Summary: A little boy throws stones into a pond and discovers a principle that may explain how the whole universe comes to exist.
And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about.
- I Kings 7, 23
So according to the Bible it's an even 3. The Egyptians thought it was 3.16 in 1650 B.C.. Ptolemy figured it was 3.1416 in 150 AD. And on the other side of the world, probably oblivious to Ptolemy's work, Zu Chongzhi calculated it to 355 / 113 . In Bagdad, circa 800 AD, al-Khwarizmi D agreed with Ptolemy; 3.1416 it was, until James Gregory begged to differ in the late 1600s.
Part of the reason why it was so hard to find the true value of Pi (p) was the lack of a good way to precisely measure a circle's circumference when your piece of twine would stretch and deform in the process of taking it. When Archimedes tried, he inscribed two polygons in a circle, one fitting inside and the other outside, so he could calculate the average of their boundaries (he calculated p to be 3.1418). Others found you didn't necessarily need to draw a circle: Georges Buffon found that if you drew a grid of parallel lines, each 1 unit apart, and dropped a pin on it that was also 1 unit in length, then the probability that the pin would fall across a line was 2/p. In 1901, someone dropped a pin 34080 times and got an average of 3.1415929.
Like all these men noticed, everyone everywhere at anytime who's tried to find p have all come up with pretty much the same number, with some more precise than others. It didn't matter if you tried it on the other side of the world, or got your brother to do it, or let the circle relax a bit before trying again, or snuck up on it when it wasn't looking, or waited until mom and dad had gone to bed before creeping down to the study on tiptoe to catch it by surprise, the ratio of a circle's diameter and circumference still was, is, and always will be 3.1415926535897932384626—and so-on and so-on and so-on...
“I had a vision this morning” the boy said to his father, “while throwing stones into a pond and watching the ripples. I saw another little boy on the furthest end of the Universe, on a planet so far away that if you'd shone a light in its direction on the second the Big Bang happened, the light still wouldn't have reached it by now. And he was throwing stones into a pond just like me, and the circles were exactly the same.”
“They're the same everywhere,” his father replied.
“So how do they know to be? Where did they learn their shape from?”
“Do you know what Pi is?”
“Yeah, we learned it in school last month.”
“Well Pi doesn't have to be told what it is, just like 4 doesn't need to be told it's the sum of 2 + 2. And as long as Pi is what it is, every circle is the same everywhere.”
“But where do Pi and 4 come from?”
“They don't come from anywhere, we just discover them whenever something happens. Numbers were always there, even if we never bothered to look for them, even if nobody ever drew a circle, or nobody ever threw a stone into a pond.”
Here's a party trick.
- Take any two numbers at random, second bigger than the first (such as 214 and 346)
- Add them together
- Add the result to the second largest number
- Repeat from step 2 for a while (five or six times, maybe)
- Divide the last result you got by the second-to-last result
I'll bet your answer is somewhere close to 1.618. Let's try it:
- 214 + 346 = 560
- 346 + 560 = 906
- 560 + 906 = 1466
- 906 + 1466 = 2372
- 1466 + 2372 = 3838
Okay, that's enough, let's see what happens when we divide the last by the second-to-last:
- 3838 / 2372 = 1.618
(You may not always get it exact to three decimal places, but you should at least get it at two decimal places, or 1.61—something.)
1.618 is known as Phi (f) (pronounced fee) or sometimes “The Golden Ratio”, and joins p as one of the great mathematical constants. The earliest description we know of comes from Euclid in 300 B.C., and he described it as a geometry problem: “A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser.” Or like this:
The segment AC is longer than CB, but it's also been drawn so that ratio of AC to CB is the same as the ratio of AB to AC. There's only one ratio where this geometrical arrangement can be so, and it's approximately 1.6180339887. Like p, its precision also goes on forever.
The party trick is just another way of generating Fibonacci numbers, which are nothing more than the sequence of numbers you get when you add the last two numbers in the sequence together. The official sequence begins as: 1, 1, 2, 3, 5, 8, 13 and so-on. It starts off with two ones to get it going, but you can get a sequence that's Fibonacci-like with any two starting numbers. After only a few iterations, the ratio of the last number to the second-to-last converge towards the magical Golden Ratio, or f. The more iterations you go, the closer to the exact precision of f you get.
