Special Pi Day Edition, by Carl Sagan

In his novel Contact, Carl Sagan described his character discovering a hidden message in the digits of pi:

"The Argus computer was so persistent and inventive in its attempts to contact Eleanor Arroway that it almost conveyed an urgent personal need to share the discovery.

The anomaly showed up most starkly in Base 11 arithmetic, where it could be written out entirely as zeroes and ones. Compared with what had been received from Vega, this could be at best a simple message, but its statistical significance was high.

The program reassembled the digits into a square raster, an equal number across and down. The final line was an uninterrupted file of zeros, left to right. The second line showed a single numeral one, exactly in the middle, with zeros to the borders, left and right. After a few more lines, an unmistakable arc had formed, composed of ones. The simple geometrical figure had been quickly constructed, line by line, self-reflexive, rich with promise. The last line of the figure emerged, all zeros except for a single centered one. The subsequent line would be zeros only, part of the frame.

Hiding in the alternating patterns of digits, deep inside the transcendental number, was a perfect circle, its form traced out by unities in a field of noughts.

The universe was made on purpose, the circle said. In whatever galaxy you happen to find yourself, you take the circumference of a circle, divide it by its diameter, measure closely enough, and uncover a miracle — another circle, drawn kilometers downstream of the decimal point. There would be richer messages farther in. It doesn’t matter what you look like, or what you‘re made of or where you came from. As long as you live in this universe, and have a modest talent for mathematics, sooner or later you’ll find it. It’s already here. It's inside everything. You don‘t have to leave your planet to find it. In the fabric of space and in the nature of matter, as in a great mark of art, there is, written small, the artist's signature. Standing over humans, gods, and demons, subsuming Caretaker: and Tunnel! Builders, there is an intelligence that antedates the universe.

The circle had closed.

She found what she had been searching for."

Zeno and the Fundamental Structure of the Universe

The universe is a weird place. I’m not the first person to notice this. A guy named Zeno also noticed it more than 2,000 years ago. Now, you wouldn't think there would have been a lot of people 2,000 years ago named Zeno, but it turns out there were. The Zeno I’m talking about is Zeno of Elea, a Greek philosopher.

We don’t know much about Zeno. Apparently, he’s mentioned, along with a buddy of his called Parmenides, in a book Plato wrote called Parmenides. That doesn’t help us much, because no copies of that book survive to our time. All we know about that book is what other people say about it when they refer to it in their own writing. People like Aristotle.

Anyway, Zeno and Parmenides had this idea about the universe, namely, that nothing changes. They didn’t just mean that you hit the exact same traffic every time you drive to Boston, or that the stories on TV all seem to be the same: they meant that NOTHING changes.

Ah, you might say, well, I took tuna for lunch today, and yesterday I took turkey, so that’s a change, which kind of refutes that whole theory, doesn’t it?

No, they’d say, that’s all an illusion. Really, nothing changes. And we can prove it, because the whole concept of change leads to logical contradictions. And if something leads to logical contradictions, it CAN’T be true.

Zeno is said to have concocted 40 of these demonstrations of how change leads to logical contradictions, which are called paradoxes. As I’ve said, we don’t have anything about him except from what others tell us, so we know of about 9 of these paradoxes, 5 of which are the most famous. These have to do with how the idea of motion—which is a change in position—leads to logical contradictions. Let me tell you about 2 of them.

The first one is called the Dichotomy. Suppose you want to walk up to a wall. To do this, you first have to move half the distance to the wall. Then you have to move half of the remaining distance. Then you have to move half of that remaining distance. Then half of that. And half of that. And so on, infinitely. In other words, before you can get to that wall, you have to do an infinite number of things first. And, of course, you will never finish that infinite number of things, so you will never get to the wall.

Ah, you might say, well, I’ve actually walked right up to walls with no trouble, so that’s not quite true, is it?

Exactly, Zeno would say. Logically, you cannot reach the wall. But you did actually reach the wall. That’s the logical contradiction I was talking about. Ergo, motion—and any other change—is actually impossible and is only an illusion.

These days, we can resolve paradoxes like this one using mathematics that was unknown in Zeno’s day. For example, we now know how to add up an infinite series like the one Zeno is presenting us with, namely ½ + ¼ + 1/8 + 1/16 + … You learn how to do this in calculus or pre-calculus. It turns out that this infinite series adds up to 1, so we don’t have to cover an infinite distance to get to the wall, just an ordinary finite distance.

