Breaking Pi

I love working with others to discover new mathematics, but there’s a kind of research I’d love even more: helping decode a Message from an extraterrestrial civilization. The chance to do that would make me drop all my mathematical projects — though in a way it wouldn’t, since decoding the Message would almost certainly involve a lot of math.

As a teenager I was captivated by a 1973 book called Communication with Extraterrestrial Intelligence. It was edited by a not-yet-world-famous astronomer named Carl Sagan who was interested both in sending messages to the stars and in seeking messages from the stars to us. He went on to host the incredibly popular TV program “Cosmos” and to write several best-selling books, including the novel Contact which I’ll have a lot to say about later.

The reason I’m writing this particular essay this month is because almost exactly two centuries ago, the mathematician and astronomer Carl Friedrich Gauss proposed sending a message to the moon. (Gauss’ ideas about life on other worlds had a respectable pedigree in European thought; see the excellent articles by Aldersey-Williams and Dillard listed in the References.) Gauss had invented a kind of signaling device he called the heliotrope, and on March 25, 1822, he wrote a letter to the astronomer Heinrich Olbers, saying “With 100 separate mirrors, each of 16 square feet, used conjointly, one would be able to send good heliotrope-light to the moon. … This would be a discovery even greater than that of America, if we could get in touch with our neighbors on the moon.”

Gauss (or perhaps a contemporary of his) made a related proposal to install in the wheat fields of Siberia an enormous diagram of a 3-4-5 right triangle, embellished with extra lines in the manner of the proof of the Pythagorean theorem given in Euclid’s Elements. A big enough diagram would be visible from the moon and would prove to the moon-dwellers that there is intelligent life on Earth. (See Endnote #1.) It’s like aliens announcing their presence to us using crop-circles, in reverse. But speaking of circles, it’s worth noting that a giant circle would not have served the purpose of announcing the existence of intelligent Earthlings since many natural processes give rise to circles (such as the impacts that created circular craters on the Moon). On the other hand, Nature has shown no interest in proving theorems, and no physical process has ever been discovered that creates diagrams like Euclid’s.

Nowadays we know enough about other planets in our solar system and their moons to know that they are all inhospitable to life as we know it. If we seek cosmic company, we must look to solar systems light-years away from our own. At interstellar distances, even planet-sized pictures are illegible. So we must give up on the 19th century idea of communicating through pictures.

Or must we? For nearly a century, humans have contrived to convey pictures across great distances through the magic of television, which divides an image up into pixels and transmits those pixels through the air via radio waves. There’s no technical obstacle to our sending television signals into outer space. Indeed, humankind began doing so, unthinkingly, at the dawn of the TV age. So if we wanted to, we could start broadcasting programs specifically designed to make a good impression on our cosmic neighbors (to compensate for the fact that we’ve already sent them all 98 episodes of “Gilligan’s Island”).

Of course, the aliens might be very different from us. They might have four arms, or two heads. Hard as it is to imagine, they might not even watch television. The history of science fiction shows a steady broadening of our conception of what a sentient being could be: a hive-mind, a world-mind, a super-intelligent shade of the color blue… you name it. There could be minds that don’t use language or understand the world through pictures, and we might have a lot to learn from those sorts of minds. But if we humans want to be less isolated in the universe, what better aliens for us to reach out to, in our first attempts at interstellar communication, than ones who resemble us? You have to start somewhere.

CONTACT: A DEFINITIVELY INACCURATE SUMMARY

I’m now going to summarize the plot of Sagan’s novel. Sort of. You’ll want to read my summary even if you already read the book because I take some liberties with what Sagan wrote, and the main theme of my essay hinges on those deliberate discrepancies. And if you only saw the movie, definitely read my summary, because the book and movie differ in some key respects.

In the novel “Contact”, humanity receives a signal from Vega, a star twenty-six light years away. The signal seems to be just the base-two representations of the first 261 primes (see my essay “The Clatter of the Primes” from last month), and since no known physical process generates base-two representations of primes, the message seems to indicate the presence of a mind.