Apart from tricks easily performed on the back of a napkin, f also shows up everywhere in nature. The chambers of a Nautilus shell, for example, increase in size according to the same rule as the Fibonacci sequence, and therefore converge to f. Poetically, it's also seen in the spiral vortex of galaxies.
The following morning, at breakfast, the boy revealed what he'd been thinking about the night before. “Dad, you said that numbers didn't need to come from anywhere. They're just always, like, there?”
His father nodded and spread marmalade across his toast. “It's like when you pick the biggest number you can think of, and then add one to it to get an even bigger number. You might ask yourself, did that bigger number not exist until you thought of the biggest you could, and added one to it? Or was it always there? It's kind of silly to think about it, really.”
“Yeah, but...” and the boy frowned with the trouble of putting his mind into words, “was there ever a time when they didn't exist?”
“Of course not, numbers aren't something are born and age.”
“So all numbers exist all the time?”
“Yep. Simultaneously, even.”
“Even if the Big Bang never happened and the universe never existed?”
The father put down the spreading knife and munched his toast before he answered again, “yes. According to Plato, mathematics is discovered and not invented. Two plus two is still four, even if you don't have four physical things to prove it, and even if there's nobody around to think it. So a physical universe isn't necessary for a number to just be.”
Although Pi and Phi's precision went on forever, and the “birthplace” of all those digits unknown, mathematicians still thought, for centuries, that simple equations led to simple results. The golden age of mathematics began with Newton's Principia Mathematica, a book which laid out the simple equations that described the movement of the planets, or of cannonballs, or of anything else that had mass and flew through space. To the limited degree in which such phenomena could be observed and measured by man at the time, Principia harmonized the universe perfectly.
The complacency of these men wouldn't be completely broken until only forty years ago by a weather forecaster. His name was Edward Lorenz, and he'd been working on a model of weather systems that seemed to do a pretty good job of predicting actual outcomes. Only one day he sat down at his computer terminal, wanting to pick-up from the middle of a previous run, and re-typed the data he'd been working with the day before. The simulation proceded by duplicating the results of the previous run—exactly as expected—but then diverged wildly, as time went on, to the shock of Mr. Lorenz.
Edward had made an error when re-entering his data, and had stumbled upon the concept of “sensitive dependence upon initial conditions”.
Lorenz's simulation worked by processing some numbers to get a result, and then processing the result to get the next result, thus predicting the weather two moments of time into the future. Let's call them result 1, which was fed back into the simulation to get result 2. result 3 could then be figured out by plugging result 2 into the simulation and running it again. The computer was storing result n to six decimal places internally, but only printing them out to three. When it was time to calculate result 3 the following day, he re-entered result 2, but only to three decimal places, and it was this that led to the discovery of something profound.
Given just an eentsy teensty tiny little change in the input conditions, the result was wild and unpredictable. This was bad news for weather forecasters, because it meant that even if your model of how weather worked was perfect, being off by a billionth of a decimal point in your temperature or wind measurements would still, eventually, result in a wild departure from the real world. Lorenz would go on to write a paper that introduced the famous Butterfly Effect, where a “butterfly flapping its wings in Tokyo makes the weather in New York different”.
But Lorenz is also famous for another butterfly:
Fig 2. A Lorenz butterfly, also called a strange attractor
Lorenz simplified his simulation down to three formulas that describe the flow of fluid in a layer of fluid that has uniform depth and a constant temperature difference between the top and the bottom, formulas that could be used as the building blocks for a weather simulation. You don't have to understand math to look at these formulas and see that they're all pretty simple:
dx/dt = 10(y - x)
dy/dt = xz + 28x - y
dz/dt = xy - (8/3)z
These three formulas are used to find coordinates to plot on a three-dimensional graph (dx, dy and dz, while dt refers to how much time you want to cover with each step). A sample of one such graph can be seen in Fig 2, which was made by picking an arbitrary starting point somewhere on the X, Y and Z coordinates, and then using result 1 to find result 2 and so-on. The line it draws through space is a long, curving string of mathematical spaghetti. If you'd like to draw one for yourself, and you have Java on your computer, you can make all the Lorenz butterflies you want here.
