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Mathematics Goes to the Movies
by Burkard Polster and Marty Ross
In Pi the brilliant (fictitious) mathematician Maximillian (Max) Cohen, played by Sean Gullette, is struggling to discern a pattern behind the numbers that make up the stock market. Along the way, he comes across some of mathematics’ all time favourites: Pi, the Fibonacci numbers and the Golden Ratio. He also gets mixed up with the numerology of Jewish mysticism. In fact, the answer to his question seems to be hidden in a 216 digit decimal number, that the mystics believe to be the name of God. The number seems to be the key to everything and has to power to wreck havoc with whoever gets close to it. Max’s computer crashes repeatedly while analysing patterns that supposedly conceal the number (the stock market, the Torah in number form, and Pi), Max suffers debilitating migraine attacks, and his former PhD supervisor at Columbia University Sol Robeson, played by Mark Margolis, suffers two strokes, the first one leading to his retirement, the second one, towards the end of the movie, to his death. Eventually, Max ends up lobotomising himself by drilling into the math part of his brain to eradicate all traces of the number.
Although this is not apparent from the movie, background information on the movie reveals that Max is using chaos theory to analyse patterns in nature. In the movie he identifies himself as a number theorist and from the choice of the examples of the math in the movie it appears that what he is really using is number theory. Sixteen minutes into his director’s commentary on the DVD Darren Aronofsky has the following to say about the math in the movie:
“So, all this math stuff is real stuff. Fibonacci is a real dude or was a real dude and the golden spiral was real, we did not make it up for the film. It is just a lot of plagiarism from the bible from math text books, from Pythagoras, we kind of all stuck it together.”
The credits of the movie include consultants for Go, Judaica and Medicine, but none for mathematics. From the director’s commentary we know that the director’s father is a scientist. The story about Archimedes in the movie was told to the director by his father.
In his commentary Sean Gullette, the actor who plays says that the math material was compiled by the director.
The director’s shooting script includes the deleted scenes, more math, and at the time of the writing of this book was freely available from various websites. Highly recommended reading.
The commentary on the DVD by Sean Gulette, the actor who plays Max, gives a nice summary of Max’s story and includes a lot of the background story for Max and Sol. Most importantly, about fourteen minutes into his commentary he says:
“He (Max) was a kid who had grown up on a commune in northern California and after his father joined the CIA and vanished he and his mum moved to Berkeley together and Max emerged very young as a maths prodigy. Seems that with these math geniuses that there is some sort of biochemical neurological thing that makes them excel. And so he was recognized at an early age as somebody with math potential and that was his way out of that life. He went to UC Berkeley, hooked up with Sol who was a brilliant Soviet expatriat working on number theory and writing algorithms to recognize patterns in large groups of numbers. And his thesis had been on the premise that the stock market contains a natural order because it’s a chaotic system and quantum theory teaches us that whenever there is a true natural chaotic system there must be a natural order within it.
(29:30 minutes into the commentary) Max is brought to New York by Columbia to study with Sol and for a while works under his tuition while Sol is studying the numbers of Pi but after Sol’s encounter with the strange forbidden truth that he thinks caused his stroke Sol quits and Max is forced to continue his work alone.”
Throughout the whole starting sequence of the movie, we see the decimal expansion
of Pi scrolling up the screen. According to the director’s commentary,
this sequence is supposed to show us what is going on in Max Cohen’s mind
while he is having one of his migraine episodes.
(This is only correct up 3.14159265 here (next correct digits are 35)
This scene introduces Max as someone who is brilliant with numbers.
As Max leaves his apartment, a young girl named Jenna runs up to him.
JENNA: Max, Max! Can we do it?
JENNA: Three hundred and twenty-two times four hundred and eighty-one.
This is a mistake. It should be …ninety-one according to the director’s shooting script and also for the product to give the result that Max comes up with.
Jenna types it into her calculator.
MAX:(instantly) One hundred fifty-eight thousand, one hundred two. Right?
Max heads down the staircase and Jenna screams after him.
JENNA: Okay, seventy-three divided by twenty-two.
MAX: (instantly again) Three point three one eight one eight one eight...
His voice trails off into the distance as he keeps walking down the stairs. He keeps saying “one” on one step and “eight” on the next, … a very nice portrayal of the infinite decimal expansion of this number.
MAX voiceover (V.O.):
Restate my assumptions:
1. Mathematics is the language of nature.
2. Everything around us can be represented and understood through numbers.
3. If you graph the numbers of any system, patterns emerge. Therefore there are patterns everywhere in nature.
Evidence: the cycling of disease epidemics, the wax and wane of Caribou populations, sunspot cycles, the rise and fall of the Nile. So what about the stock market? The universe of numbers that represents the global economy. Millions of human hands at work. Billions of minds, a vast network screaming with life, an organism, a natural organism.