At first it appears that the message from the stars might be nothing more than a Cosmigram saying “I like primes. Do you like primes too? Send proofs.” (See Endnote #2.) But the message turns out to be much more than that; it’s composed of several layers of increasing complexity. Beneath the All-Primes,-All-The-Time layer is a faithful echo of Earth’s first TV broadcast, and beneath that is an instruction manual for building a machine, written in a language that we’re able to decipher because it’s heavily based on mathematics.

The Machine turns out to be a single-use round-trip transportation system that takes five lucky humans to an amazing chocolate factory in the middle of our galaxy, with other stops along the way, by way of wormholes (aka Tunnels) through spacetime. At the climax of their tour, the five humans encounter mind-reading, shape-shifting aliens (or maybe just aliens who’ve developed super-advanced technology for messing with people’s heads). The aliens describe themselves as mere Caretakers who hope that someday the Tunnel Makers will return and explain the meaning of Life, the Universe, and Everything, and while they’re at it, resolve certain mathematical mysteries that have them stumped. Even though the Caretakers are much smarter than us, the universe still fills them with feelings of wonder and awe, or to use a fancier phrase, A Sense Of The Numinous. As one of the Caretakers explains to Dr. Arroway, the novel’s protagonist:

“I don’t say this is it exactly, but it will give you a flavor of our numinous. It concerns the number one-third: the ratio of the counting number one to the counting number three. You know it well, of course; it is a non-decimal fraction, and you also know that you can never come to the end of its decimal expansion. Our scientists have found patterns in the digits of one-third in various bases, and we think it contains a message from the Creator of the Universe.”

At the end of the book, Arroway, having returned to Earth, pulls up the computer code that helped decipher the message from Vega and repurposes it to compute the base-eleven expansion of 1/3 and look for patterns. And sure enough, billions of digits out, she finds a string of 0’s and 1’s whose length is exactly one million. Turning that string into a thousand-by-thousand square of black and white pixels, she discovers a picture of a circle divided into three equal pieces, in perfect concordance with the concept of one-divided-by-three. Arroway has broken the code in the number one-third and has glimpsed the Unity underlying Reality; her long journey of discovery is at an end. Or is it only beginning?

AN APOLOGY, SORT OF

I’m sorry; I lied in several places. I did warn you about that, didn’t I? The first lie I told was about that chocolate factory in the middle of the galaxy (if you missed it, you were skimming way too quickly). The second lie was my rendering of the conversation between Arroway and the alien. Here’s what the alien actually says (see Endnote #3):

“I don’t say this is it exactly, but it will give you a flavor of our numinous. It concerns pi, the ratio of the circumference of a circle to its diameter. You know it well, of course, and you also know that you can never come to the end of pi.”

Ah, so the aliens are entranced by pi! That makes so much more sense than having them go all gooey over one-third!

Don’t get me wrong, one-third is a pretty interesting number, but it lacks the cachet of pi. And not just because of pi’s Aegean mystique. Although both 1/3 and pi have decimal expansions that go on forever, they go on forever in entirely different ways. The digits of pi are varied and enigmatic, whereas the digits of the fraction 1/3 are dully predictable: it’s just 3’s all the way out. Other fractions are more interesting, like 22/7 (not far from pi on the number-line); its decimal expansion promisingly begins 3.14 but it too repeats eventually: 22/7 equals 3.142857142857142857… What’s more, 1/3 isn’t repetitive merely in base ten; its expansion in any integer base must repeat.

On the other hand, it’s known that pi is irrational, which means that the digits never stop offering novelty of some sort or other. In fact, mathematicians have found no discernible patterns in the digits of pi, and it’s believed that every sequence of digits will eventually turn up if you compute pi to enough decimal places. (See Endnote #4.)