Lorenz had discovered a Strange Attractor, which is a solution to an equation that converges to a single point—called the attractor—no matter what the initial conditions were. The whirlpool in Fig 2 shows off two points of strange gravity, around which the “particle” was compelled to orbit between. These points weren't imagined by an artist, they weren't the result of a malfunction in the computer, they were just there, waiting to be discovered.
Dad was waiting to pick up his son outside of school that day. And after a car ride together they arrived at the large building where dad worked. They went inside to a room full of computer screens and a door to a warm and noisy place filled with cabinets and masses upon masses of aqua blue and cadmium yellow wires that coursed up out of the cabinets and through conduits hung from the ceiling.
“We do some very exciting stuff here at the university, son,” he told his child. “In that room with all the cabinets is our big computer, a lot more powerful than the one you use to talk to your friends with, and in here we see pictures of what it's working on.”
The boy had his nose pressed against the glass to the machine room, “what does it work on?”
“The work we're doing now is paid for by the government to simulate what goes on in nuclear reactions. The computer in there is pretending to be a nuclear reactor, but without having actual nuclear fuel. It works because we have a pretty good idea of how atoms work mathematically, so the computer can simulate them as if they were real, but they're not. Inside that room, they're just numbers.”
“How can they be the same as a real reactor?”
“‘Every finitely realizable physical system can be perfectly simulated by a universal model computing machine operating by finite means’”
“What does that mean??”
“That's the Church-Turing principle, son. It means that anything physical can be simulated on a computer, just as long as you know how that thing works. We know how atoms work, so we can program that knowledge into a really big computer like that one on the other side of the glass. What's going on inside those cabinets is exactly the same thing that goes on inside a real reactor, except the fuel it burns and the energy it produces is all imaginary.”
He then took his son to one of the biggest displays in the control room, a projector that shone an image onto a large screen taking up most of a wall. It was currently showing some windows and icons, but dad whisked the pointer around and opened up a file that started a movie playing. In the corner of the movie, some numbers ticked by to show the fractions of a second that had elapsed. Starting with a brilliant point of light at the top of a tower, the light expanded into a mottled cloud that engulfed the tower, turned the guy wires into fiery spikes, and continued until it was now pushing clouds out of the sky, rising up into the air on the end of a smokey stalk.
“That's a reproduction of the Trinity atomic bomb explosion,” dad said, “one of the earlier simulations we ran on that machine before we switched over to looking at fission reactors. It's almost exactly the same as the real one, but not quite.”
“I thought you said it was the same as the real thing, just now?”
“Well, we can't simulate something that's exactly the same as an event that's already happened. That's because even the tiniest little mistake in the initial data would slowly expand into a huge difference. However, we don't need to simulate a specific event, we just need to get an idea of how reactors behave in general. That way we can build better and safer ones. Since we control the initial data, we can run the exact same simulation over and over, and so can the guys on the west coast when they want to verify that our computer isn't broken.”
The boy pressed his nose against the glass again, “could that simulate a kid like me, then?”
His father laughed, “sure, if only we knew how you worked!”
“But it's not impossible, in theory” the boy said, appending a phrase he'd heard many times from his dad.
“No, not impossible. But he'd grow up to be different than you. He wouldn't experience exactly the same things, and he wouldn't have exactly the same thoughts.”
“So does that mean you could simulate everybody in there? The whole world?”
His father hmm-ed and rubbed his chin, “I think I know what you're getting at, but this computer isn't nearly powerful enough. In fact, a computer that powerful probably won't get built for a hundred years.”
His son was grinning, “yeah, but say one does, the people in there would all be numbers, right?”
In mathematics, a set is a collection of numbers that fall within some kind of boundary. So if we say that a set called Z includes every number between 0 and 10, then Z must include 1, 2, 3, 4, 5, 6, 7, 8, and 9. But it should also include 1.1, 1.01, 1.001, 1.0001 and so-on to infinite precision, so to make it easier to write we could simplify it to: Z n = 0 < n < 10.