My hypothesis: Within the stock market there is a pattern as well, right in front of me, hiding behind the numbers. Always has been.
Max is playing Go with his former supervisor at Columbia University, Sol Robeson.
SOL: You haven't taken a single break.
MAX: I'm so close.
SOL: Have you met the new fish my niece bought me? I named her Icarus. After you, my renegade pupil. You fly too high, you'll get burned. The more I see you, the more I see myself thirty years ago. My greatest pupil published at 16, Ph.D. at 20. But life isn't just mathematics, Max. I spent forty years looking for patterns in Pi, I found nothing.
MAX: You found things.
SOL: I found things, but not a pattern.
Max sits in New York City subway car.
MAX: (V.O.) Not a pattern …. Sol died a little when he stopped research on Pi. It wasn’t just a stroke. He stopped caring. How could he stop when he was to close to seeing Pi for what it really is. How could you stop believing that there is a pattern, an ordered shape behind those numbers when you are so close. We see the simplicity of the circle, we see the maddening complexity of the endless string of numbers, three point one four, off to infinity
While he says this draws a circle with its diameter on his Wall Street journal, then he writes "A=Pir2" and "C=2 pi r." Next he writes "pi=3.14159..."
Scribblings on paper in coffee shop
Max meets Lenny Meyer, a Hassidic Jew.
LENNY: So, what do you do?
MAX: Um, I work with computers, ...math.
LENNY: Math? What type of math?
MAX: Number theory. Research, mostly.
LENNY: No way, I work with numbers myself. I mean not traditional, though I work with the Torah. Amazing. You know Hebrew is all math, its all numbers. You know that? Here look. The ancient Jews used Hebrew as their numerical system. Each letter is a number. Like the Hebrew A, Aleph is one, B, Bet, is two, understand? Look at this. The numbers are interrelated. Like take the Hebrew word for father Ab, Aleph, Bet, one (plus) two equals three, all right. Hebrew word for mother Am, Aleph, Mam, one (plus) forty equals forty one. Sum of three and forty-one, forty-four, right? Hebrew word for child, mother, father, child, (ye)(le)d, that’s ten (plus) thirty and (plus) four, forty four (!) The Torah is just a long string of numbers. Some say that it is a code sent to us from God.
MAX: That’s kind of interesting.
LENNY: Yeah, that’s just kids’ stuff. Check this out. The word for the Garden of Eden, Kadem. Numerical translation one forty-four (The numbers 4, 30 and 10 that he scribbles do NOT add up to 144. In the shooting script we find the proper numbers are Kuf hundred. Daled, four Mem, forty.) Now the value of Tree of knowledge...in the garden, right, Aat Ha Haim, two hundred thirty-three. One forty four, two thirty three. Now you can take those numbers..
MAX: Those are Fibonacci numbers.
MAX: The Fibonacci sequence.
Fibonacci is an Italian mathematician from the thirteenth century.
If you divide a hundred and forty-four into two hundred and thirty-three, the result approaches Theta.
MAX: Theta, the Greek symbol for the golden ratio, the golden spiral.
LENNY: Wow, I never saw that before. That's like that series that you find in nature, like the face of a sunflower?
MAX: Wherever there's spirals. (Here, he looks at a spiral created a dropping some cream into his coffee)
LENNY: See, there's math everywhere.
Lenny's smoke drifts by MAX:'s eyes. We see smoke spirals spinning in front of us.
MAX (v.o.): Restate my assumptions:
1. Mathematics is the language of nature.
2. Everything around us can be represented and understood through numbers.
3. If you graph the numbers of any system, patterns emerge. Therefore there ARE patterns everywhere in nature.
As he says this he draws a square spiral.
So what about the stock market? The universe of numbers that represents the global economy. Millions of human hands at work. Billions of minds, a vast network screaming with life, an organism, a natural organism.
My hypothesis: Within the stock market there is a pattern, right in front of me, playing with the numbers. Always has been.
Following this he uses Euclid, his computer, to predict the stock market. The results seem absurd. Before the computer crashes it prints out a long string of numbers.
Sol’s apartment. Max and Sol are playing Go.
MAX: Euclid crashed. I lost all my data, my hardware.
SOL: Your mainframe?
SOL: What happened?
MAX: First I get these crazy low picks. Then Euclid spits out this long string of numbers. Never saw anything like it and then it fries. The whole machine just crashed.
SOL: You have a printout?
MAX: Of what?