My third lie (an elaboration of my second lie) concerns what happens when Arroway does her own computer experiments. Here is what she finds in the book’s (real) concluding paragraph when her computer computes enough digits of pi in base eleven:

“Hiding in the alternating patterns of digits, deep inside the transcendental number, is a perfect circle, its form traced out by unities in a field of naughts. 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 from the decimal point. There would be richer messages farther in. 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 work of art, there is, written small, the artist’s signature. Standing over humans, gods, and demons, subsuming Caretakers and Tunnel builders, there is an intelligence that antedates the universe.“

Ah yes, “transcendental”: what a wonderful math-word! It gives you goosebumps even if you don’t what it means. Irrational numbers like the square root of two already stir awe, but transcendental numbers bring that awe to a whole new level. A number like the square root of two, when expressed as a decimal, has the same enigmatic aspect as pi, but the square root of two can be described by a simple algebraic equation: x2 = 2. Pi, on the other hand, satisfies no such equation; it transcends mere algebra. How much more fitting an object of veneration it is than a pedestrian number like one-third! How worthy a vessel it is for a message from an Artist who transcends mere Tunnel builders, who in turn transcend mere Caretakers, who in turn transcend merest us! And what better way to show one’s power than to bend an entire universe?

A famous cartoon by Randall Munroe; taken from https://xkcd.com/10/. Permission pending.

But not so fast. There’s something not quite right with Sagan’s conceit, and the best way to explain what’s wrong is to go back to my rewrite.

THE ARITHMETIZATION OF PI

My version of the plot (the version in which pi gets replaced by 1/3) is dramatically unsatisfying, but there’s something worse about it from a mathematical point of view. After all, when you apply long division to divide 1 by 3, you keep computing 10 ÷ 3 = 3r1 (that is, three goes into ten three times, leaving a remainder of 1), and that 1 feeds back into the process by becoming a 10, over and over, in a vicious cycle. It’s hard to imagine a universe in which the calculation process burps and suddenly starts producing a string of 0’s and 1’s before going back to producing 3’s. The scenario becomes even less plausible when you stop to consider how many different mechanisms there are for computing in our world. What alternative laws of physics would cause all the different machines that might compute the digits of 1/3 in all kinds of different ways to burp in unison?

If you think (as I do) that the decimal expansion of 1/3 doesn’t depend on what universe you happen to be in, then I ask you: How are the digits of pi different?

You might reply “1/3 is an arithmetic quantity, whereas pi is a geometric quantity derived by measuring circles, and we can certainly imagine a Creator who warps those circles.”

If you say that, then you’re in good company; most mathematicians up until the Renaissance would have tended to view pi as a purely geometrical construct. And Sagan seems to think so too (see Endnote #5) when he says you can approximate pi if you “measure closely enough”.

But Arroway’s computer doesn’t measure circles. It’s a digital computer, doing digital calculations. And in fact, if our current theories of the universe are correct, there is no way, even in principle, to get more than a few dozen digits of pi by doing measurements; between the Scylla of general relativity and the Charybdis of quantum effects, you can’t just build a big circle and measure it ultra-accurately. Gravitational warping of space-time or the uncertainty principle or both will thwart your efforts.

What exactly is Arroway’s computer doing? Probably something like computing a really long decimal approximation to 16 arctan 1/5 and subtracting an equally long decimal approximation to 4 arctan 1/239 from it, where arctan x can be approximated by taking partial sums of the infinite series x – (1/3) x3 + (1/5) x5 – (1/7) x7 + … This series was first discovered by the Indian mathematician Madhava of Sangamagrama in the late fourteenth century, though it is often attributed to the European mathematician James Gregory who rediscovered it two and a half centuries later. The sum appears to have nothing to do with circles. Geometry has been banished; instead, we have an infinite arithmetic expression. The switcheroo can be justified by calculus if we assume that Euclid’s axioms are correct and that arc length satisfies an additional axiom due to Archimedes. (See Endnote #6.)

If you have trouble imagining a universe in which 1/3 has a different decimal expansion than the one you were taught in school, then you should likewise have trouble imagining a universe in which arctan 1/5 or arctan 1/239 has a different decimal expansion, since the same sort of mechanical processes are involved; there’s a difference in degree of complexity, but not a difference in kind. So you should also have trouble imagining a universe in which 16 arctan 1/5 minus 4 arctan 1/239 has a different value. But that’s the universe of Contact.

The arctan formula for pi that I gave above is far from unique; at this point in mathematical history we know thousands of other formulas like it. Although the number pi originated in geometry, it has been fully liberated from geometry and from the peculiarities of the specific kind of geometry that we find in the physical universe. Or maybe some would say that pi has been imprisoned in calculus! Either way, it’s been transformed, and Arroway seems unaware of this transformation when she muses “If there was content inside a transcendental number, it could only have been built into the geometry of the universe from the beginning.”