In the 1970s a mathematician named Benoit Mandelbrot was studying this set:
Z n = Z n-1 2 + C
It had come out of the work of French mathematician and WWI soldier Gaston Maurice Julia, a poor fellow who lost his nose to a battle wound and had to wear a leather strap over his face the rest of his life. He'd worked on mathematics inbetween operations at the hospital, finally publishing a paper in 1918 that made him famous, and from which Benoit Mandelbrot began a journey into a new branch of fractal mathematics.
Benoit had a big advantage over Julia, which was that he had the sense to be born several decades later and make himself a fellow at IBM's Thomas J. Watson research center, giving him access to enormous computing power. He had developed a function for expressing all possible Julia sets. When graphed on some of the earliest computers capable of complex graphics, Mandelbrot's set looked something like this:
Fig 4. Zoomed in somewhere in the vast Mandelbrot set.
And as Mandelbrot's set was rendered on better and better computers, with some tricks to apply an arbitrary pallette of colors, the graph became a staple of pop culture. A colorful finish for lunch boxes, glossy file folders, posters, t-shirts, and backgrounds. Anybody with a reasonably fast home computer could find a Mandelbrot program and sit there, in front of a screen, clicking away with a mouse to zoom further and further and further into one of the tendrils that feathered the edges of the set, discovering complexity that was just there.
At no time did anybody's computer actually calculate the whole set (because that would have taken infinite time), even though you could still see what the general shape of the whole looked like. If you wanted to know what color the pixel should be when the center of the graph is -0.870087051727264789 and the magnification is 9.46765618640618641, you could plug those numbers into an algorithm and it would find out by running the numbers through the formula. With that as your center, the computer can find out what the neighbors look like, and you get a graphing like the one in Fig 4.
A billion different patterns could be found in the Mandelbrot set and turned into merchandise without any fear of Intellectual Property violations, because the apparent artwork on display can be extracted from the formula Benoit studied above, like a portal into another world. The detail appears from nowhere you can physically point to, and it's always the same when given the same set of coordinates, no matter who wrote the program, no matter what language they wrote it in, no matter who runs it, no matter who built the computer.
Mom was treated to a bizarre display during bathtime that evening. Her son was staring fixedly at his hand while he slowly moved it in front of his face. “What on Earth are you doing?” she asked.
“I'm trying to decide if this is really happening,” was the boy's cryptic reply.
“What do you mean?”
“Well, I don't know how to explain it. I guess I'm wondering if I'm a computer simulation, right now, right as we speak.”
His mother raised her eyebrows, “your father was telling me about that. Are you trying to decide if there's a computer ticking away somewhere, calculating the movement of your arm right now?”
“That's just it, I don't know if it's really right now.”
“I think you lost me there, kiddo.”
“See, dad said that every possible number, all infinity of them, exist all the time. But if I'm just a computer simulation, where numbers describe everything that is and everything that will happen, then when is now?”
His mother clucked, “I guess now is now. If you're a computer simulation, then by the fact that you're experiencing this soapy water means that the computer, at this moment, is going through the gyrations of calculating fluid dynamics and their effect on little boys.”
“Yeah, but...” and the boy hit that mental block of trying to convert ideas into the words he's learned so far, so he struggled; “if there was another computer, and it was faster, and it ran exactly the same simulation, like dad's friends on the west coast, then they'd see me a few minutes from now, and I'd be thinking the same thoughts that I will be in a few minutes. I mean, that the other me will think the same things this me will think. So his now be my future, but I'm still thinking the same thoughts?”
“Hmm, you're trying to say that your sensations could have been calculated centuries ago, but that this was the conversation that just happened at that point in the simulation.”
“Yeah, like if you look at a movie on tape, and somewhere at the beginning of it the actors are saying ‘Is now now?’ or ‘When will then be now?’”
“‘Soon!’” his mother grinned, but the boy didn't get the joke.
Many have speculated that you could simulate a working universe inside a computer. Maybe it wouldn't be exactly the same as ours, and maybe it wouldn't even be as complex, either, but it would have matter and energy and time would elapse so things could happen to them. In fact, tiny little universes are simulated on computers all the time, for both scientific work and for playing games in. Each one obeys simplified laws of physics the programmers have spelled out for them, with some less simplified than others. But one thing that holds true for all those toy universes is that 2 + 2 still equals 4, Pi is still 3.14159, Lorenz's formulas still explore strange attractors, and Mandelbrot's set still explodes with detail.