SOL: Of the picks, the number?
MAX: I threw it out.
SOL: What was the number it spit out?
MAX: I don’t know, just a long string of digits.
SOL: How many?
MAX: I don’t know.
SOL: What was it, a hundred, a thousand, two hundred sixteen (this is important)!? How many?
MAX: I don’t know. Probably around two hundred. Why?
SOL: I dealt with some bugs back in my Pi days. I was wondering if it was similar to one I ran into. Have you met Archimedes (one of his fish). The one with the black spots. You see?
SOL: You remember Archimedes of Syracuse? The King asks Archimedes to determine if a present he’s received is actually solid gold. Unsolved problem at the time. It tortures the great Greek mathematician for weeks. Insomnia haunts him and he twists and turns in his bed for nights on end. Finally, his equally exhausted wife, she’s forced to share a bed with this genius, convinces him to take a bath, to relax. While he is entering the tub Archimedes notices the bathwater rise. Displacement. A way to determine volume. And thus, a way to determine density, weight over volume. And thus, Archimedes solves the problem. He screams “Eureka!” (Greek for “I found it!) and is so overwhelmed he runs dripping naked through the streets to the King’s palace to report his discovery. Now, what is the moral of the story.
MAX: That a breakthrough will come...
SOL: Wrong. The point of the story is the wife. You listen to your wife, she will give you perspective. Meaning, you need a break, you have to take a bath, or you’ll get nowhere. There will be no order, only chaos. Go home Max and you take a bath.
In a synagogue.
LENNY: You know when you told me that you were Max Cohen I didn’t realize that you were the Max Cohen. Your work is revolutionary you know that. It’s inspired the work that we do.
MAX: It has?
LENNY: Yes, very much so. The only difference is, we’re not looking at the stock market. …..We’re searching for a pattern in the Torah.
MAX: What kind of pattern?
LENNY: We’re not sure. We only know, it’s 216 digits long. 28:00
Back in Sol’s apartment.
MAX: What's the two hundred and sixteen number, Sol?
SOL: Excuse me?
MAX: You asked me if I had seen a two hundred and sixteen digit number, right?
SOL: Oh, you mean the bug. I ran into it working on Pi.
MAX: What do you mean "ran into it"?
SOL: What is this all about,
MAX: There's these religious Jews that I’ve been talking to.
SOL: Religious Jews?
MAX: Yeah, you know Hassids the guys with the beards. I met one in the coffee shop. It turns out the guy's a number theorist. The Torah is his data set. He tells me that they are looking for a two hundred and sixteen digit number in the Torah.
SOL: Come on, it’s just a coincidence.
MAX: There's something else, though.
MAX: You remember those weird stock picks I got.
SOL: Yesterday's stock picks, yes?
MAX: It turns out they were correct. I got two picks on the nose. Smack on the nose Sol. Something's going on. It has to do with that number. There is an answer in that number.
SOL: Come with me.
Sol and Max are playing Go again.
SOL: The Ancient Japanese considered the Go board to be a microcosm of the universe. Although, when it is empty it appears to be simple and ordered, the possibilities of game play are endless. They say that no two Go games have ever been alike. Just like snowflakes. So, the Go board actually represents an extremely complex and chaotic universe and that’s the truth of our world, Max. It can't be easily summed up with math. There is no simple pattern.
MAX: But as a Go game progresses, the possibilities become smaller and smaller. The board does take on order. Soon, all moves are predictable.
SOL: So, so?
MAX: So, maybe, even though we're not sophisticated enough to be aware of it, there is a pattern, an order, underlying every Go game. Maybe that pattern is like the pattern in the stock market, the Torah. This two sixteen number.
SOL: This is insanity, Max.
MAX: Or maybe it's genius. I have to get that number.
SOL: Hold on, you have to slow down. You're losing it, you have to take a breath. Listen to yourself. You're connecting a computer bug I had, with a computer bug you might have had, and some religious hogwash. If you want to find the number two sixteen in the world, you'll be able to find it everywhere. Two hundred sixteen steps from your street comer to your front door. Two hundred sixteen seconds you spend riding on the elevator. When your mind becomes obsessed with anything, you will filter everything else out and find that thing everywhere. Three hundred and twenty, four hundred and fifty, twenty-three. Whatever! You've chosen two hundred sixteen and you'll find it everywhere in nature. But Max, as soon as you discard scientific rigor, you are no longer a mathematician. You are a numerologist.
On the beach finds sea shell in the form of spiral.
Later he has a look at the sticky substance he found in his crashed computer under a microscope–another spiral.
In his apartment.
