The first part of this series surveyed previous attempts at contacting aliens. The second part proposed a base-neutral notation system for encoding messages to aliens.In the previous post in this series, we used continued fractions to represent any real number as a unique sequence of whole numbers. Such sequences are ideal for sending real numbers to aliens via radio signals. Now we have to choose which numbers to send them, out of an infinite choice of candidates.

I propose sending them two numbers. The first of these is the topic of this post: Khinchin’s Constant, KK equals 2.685452001065306445… in base ten, or [2,1,2,5,1,1,2,1,1,3,10,…] as as the sequence corresponding to its continued fraction..

What is so special about K? It is one of the very few numbers capable of giving the driest of mathematical texts exclamatory hiccups. Mathworld prefaces its introduction to K with “Amazingly, …“. The bible of mathematical constants, the stolidly named Mathematical Constants, irrupts with “Here occurs one of the most astonishing facts in mathematics.”

And yet K is virtually unknown to a wider audience. Pi, e, i, the golden mean and the square root of two are all well ensconced in high school maths curricula, though not K.

To explain why I think K would make a excellent number to send to aliens, it will help to first derive it. This is easy to do, because we’ve already done all the hard work exploring continued fractions in the previous post. Pick any random real number — you know, one that in base ten would look something like 14.7631809156… with additional random digits continuing off to the right ad nauseam. Then, represent this number as a continued fraction to find the unique sequence of whole numbers that corresponds to it, just as we’ve done before.

Now consider the first n terms of this sequence — that’s n whole numbers, starting with 14,1,3,4,… in our example. To find the geometric mean of this group of n numbers, we multiply them together and then take the nth rootTo find the arithmetic mean of n numbers, you add them up and divide the sum by n. To find the geometric mean of n numbers, you multiply them and then take the nth root.. What Aleksandr Yakovlevich Khinchin proved in 1934 is that as you make n larger and larger, the geometric mean of the first n terms of this sequence converges on our constant K, 2.685452001065306445…, regardless of the number we picked.

That, to mathematicians, was an utterly unexpected result. There are two reasons why, I think. First, most of the numbers we use every day correspond to sequences whose geometric means evidently do not converge on K. All rational numbers correspond to finite sequences, and therefore cannot possibly lead to KThe rational number 1.23, for example, corresponds exactly to [1,4,2,1,6,1]. (To be tedious but thorough: What if the number lies between 0 and 1? Divide the number into 1 first to get the same sequence without a zero as the first term. This works because there is a unique, one-to-one correspondence between a number and that number divided into 1. Or else just ignore the zero and start from the second term.). Nor can any irrational number that is not transcendental, because its sequence always obeys a pattern: The Golden Mean, for example, corresponds to the sequence [1,1,1,1,1,1,1,1,1,…], whose geometric mean is obviously 1. The sequence corresponding to the square root of 8, [2,1,4,1,4,1,4,1,4,…] converges on 2, not K.

Second is that by inspection, we can easily construct an infinite number of infinite sequences (all of them having a unique correspondence to a real number) that clearly do not converge on K. [1,2,3,4,5,6,7,8,…]’s geometric mean will grow to infinity. [100,100,100,100,…]’s geometric mean is 100. [101,101,101,101,…]’s geometric mean is 101, and so on.

So how can Khinchin’s proof hold? It can because there are innumerably more real numbers that do obey Khinchin than do not. And if I asked you to choose one real number at random (as I did), the probability that you’d pick one that does not obey Khinchin is zero. Zilch. GuaranteedIf you really want to know more about why that is so, the answer involves countable vs. uncountable sets and Georg Cantor..

A couple of things about K, then:

Although it’s been proven that virtually every number obeys K, nobody has ever proved that a specific number obeys. Pi very much looks like it doesThe convergence of pi to K, foisted from Mathworld (see link).
, because we’ve crunched enough numbers and had a look, but we don’t know for sure. A few other useful mathematical constants seem to as well. Still, most of our workaday numbers do not converge on K, no doubt for the same reason that these numbers caught our eye in the first place — they concern themselves with ordered systems.

All this is very interesting. But the real clincher as to why we should beam K to aliens — the thing about this number that takes it to a whole new level, as it were, is this: It would appear that Khinchin’s Constant obeys itself. The geometric mean of the first n terms of K, [2,1,2,5,1,1,2,1,1,3,10,…], also converges on K, for as far as we’ve looked. Khinchin’s Constant appears to be autological.