So let's say you wrote a program that simulated a simple little universe. You base it on a model that can be expressed with some mathematical formulas, and then start it running with an initial set of data. Meanwhile, a colleague of yours writes another program and bases it on the same mathematical model and starts it running with exactly the same set of initial data. You let the two simulations run for hours, days, months, even years, and they never diverge from each other. They'd match each other with 100% accuracy, forever.
How can this be so? Well even Lorenz's butterfly will plot the same curved string forever if you run it again with exactly the same initial conditions, if you don't make the mistake that Lorenz did; if you re-enter the numbers with the same precision that you did the last time. Lorenz and Mandelbrot and the ancient computers of Pi discovered that infinite diversity can be expressed with fiendishly simple math, but the numbers do not come out differently for no good reason. Math is universal, and its rules don't change.
So if Pi was always there, whether we bothered to look for it or not, and the Mandelbrot set was always there, indifferent to which mathematicians were born, and the number '4' was always there, waiting for us to add 2 plus 2, then just where exactly is that simulated universe? Is it inside the computer? If so, how can it be in both the first computer you set up, and also in the second one your colleague built? And would it still be “there”, even if you and your colleague had never bothered to write the program?
Take away the initial set of data, and you'll know that if you stumbled on it again, your toy universe would play itself out again the same way. Take away the program, and you'll know that if you ever wrote it again to run the simulation by the same rules, then it'll “re-discover” the same toy universe all over again. Take away the computer, or take away the programmer, or take away the entire universe, right down to the Big Bang, just take away everything, and the math that would drive that simulation is still just there.
“Dad, why would you need a computer bigger than the one at work to simulate the whole world?”
“There'd be so many variables, so much data, that you'd need an enormous amount of memory to store it all, and the computer itself would have to be outrageously fast for us to see anything interesting happen within our lifetimes.”
“Okay, so pretend you built one, and you simulated a world for ten seconds of its time, and you recorded that in a file just like the one of the Trinity explosion. Isn't that file just one big huge number?”
“Y—es, pretty much. It's just a string of ones-and-zeroes recorded on a hard drive.”
“Okay,” said the little boy, ”so did that number exist before you built the computer, ran the simulation, and recorded it?”
Many have speculated that the universe as we know it is just a simulation running on a computer, but what if it ever broke down? The same simulation run on a resurrected computer could pick it up where it left off, so long as the programmer remembers not to make Lorenz's mistake, and re-enters the initial data with the same precision as before. As simulacrums running inside this hypothetical machine, we couldn't know.
Many have speculated that a computer running this universe would have to be massive, and it's extremely unlikely that it would be built by any Meta-Beings, but would it have to be? Would the theory for the computer need to be written down? Would the materials to build it need to be assembled? Would the Meta-Universe even need to exist at all?
Mathematicians have been playing with toy universes for decades now, thanks to simple and limited computers that can sustain a simple and limited simulation. One kind of universe is called Cellular Automata (CA), which imagines a universe divided into a grid, and dots of matter that may exist in the squares between grid lines. Dots are born and die according to very simple rules, like: “a dot will be born in any empty square with at least two inhabited neighbors”, and “a dot will die (of loneliness?) if there are no inhabited neighbors”.
You can standardize the way that rules are expressed just by showing a diagram of all possible neighbor-combinations, and what state they will lead to in the next “second” of the toy universe's life. Steven Wolfram's book A New Kind of Science, shows how along with examples such as this one.
In a one-dimensional CA universe there are eight possible neighbor-combinations, each of which can lead to only two possible outcomes. This means you can express all of the rules that make the toy universe work in a single binary number. Like this:
(The squares at the top of each box represent the neighbor combination, while the square at the bottom shows what color to fill the square for the next iteration of the CA)
If each white square represents 0, and each black square represents 1, then the above ruleset can be expressed as the number 110 (or 01101110 in binary). Wolfram was fascinated with this particular ruleset, because it evolved such a complex and infinitely progressive pattern, barely the tip of which is shown here. Anybody can re-discover this toy universe anytime and anywhere just by following the same rules, painting-by-numbers a pattern which is just there.
