordered.

 

 

Entropy

 

In the late 1800s, a new concept was invented to attempt to quantify disorder, and it was given the name “entropy,” from the Ancient Greek entropia, meaning “transformation” or “turning towards.”

The statistical definition of entropy, due to Ludwig Boltzmann, is related to the number of ways a system can be configured subject to a constraint, such as temperature or volume in the case of gases of molecules.

Like the number of ways that ten pencils can point, entropy relates to the number of configurations that a certain number of molecules can have in a gas.

 

 

 

 

 

The statement that “Entropy always tends to a maximum” that has made its way into common parlance states that a system is most likely to exist where it has the largest number of possible configurations.

Pencils will always point all over the place. Disorder strongly tends to increase. 

While it is theoretically possible that a tornado can blow through a junkyard and, given enough aluminum and copper and plastic in the yard, assemble a fully operational airliner, the probability is vanishingly small.

There’s enough energy and enough raw materials, but that is not enough by itself.

 

 

 

 

So the assembly of an airliner is clearly “artificial,” meaning due to effects that are not subject to simple laws of probability. 

And art is “artificial,” as it’s not by chance that those two words share the same root.

Art is clearly not random, and humans call tell that this is so. Usually there are other clues besides it hanging on a gallery wall. 

You might be wondering that, if disorder tends to increase, how are there living creatures and buildings and mountains and sunshine?

We’ll get to that in the next episode. Stay tuned.

 

 

 

 

With gratitude from our studio to yours,

Nancy & Bruce

Click on the arrow to watch clips from the Book Unboxing!

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!

 

 

 

 

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