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Metaphorical Gesso & The Anti-Problem In Art- Nancy Hillis MD & Bruce Sawhill PhD)

Metaphorical Gesso & The Anti-Problem In Art- Nancy Hillis MD & Bruce Sawhill PhD)

Metaphorical Gesso & The Anti-Problem In Art


In grad school, I got an early dose of skepticism towards art criticism. I was not quite an Art Philistine, having traipsed through the Metropolitan Museum of Art and the Louvre with all of the other shuffling hordes.

I found art that was compelling to me and began to develop opinions about it, but I always had a hard time with people writing about art.



Entrance area of the Louvre, Paris



During that time, a sculpture was erected in front of our Physics Department building at Stanford. It was created by a sculptor I had never heard of at the time, Josef Albers. Other people clearly had heard of him. 

It was a geometric set of straight lines created from strips of shiny metal attached to a wall that was about head high, at the center of a grassy circle.



Albers Wall, Stanford University



Two friends and I decided to write a tongue-in-cheek art review for the campus paper. One of us had a background in art history but was now a grad student in chemistry.

The sentence I remember from the faux review was, “Are the pieces of metal glued to the wall “art” or is the whole thing art?” 

Since Nancy and I often talk critically about art criticism, this long-ago review came to mind, and that sentence in particular. 


Art lives in a context, and that context influences us, often subconsciously.


Art & Context


When we think of art, it is often in the form of historical or environmental contexts-


  • What was going on politically when this artwork was created?
  • What was happening in the artist’s life when the painting was painted?
  • Did the artist have a patron and how did that patron influence his/her creations?


But context can extend right into the creation itself.

I think of that Albers sculpture on a wall. The wall provides context. A wall by itself is not usually considered art, but it is an important context for this particular sculpture. Without it, the artwork could not be presented as envisioned.


Art is a balance between the active and the passive, the foreground and the context, even within a particular sculpture or canvas.


In music, it is the “spaces between the notes,” a saying attributed to many musicians, from Claude Debussy to Miles Davis to Alfred Brendel.


Music is the contrast between sound and silence, between sound and not-sound.


You didn’t think you’d get this far without some math or physics or complexity science, did you?


Metaphorical Gesso & The Anti-Problem


Many mathematical problems, particularly in our favorite field of combinatorics, are best solved by considering the “anti-problem.”

This means you think about identifying that which is not what you’re interested in, then subtract it out, leaving what you’re interested in.

You metaphorically gesso over what you don’t want and see what’s left.


Artists Lead The Way


Or, like Michelangelo, you release the emerging masterpiece by subtracting marble with your chisel. Michelangelo removed “not-David” from the stone, leaving David.

He was way ahead of the mathematicians.



Bruce Sawhill- David- The Accademia

Bruce Sawhill- David- Michelangelo- The Accademia, Florence, Italy




And the great scientist, Galileo Galilei, observing the moon through his telescope, used his considerable skills as an artist to depict what he saw through the lens.

His knowledge of perspective, gained from painting, allowed him to infer that there were mountains on the moon. This was not known, indeed, it was assumed to be otherwise- that heavenly bodies had to be perfectly geometrical, an idea dating back to Plato.

Indeed, creating art took Galileo into the adjacent possible- connecting astronomical observation with earthly experience.


The Art of Mathematics & Birthdays


Notably, Galileo claimed he was born on the day of Michelangelo’s death.

This is interesting for two reasons: 1) it reinforces a contemporary belief that genius never dies, but passes from one individual to the next and 2) Galileo chose, as his spiritual predecessor, an artist, not a scientist.

Now, people played fast and loose with birth and death dates back then…but that’s beside the point.

Galileo, one of the greatest minds of all time, valued art so deeply. He intuited then what we experience now…


Art takes us somewhere new- and science can learn from artists.


The Birthday Problem


But back to mathematics. Math is notorious for problems that are easy to state but hard to solve. A classic example is the “Birthday Problem.” 

The question to be answered is:

How many people do you need to have in a room to have a good chance (say 50/50, but it can be anywhere from 0 to 100%) that two of them have the same birthday?

This is the “foreground” problem- it’s like the paint on the canvas or the mark making or, in the case of music, the notes to be played- not the blank space on the canvas or the silence between the notes.



Birthday balloons in combinatorial abundance


Birthdays & Combinatorics


I once taught a statistics class at University of California Santa Cruz of about 200 people. The auditorium had four sections of about 50 students each.

I gave the student sitting at the bottom left of each section a copy of a year’s calendar and said, “Circle your birthday and pass it on. If your birthday falls on a circle that’s already there, stop and count the circles that have been drawn up to that point.”

To the surprise of some, it didn’t take long to discover that every section stopped before the last student in that section. Answers were between 13 and 37 students before a birthday was duplicated. It turns out you didn’t need anywhere near 365 people to duplicate a birthday.