Also, it doesn’t take us an infinite time to do it, either. We know that the time it takes to cover each of these distances is also an infinite series of the same kind, namely ½ + ¼ + 1/8 + 1/16 + … This also adds up to 1, so it doesn’t take us an infinite time to perform this infinite series of tasks, just an ordinary time.

Easy, right?

Well, the other paradox I want to tell you about is a little trickier. Spoiler alert: it turns out that the math for this one also explains it pretty easily. But the physics of it: that’s where the fun starts.

This other paradox is called “Achilles and the Tortoise”, and it has a little story attached to it. The Tortoise, slowest of all creatures, challenges Achilles, the mighty Greek hero of the Iliad, to a race. Achilles, a very speedy runner, agrees, and even graciously gives the Tortoise a generous head start. The race begins, and both racers start off, Achilles very fast and the Tortoise very slow. Achilles rapidly gets to where the Tortoise started from (call this location L₁). By the time Achilles gets there (L₁), however, the Tortoise has, of course, moved on a ways (to L₂). Achilles again gets to where the Tortoise was (L₂), but by the time he does so, the Tortoise has moved on a slight distance (to L₃). Again, Achilles catches up to where the Tortoise had been (L₃), but the Tortoise has again moved on (to L₄). This happens over and over. Indeed, instead of passing the Tortoise, Achilles can’t even catch up with him. Achilles stops to ponder this, as Greeks tend to do, and he takes so long pondering that the Tortoise crosses the finish line and wins the race.

What is going on here? We are surely all familiar with a faster racer overtaking a slower racer. It happens in every single race. But, the way Zeno tells it, that seems impossible. Every time fast Achilles gets to where the slow Tortoise was (L_n), the Tortoise has moved on a short distance (to L_n+1).

Now, as I said, this also has a fairly simple mathematical explanation. Let’s think about the times in this situation, rather than the distances. In the time it takes the Tortoise to move from location L_n to location L_n+1, Achilles is moving much further than the Tortoise is. Eventually, the distance that Achilles covers in this time is going to surpass the distance that the Tortoise covers, Achilles will pass him by, and win the race, just as we suspected all along.

Whew! Glad that’s straightened out!

But that was just the math part of it. What about the physics?

The physics of the situation is more complicated. As long as we can keep dividing the distance into tinier and tinier pieces to represent the Tortoise’s motion, there doesn’t seem to be a way for Achilles to catch up. Whenever he covers the tiny distance to where the Tortoise was (L_n), the Tortoise isn’t there anymore: he’s traveled the ever-tinier distance to L_n+1.

So, how can this paradox be resolved in physics?

One way is if that previous assumption isn’t really true: maybe we can’t really keep dividing the distance into tinier and tinier pieces. Maybe there’s some limit to how much you can divide distance.

Look at it this way. We’ve been thinking of this race as taking place on something like a wide open soccer field: no divisions, no limitations, nothing to suggest you CAN’T keep dividing the distance into tinier and tinier pieces.

But what if the race isn’t on a soccer field? What if it’s on a checkerboard, an enormous checkerboard, and Achilles and the Tortoise are checkers or chess pieces on the checkerboard? Achilles is a much faster kind of piece than the Tortoise is: he can move 10 squares to the Tortoise’s 1. But the thing about a checkerboard is this: the pieces can’t be just anywhere. They have to be in one square or another, not somewhere in between. In other words, on a checkerboard, you CAN’T keep dividing the distance into tinier and tinier pieces. Eventually, you come down to one single square, which cannot be divided any smaller.

How does running on a checkerboard affect the race between Achilles and the Tortoise? In this way: the Tortoise CAN’T keep moving tinier and tinier distances ahead. The Tortoise can either move 1 square ahead, as the tiniest move, or no squares at all. Either way, Achilles will eventually land on the square next to the Tortoise, and then pass the Tortoise in the next moment.

Think about what this means: this suggests that distance in the universe CAN’T be divided tinier and tinier. There is some lower limit to distance in the universe, some tiniest distance—like a square on a checkerboard—and there’s nothing smaller than that. The universe is granular, not continuous. You can be in square n or square n+1, but never in between the two.