NO ESCAPE

But couldn’t there be universes built on fundamentally different geometries than ours? Indeed there could, and I wrote about some of these geometries in my essay “Three-point-one cheers for pi”. The problem with all of the geometries I know about, in terms of grounding the thrilling conclusion of Sagan’s novel in some mathematical plausibility, is that they don’t really take you away from the mathematicians’ pi; you’re still stuck in pi’s gravity-well. Sure, you can imagine a Creator who can bend space everywhere, in the fashion envisioned mathematically by Gauss and Riemann and physically by Einstein. But these sorts of geometries are still locally Euclidean, which means that as you probe at smaller and smaller scales, you recover Euclidean geometry and its pi. Indeed, in these geometries, the ratio of a circle’s circumference to the circle’s diameter varies as the circle grows or shrinks, so you could say that in those spaces, pi ceases to exist as a constant. Bending space breaks pi.

A more radical adjustment of geometry involves bending the exponent 2 in the Pythagorean theorem. Mathematicians who study such spaces call the exponent p and call the spaces Lp spaces. Just because the space we’re living in happens to be an L2 space (putting aside Einstein’s corrections) doesn’t mean we can’t imagine other possibilities! But there’s no escaping pi. If you were a scientist living in an Lp space, there’d be nothing to stop you from considering values of p other than the one governing your universe, and you’d be naturally led to ask “Which value of p makes the circumference-to-diameter ratio as small as possible?” You would discover that the answer is p=2, which would inexorably lead you to discover the pi of calculus. For that matter, you can build a computer in the L universe using John Conway’s Game-of-Life rules and program it to compute “the pi of calculus”; even though “geometrical pi” in the L universe is 4, the machine will spit out 3.1415…, not 4.0000…

If you want to imagine designer universes bearing the signatures of the artists who made them, by all means do so! Math is about imagination, after all. But don’t expect anything as humdrum as circles in Riemannian manifolds with variable curvature or Lp spaces to make this numinous vision concrete. Something weirder will be required. (Anyone know of any math like that? I’m willing to admit that such a mathematics might exist, but there’s a difference between imagining that something could be true and imagining how it could be true.)

ALPHA AND OMEGA

I can think of two other constants suited to the kind of numinous treatment Sagan describes, and as it happens, they are denoted by the Greek letters at opposite ends of the alphabet.

The first is the dimensionless physical quantity alpha known as the fine structure constant. It’s roughly 1/137. Physicists know of no reason why it couldn’t have a different value than it does, though if it were much different we wouldn’t be here pondering it, since a universe with a different value of alpha wouldn’t support our kind of life or even our kind of chemistry. One can imagine a God with the power to twirl a Dial and make alpha anything She wants, and who chooses a specific value of alpha that pleases Her, perhaps one that conveys some sort of message of love to the inhabitants of the universe She created. Unfortunately, we can’t currently measure alpha to more than about a dozen digits, and there are likely to be fundamental limits to how closely anyone can ever determine it, so if the value of alpha contains a message from our Creator, it’s a very short one. On the other hand, if we ever find a formula for alpha, so that we can determine more of its digits through computation than we could by measurement, then alpha will become imprisoned in calculus just like the modern mathematician’s pi. Alpha will become a mathematical constant, not a physical constant; it’ll be the answer to a specific mathematical question, and not something whose value we might imagine being adjusted by a Dial.

The mathematical constant pi does not transcend human understanding, but I can tell you about another number that does, a truly transcendent number, a number whose merest operational parameters pi is not worthy to calculate. I speak of the number Omega. (See Endnote #7.)

We humans have many algorithms for approximating pi as closely as we wish, but we do not have, and indeed cannot have, an algorithm that approximates Omega as closely as we wish. That’s because Omega was defined by Gregory Chaitin in a fashion that’s intimately tied up with the unsolvability of the halting problem; this result, proved by Alan Turing, is a near-relative of Kurt Gödel‘s incompleteness theorem. To have an algorithm that computes Omega to any specified accuracy, we’d have to have a way to solve every mathematical problem that ever has been or ever could be formulated, and Gödel’s theorem bars our entrance to that paradise of omniscience. The same goes for any alien species, no matter how intelligent. Each will ultimately be defeated by the riddles the Omega-sphinx asks travelers, for she knows infinitely many riddles, each harder than the one before; no finite being can pass her.