Lecture hall



Mathematicians & Symbols


Mathematicians, unlike their reputation, often don’t like numbers. They like symbols representing numbers. It is a Kabbalistic form of abstract art, with strict rules for symbol manipulation. It is more formal by far than Olympic ice skating or classical ballet.

To get a handle on the birthday problem, they would create an abstraction where there are N people in a year that has D days, and then start reasoning about the interaction of D and N. Since this is a family newspaper, we won’t do that here.


Mathematical Maquettes


They might also create a kind of mathematical maquette, analogous to an artistic maquette, a very simplified problem where D and N are such that it is easy to think about it in your head and doesn’t require tedious calculation.

Maybe a simplified year consisting of two halves, with the question being about multiple people having birthdays in the same half of the year.

An example of this could be:


  • Imagine 1 person in a “year” consisting of 2 days. Their birthday is either on Day 1 or Day 2.
  • If a second person comes along and has an equal probability of a birthday on either day, they either have their birthday on a different day than Person 1 or on the same day, each with probability 50%.
  • So you need 2 people to have a 50/50 chance of birthdays on the same day as each other in a “year” of 2 days.


Simple enough to not require pen and paper or computer. That’s a high enough probability to order extra cake on your birthday.

If you think about 3 people and 3 days, it starts getting complicated enough to require pen and paper already. They could have all the same birthday! Two could have the same birthday and one not. All three could be different. It starts to become a headache.

You just want it to go away and birthdays are overrated anyway. You just want your own birthday and everyone else can get lost.

And that petulance gives rise to an idea! 

What if I just try and calculate how many ways everyone can have their own birthday and then subtract it from 1?

It’s a tedious calculation, but you find that it takes far fewer people than 365 to have more than one person sharing a birthday. It takes about 23.

Trying to do this the obvious way by enumerating all the ways multiple birthdays could land on 365 days is overwhelmingly difficult, too hard to do by hand, but the anti-problem is not hard at all by comparison.

It pays to think about the anti-problem when things are looking tough. This strategy doesn’t always work, but when it does, it can be spectacular.


A Solution Hidden For Decades


Amazingly enough, this problem was not brought forth as being solved until 1927 by a British number theorist, Harold Davenport. Perhaps it was solved centuries before but felt to be so obvious as to not warrant visibility or publication.

Davenport did not publish it because he couldn’t believe it hadn’t been figured out earlier, perhaps by someone like Newton or Euler- though he did speak about it and his name is now associated with it. Indeed, it is so simple to state that it certainly could have been thought of centuries earlier.

Math can be like this, gems hiding in plain sight.



Solution of birthday problem



The Hat Problem


Consider the hat problem. Imagine concluding your college graduation ceremony. Celebrating four years of toil, friendship, and unfolding young adulthood- you toss your mortarboard as high as possible.

What do you imagine is the probability of anyone catching their own hat back in a sea of hats?

Well, it turns out not to matter whether there are a few people or a thousand around you. The probability is about the same, a little over one-third. You would think the probability would grow with so many possibilities!

This is known as a “veridical” paradox, one that looks wrong but is in fact true.

The answer, like the Birthday Problem, is found by thinking about the anti-problem. It’s like Michelangelo chiseling away the stone. Calculating the probability that nobody gets their hat back is best done by eliminating the probability that one or more people gets their own hat back.



Hats being tossed at graduation


In the artistic realm, trying to solve the Birthday Problem instead of the anti-problem is like marking up every square inch of canvas to find a solution. You wouldn’t want to miss anything!

Another equivalent of the Birthday Problem and other veridical paradoxes in the artistic realm is a trompe l’oeil (Fool the eye) painting. You see something that can’t be true, but there it is on the canvas.


MC Escher


The artistic equivalent of these sorts of problems that contrast X and not-X  and fool the eye is nowhere better illustrated than in the works of MC Escher, (1898 – 1972) an artist whose reproductions are to be found on dorm room walls everywhere.


Day and Night, MC Escher, woodcut


Escher delighted in veridical paintings that employed symmetries and contrasts to create logical conundrums. And he even (perhaps unintentionally) created the concept of the “black swan” long before it became a common concept. They are seen flying in the above image.

The black swan concept was introduced by the philosopher/financier Nassim Taleb as an event that no prior experience prepares you for.

It’s the adjacent possible! There are swans and there is black, so it could happen in principle.


With gratitude from our studio to yours,

Nancy & Bruce



P.S. If you enjoy this blog, pair it with The Artist’s Journey & The Adjacent Possible books:


The Artist's Journey & The Adjacent Possible books

The Artist’s Journey & The Adjacent Possible books available on Amazon


The Artist’s Journey: Bold Strokes To Spark Creativity

The Artist’s Journey: Creativity Reflection Journal

The Adjacent Possible

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