Seems strange, doesn’t it?

And it gets worse.

We know, from Einstein’s theory of relativity, that time and distance are intimately related. In the equations, one can become the other. But if distance is granular, and there is a smallest distance like a square on a checkerboard, then time must be that way, too. There must be a smallest time, a shortest moment, and nothing smaller than that.

In this view of things, time isn’t a continuous flow of events. It’s more like a movie, which, despite the illusion of continuous flow, is actually many individual pictures running too fast for us to perceive the gaps in between. That’s what the universe is like.

Notice that this has nothing to do with quantum mechanics, which also suggests that space and time come in clumps, not continuously. This current discussion is deeper than quantum mechanics: you get to where we are so far using only classical physics, no quantum about it.

Well, so what? What does it matter if the universe is continuous or discontinuous? What’s the significance of this? The significance is that the phenomenon that most closely matches what we’ve been describing is: a computer game. The characters in a computer game don’t live in a world of continuous space and time either: their world is as granular as a pixel and as discontinuous as whatever small time-step the programmer has chosen. The illusion of computer games is very convincing: characters and places and objects and motion look continuous. But we know they aren’t.

Meaning what? Meaning this: our universe could actually be a constructed thing. Perhaps not a computer simulation, but something like that. Perhaps not “The Matrix”, but something like that. A place that was deliberately built. A place that exhibits this checkerboard kind of behavior.

“Could be.” Remember that, when we began our explanation of the “Achilles and the Tortoise” paradox, we said that this was one way to resolve the paradox. There are definitely others. But this is an interesting possibility, isn’t it?

That Zeno was pretty smart.

The Coincidence of the Moon

Just a quick one, today. And not even my own words.

Isaac Asimov, author of more than 500 books, and countless essays on science, had this to say about the Moon and its perfect fit over the Sun during a solar eclipse:

"What makes a total solar eclipse so remarkable is the sheer astronomical accident that the Moon fits so snugly over the Sun. The Moon is just large enough to cover the Sun completely (at times) so that a temporary night falls and the stars spring out. And it is just small enough so that during the Sun's obscuration, the corona, especially the brighter parts near the body of the Sun, is completely visible.

“The apparent size of the Sun and Moon depends upon both their actual size and their distance from us. The diameter of the Moon is 2160 miles while that of the Sun is 864,000 miles. The ratio of the diameter of the Sun to that of the Moon is 864,000/2160 or 400. In other words, if both were at the same distance from us, the Sun would appear to be 400 times as broad as the Moon.

“However, the Sun is farther away from us than the Moon is, and therefore appears smaller for its size than the Moon does. At great distances, such as those which characterize the Moon and the Sun, doubling the distance halves the apparent diameter. Remembering that, consider that the average distance of the Moon from us is 238,000 miles while that of the Sun is 93,000,000 miles. The ratio of the distance of the Sun to that of the Moon is 93,000,000/238,000 or 390. The Sun's apparent diameter is cut down in proportion.

“In other words, the two effects just about cancel. The Sun's greater distance makes up for its greater size and the result is that the Moon and the Sun appear to be equal in size. The apparent angular diameter of the Sun averages 32 minutes of arc, while that of the Moon averages 31 minutes of arc.

“These are average values because both Moon and Earth possess elliptical orbits. The Moon is closer to the Earth (and therefore appears larger) at some times than at others, while the Earth is closer to the Sun (which therefore appears larger) at some times than at others. This variation in apparent diameter is only 3 per cent for the Sun and about 5 per cent for the Moon, so that it goes unnoticed by the casual observer.

“There is no astronomical reason why Moon and Sun should fit so well. It is the sheerest of coincidence, and only the Earth among all the planets is blessed in this fashion. Indeed, if it is true, as astronomers suspect, that the Moon's distance from the Earth is gradually increasing as a result of tidal friction, then this excellent fit even here on Earth is only true of our own geologic era. The Moon was too large for an ideal total eclipse in the far past and will be too small for any total eclipse at all in the far future."

What’s significant about this is that it is by the observation of—and the explanation of—solar eclipses and similar phenomena that we came to create the science of astronomy, which led to the science of physics, science in general, engineering, and our modern civilization. Without that coincidence of the size of the moon and the sun, none of that would have happened as it did.