Have I whetted your appetite for more information on Omega? If so, good! I think that in some places in his book Sagan makes the mistake of explaining too much, so let me maximize your sense of wonder over Omega by explaining too little.

If I were writing a book like Sagan’s, I’d do something different near the end. I’d have the alien say:

“We’ve received a different signal, not using any kind of radiation you humans know about, emanating from everywhere at once or from beyond the universe (which are two different ways of saying the same thing). It’s an infinite string of 0s and 1s, and it seems to be the base-two representation of the number you humans call Omega. That is, it seems to reveal, for each finite computer program, whether that computer program ever halts. We can’t verify that that’s what it’s truly doing, but it sure seems like it. For the last million years we’ve been stuck at verifying the 97th bit; we’re hoping to find new approaches to understand why a particular program that seems to run forever actually does. It’s related to a simple-sounding yet fiendishly difficult problem that your species solved in 1994, so we’re hoping that someday you’ll be able to help us with it! Anyway, if the Message really is the binary representation of Omega, then it can’t have been created by any finite mind in our universe. It must be the product of some sort of infinite mind outside of Time.”

I agree that Omega is a lot more arcane than pi. But I think it’s a lot more mind-blowing. In fact, I’d call it pretty effing ineffable.

IN THE MEANTIME

Mathematicians have mined a ton of numinosity from the true mathematics of pi, even without finding patterns in its decimal digits. For instance, consider the way pi comes up in statistics in the definition of the Gaussian distribution. Eugene Wigner, in his famous 1960 essay The Unreasonable Effectiveness of Mathematics in the Natural Sciences wrote:

“There is a story about two friends, who were classmates in high school, talking about their jobs. One of them became a statistician and was working on population trends. He showed a reprint to his former classmate. The reprint started, as usual, with the Gaussian distribution and the statistician explained to his former classmate the meaning of the symbols for the actual population, for the average population, and so on. His classmate was a bit incredulous and was not quite sure whether the statistician was pulling his leg. “How can you know that?” was his query. “And what is this symbol here?” “Oh,” said the statistician, “this is pi.” “What is that?” “The ratio of the circumference of the circle to its diameter.” “Well, now you are pushing your joke too far,” said the classmate, “surely the population has nothing to do with the circumference of the circle.””

And it’s not just in statistics that pi makes an unexpected guest appearance; these cameos occur all over the place in math. You might say that pi is a Tunnel through the hyperspatial mathematical landscape, serving as a magical bridge between seemingly far-flung domains in inner space. It and other bridges across the land of pure imagination called Mathematics are good enough for me, until the aliens contact us.

When the aliens contact us or vice versa, I’ll be eager to learn what alien math is like. How different might it be? People often ask whether math is created or discovered, and I’m very sympathetic to the latter view because, despite years of strenuous engagement with the terrain of mathematics, I’ve developed no ability to bend it. Some people think that the solidity of our planet’s mathematical consensus is evidence that mathematical reality is in some deep way objective. But this sociological evidence of the objectivity of mathematics is tainted by the fact that humans have been sharing mathematical ideas from culture to culture for thousands of years, and some of the uniformity of our perceptions could be a consequence of our interactions.

But if aliens had math that looked like ours, developed entirely separately from our mathematics — now that would be very strong evidence that math is built into the fabric of our universe, and maybe built into the fabric of whatever logical infrastructure permits universes to exist in the first place.

There’s a video series on the question of whether math is discovered or invented. I haven’t watched it, and I don’t plan to. For one thing, if you look at the thumbnails of the videos, you’ll notice that the speakers aren’t just mostly old white males; they’re all Earthlings! I think I’ll want to hear from some extraterrestrials before I form an opinion on the question. But, just as importantly, I think the question is premature. Do we even know what math is yet? It’s still early days. Our species’ journey of mathematical discovery is only just beginning.