By showing a simple toy universe can be simulated on the computer, and then by showing that the principle scales to larger and more complex simulations, you can say that a simulation of any depth and complexity can be not only thought of and written down, but expressed in purely mathematical terms. And that if the starting-point of this toy universe can be expressed as a single number, then—like with the Mandelbrot set and its way of expressing every possible Julia set—you could express every possible variation of that toy universe with nothing more than a set. “Let the Universe n be 0 < n < 256”
Wolfram is convinced that every natural phenomena in this universe (the one you're reading this essay in, the Real World) may be driven by fiendishly simple algorithms, and none of them are fundamentally different from the CA he plays with on his computer. Every sea-shell, plant leaf or body organ with an interesting pattern that can be described in terms of a Cellular Automata makes the hypothesis more and more convincing.
But who needs to prove that our own universe is nothing more than a big simulation, when you can use a computer to prove another one already is? Just imagine one complex enough to eventually evolve a sentient creature, one capable of writing this essay and another capable of reading it. If a simulation like that is theoretically possible, then, according to Plato's view—where numbers and the rules of mathematics exist independently of our minds—those two creatures must already exist, have already written the essay, and already read it.
Was it necessary to sit down and design the algorithms and initial data for the simulation? Not as long as everything that happens in that simulation can be expressed as a number. Michaelangelo claimed the statue was already in the block of stone, and he just had to chip away the unnecessary parts. And in a literal sense, an infinite number of universes of all types and states should exist in thin air, indifferent to whether or not we discover the rules that exactly reveal their outcome. Our own universe could even be the numerical result of a mathematical equation that nobody has bothered to sit down and solve yet.
But we'd be here, waiting for them to discover us, and everything we'll ever do.
On a planet at the furthest end of the universe, so far away from Earth that if you'd shone a light in its direction at the moment of the Big Bang, the light still wouldn't have reached it by now, a little boy sat throwing stones into a pond and watching the ripples.
The boy had been born with a mutation in his brain that disabled his sense of time. From his perspective there was no then or now or later, nor did these concepts make sense to him when his parents explained them. When he looked up in the sky and saw an airplane in flight, he saw a great metallic ribbon of airplane stretch all the way from the horizon. When he watched the pond he saw all the stone tosses he'd already made, and all of the ripples they made. But when he looked at the ripples he didn't see circles, he saw discs that were haloed by a fuzzy fringe of interference patterns where they intersected the reeds and the shoreline of the pond.
Time was a dimension that this little boy could see, just like you could see a line stretching to infinity when you stand astride a road in a desert that stretches to the horizon. He perceived his life like an aperture—a window—that slowly got wider with every second to pass by.
The boy didn't cry when his pet dog died because he could still see it and hear it and feel the sensation of petting its fur. He didn't pine for Christmas morning, because he was still living in the same moment as the last ones. And he played with his toys only once, for a few minutes or an hour, exploring them the way you'd explore a footpath in the woods until it came to an end.
One day his dad took him to the place where he worked, to show off the huge computer his university was building. “When this is completed it'll be the largest and most powerful in the world. It'll use a special kind of memory that's holographic, which means that it'll be able to store an infinite amount of information, in theory. Do you know what a hologram is?
The little boy shook his head.
“It just means it captures a picture of the world from every possible angle, an infinite number of angles in fact. That's why holograms on stickers and fancy book-covers look three-dimensional to us, because you can move your head and see what the object in the picture would look like if you were really standing in front of it.”
“So why do they need to use it in there?” The boy asked.
“Because we're going to run a very big simulation. Your uncle came up with a very interesting mathematical algorithm several years ago—right when you were born in fact—and when we plotted the graph for it we saw a lot in common with what we know of the Big Bang. The thing is, there are an infinite number of possible starting points for this algorithm, one of which, we think, might be the starting point of our own universe. We're going to try computing all of them, and we need a holographic memory to store them in. It'd be like looking at all the possible universes there ever could have been.”
The little boy's thoughtful silence passed like seconds to his father, but like footsteps to the boy, until he had mentally walked far enough to see the conclusion of his thoughts up ahead of him. “So it's like a big telescope?” He presently said.
“Yes, son, that it is.”