MAX: Remember Pythagoras. Mathematician, cult leader, Athens, ca. 500 B.C.E. Major belief: The universe is made of numbers. Major contribution: the Golden Ratio.
Best represented geometrically as the Golden Rectangle. Visually there exists a graceful equilibrium between the shape’s length and width. When it is squared, it leaves a smaller golden rectangle behind, with the same unique ratio. The squaring can continue smaller and smaller and smaller, to infinity. (…also comment on the word “squaring”… and it should be A:B::A+B:A)
MAX (V.O.): More evidence: Remember DaVinci, artist, inventor, sculptor, naturalist scientist, Italy 15th century, rediscovered the balanced perfection of the golden rectangle and penciled it into his masterpieces. Connecting a curve through the concentric golden rectangles, you generate the mythical golden spiral. (… also comment on the word concentric…)
Pythagoras loved this shape for found it everywhere in nature: the nautilus shell, rams horns, whirl pools, tornados, our fingerprints, our DNA, and even our Milky way.
Walking around in Chinatown.
MAX (V.O.): When I was a little kid my mom told me not to stare into the sun. So, once when I was six I did. At first the brightness was overwhelming but I had seen that before. I kept looking, forcing myself not to blink and then the brightness began to dissolve. My pupils shrank to pinholes and everything came into focus and for a moment I understood. My new hyposthesis: My new hypothesis. If we are built from spirals, living in a giant spiral, then everything we put our hands to is infused with the spiral.
Max makes a deal with some stock market analysts who want to get their hands on his results on predicting the stock market. This gets him a new powerful computer chip. He also offers to help Lenny to find the number and gets the Torah in binary form off him. He starts analyzing the Torah with his fixed computer Euclid. Computer crashes and so does Max (another violent Migraine attack).
The main monitor is filled with numbers. And then a number pops onto the screen
These are actually 218 digits.
MAX: You lied to me.
SOL: Ok, sit down.
SOL: I gave up before I pinpointed it. But my guess is that certain problems cause computers to get stuck in a particular loop. The loop leads to meltdown, but just before they crash they...they become "aware" of their own structure. The computer has a sense of its own silicon nature and it prints out its ingredients.
MAX: The computer becomes conscious?
SOL: In some ways...I guess...
MAX: (To himself) Studying the pattern made Euclid conscious of itself. Before it died it spit out the number. That consciousness is the number.
SOL: No, Max, it's only a nasty bug.
MAX: It’s more than that!
SOL : No it’s not! It’s a dead end, there is nothing there.
MAX: It’s a door Sol, a door.
SOL: A door in front of a cliff. You're driving yourself over the edge. You need to stop.
MAX: You were afraid of it. That’s why you quit.
SOL: Max, I got burnt.
MAX: C'mon, Sol.
SOL: It caused my stroke.
MAX: That's bullshit. It's mathematics, numbers, ideas. Mathematicians are supposed to be out on the edge. You taught me that!
SOL: Max, there's more than math! It’s death Max!
MAX: You can't tell me what it is. You've retreated to your Go and your books and your goldfish, but you're not satisfied.
SOL: Max go home. Get out of my house.
MAX: I’m going to know what it is, I’m going to see it, I’m going to understand it. 1:07:20
Synagogue. A Rabbi tells Max why the number is important for them.
MAX: (Incredulous) You're telling me that the number in my head is the true name of God!?
MAX: It's more than God...it s everything. It's math and science and nature...the universe. I saw the Universe's DNA
MAX: It's just a number. I'm sure you've written down every two hundred sixteen number. You've translated all of them. You've intoned them all. Haven't you? But what's it gotten you? The number is nothing! It's the meaning, the syntax. It's what's between the numbers. If you have not understood it, it’s not for you. I've got it, I’ve got it and I understand it, I'm going to see it! Rabbi...I was chosen.
More glimpses of pictures of math, Hebrew writing, and Go board while he is running through the streets towards Sol’s apartment. Here just two example of the math.
Sol had a second stroke. In Sol’s apartment Max finds a piece of paper with Sol's handwriting on it. On the paper is THE number. Max slides it into his pocket. Max looks at the Go board. The pieces are arranged in a giant spiral across the board.
He burns the sheet of pager with the number on it then drills into the math section of his scalp with a power drill.
After that he cannot do any more math, seems ok otherwise, and we see him smile for the first time.
The last remark in the director’s commentary:
I always wanted to end the film with little Jenna saying: what’s the answer? And having Max not able to give the answer because for the first time he doesn’t have the answer. And, in many ways I think that’s what Pi is about, it’s about the questions, it’s about the chaos, it’s about the search for order. That’s were Max finds beauty, he finds it in now (?), he finds it in the world around him