REFERENCES

Hugh Aldersey-Williams, “The Uncertain Heavens: Christian Huygens’ Ideas of Extraterrestrial Life” at publicdomainreview.org.

George Dillard, “A Golden Age — of Belief in Extraterrestrials” at historyofyesterday.com.

Martin Gardner, “Chaitin’s Omega,” chapter 21 in Fractal Music, Hypercards and More: Mathematical Recreations from SCIENTIFIC AMERICAN Magazine.

Carl Sagan (ed.), Communication with Extraterrestrial Intelligence, 1973.

Carl Sagan, Contact, 1985.

ENDNOTES

#1. There’s a poetic aptness to using the Pythagorean theorem as a way of contacting hypothetical Moon-dwellers, since the Pythagorean philosopher Philolaus of Croton believed the moon was inhabited. Then again, the Pythagorean Theorem was known in many parts of the world long before Pythagoras was born, and it’s not even clear whether the proof given by Euclid was discovered by the Pythagorean school.

#2. On the other hand, the plaque Sagan sent on Voyager, featuring nude humans waving hello, could be construed as a solicitation of a much less innocent kind.

#3. In reply, Arroway says “But this is just a metaphor, right?” but gets no direct answer. Also, the physicist Eda, one of the other four travelers, is told the same story, but about a class of transcendental numbers Arroway hadn’t heard of. Sagan chooses to be a bit vague here, as well he should, since nothing dispells numinosity more than revealing too much.

#4. Pi reminds me a bit of Borges’ “Library of Babel” which contains every possible book. If you convert Sagan’s novel into 0’s and 1’s, then that string of 0’s and 1’s appears somewhere in the decimal expansion of pi, if our current guesses about pi are correct. Of course, pi would also contain far more numerous erroneous versions of the novel, and many, many more copies of my bogus summary if only because it’s much much shorter. Pi also reminds me of the “Infinite Monkey Theorem”.

#5. I like to think that Sagan the scientist knew everything I say in this essay, but that Sagan the novelist pretended not to know it for the sake of crafting a more accessible and appealing story.

#6. One version of Archimedes’ axiom is that if you have convex regions A and B, with A containing B, then the perimeter of A exceeds the perimeter of B. For instance, if A is a square of side-length 2, and B is the disk of diameter 2 inscribed in A, and C is the regular hexagon of side-length 1 inscribed in B, then the perimeters of A, B, and C are 8, 2pi, and 6, respectively, proving that pi lies between 3 and 4. So the first digit of pi, at least, has some sort of intuitive geometric meaning, even if the later digits seem meaningless to our puny human minds.

Posted by JJacquelin at https://math.stackexchange.com/questions/740557/how-do-i-prove-that-3-pi4

#7. In fact, there is not a single Omega number; rather there are infinitely many, one for each prefix-free universal computable function F. But we know of some fairly simple F ‘s, so it’s common to informally assume we’ve agreed on one of them without specifying which. This doesn’t bother anyone, though it means you’ll have some trouble figuring out what day of the year to designate as Omega Day.

2 thoughts on “Breaking Pi

  1. Matt Lehman

    I think many of my students encounter alien mathematics when I lecture, and I encounter alien mathematics when I grade their work. If we mathematicians ever encounter alien mathematics, I would like to know which ratio they use for the circle constant.

    The video series I’m familiar regarding whether math is invented of discovered, is a three-part one titled BBC Magic Numbers Mysterious World of Maths. It’s hosted by Hannah Fry, who’s 2/3s of an “old white male”. She interviews mathematicians and scientists, some of whom are none of the above. However, they are all Earthlings,… probably. Here’s a link to the first episode, if I may.

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  2. jamespropp Post author

    I was too hasty in describing the videos in question (which may overlap with the ones you saw): along with Roger Penrose, George Lakoff, Steven Weinberg, Edward Frenkel and Ed Witten we can see Neil Degrasse Tyson, Po-Shen Loh, and others who are not white or not old (though I still don’t see any women). I shouldn’t have been so prejudiced against the series. Did any of you watch any of the videos, and did you learn anything you didn’t already know?

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