Take a look for yourself. Convinced?

What does this imply? It implies that the sequence of whole numbers that describes K, [2,1,2,5,1,1,2,1,1,3,10,…], is simultaneously described by KHofstadter illustrates the notion with this Escher print:

He goes on to posit that such systems are the basic building blocks for self-awareness in far more complex systems, such as ourselves.. It implies that K is infused with that essential quality of self-loopiness, of continuous folding back on itself, that Douglas Hofstadter identifies in Gödel, Escher, Bach as being at the core of all self-referential systems.

What K embodies, then, is the seemingly paradoxical ability to describe the properties of the system that produces it. It’s akin to what happens when a mind contemplates the laws of physics that govern the mind. Our aliens, which for our purposes here are really just stand-ins for complex self-aware systems, would not escape noticing this analogy — and not just because we’re giving them a massive hint by sending K as [2,1,2,5,1,1,2,1,1,3,10,…].

By including K in our message, then, we are broadcasting that we consider self-referential systems to be special — a prerequisite for the kind of complexity that underpins self-awareness, which in turn allows for the understanding of messages from outer space.

The second number I propose to send will take a completely different tack.

The Fahrenheit scale only preceded the Celsius scale by a few decades, and both were invented in the early 1700s.

Daniel Gabriel Fahrenheit, a German physicist, originally planned to place the 0 degree point at the temperature at which an equal water/salt mixture froze, the 30 degree point at where water froze and the 90 degree point at the temperature of the human body.

Unfortunately, he got those measurements wrong, and the freezing point of water was later revised to 32 degrees and the temperature of the human body to 98.6 degrees.

Swedish astronomer Anders Celsius‘s temperature scale originally placed the 0 degree point where water boiled, and the 100 degree point where water froze. This seems a bit absurd given today’s conventions, but there is a logic to it: In quotidian use we almost never deal with temperatures hotter than the boiling point of water, though we do deal with temperatures colder than the freezing point of water; it would make sense, therefore, to use the boiling point as a kind of natural origin, and measure out from there.

And if you’re a Swede inventing a new temperature scale, the idea of measuring a quantity of cold rather than a quantity of heat is not all that preposterous, certainly not if you’ve just recently been subjected to a Swedish winter.

Had Celsius’s system not been tampered with, 76 (orthodox) degrees celsius would today correspond to 76 degrees fahrenheit, and we’d all assign that temperature some magical “ideal” quality, seeing how the value would be naturally endorsed by both scales. CNN’s weather forecasts would use a special graphic to highlight 76-degree days.

Instead, in real life, both scales “endorse” -40 degrees. What happened? Sweden’s most famous scientist ever, Carl Linnaeus, was heavily into plants, and since these tend to die around where water freezes, he felt this point was a more natural zero point. So he switched the temperature points around soon after Celsius died, in 1744.

This proved to be a good idea, in the long run. The concept of a quantity of heat would prove far more useful, scientifically, than a quantity of cold, as it would later lead Lord Kelvin to the Second Law of Thermodynamics, the concept of absolute zero, and the necessity of a scale that used it as a zero point. Whence kelvins.

If you want to, you can read Part I: Prelude first.Mathematical bases sure are a convenient shorthand when adding, multiplying and subtracting, and their invention was necessary for the kind of recordkeeping that allowed ancient civilizations to blossom — amphoras of wine shipped, monies owed to the emperor, sacrifices made to the gods — let’s hear it for property rights and central planning.

But bases are arbitrary. The Babylonians used base 60. The Mayans used base 20. The Greeks and others used base 10, while the geeks have adopted binary notation, base 2, as their own.

All these number systems have quirks, too. Modern systems use place to determine the value of a number (i.e the 5 in 350 stands for 5 groups of 10); the Romans used different numeric symbols (XXX stood for three tens, CCC stood for three hundreds). Without prior access to such rules, a stranger would need a good number of examples from which to glean patterns. It wasn’t until the 1820s that we deciphered the Mayan number system, for example, and it took all the lateral thinking the splendidly named Constantine Samuel Rafinesque-Schmaltz could muster.

Number systems have another weakness: Division. They have no neat way of precisely representing all rational numbers — numbers that can be represented by a fraction, such as 6/2, 7683/99746 or 1/7. All bases can represent some rational numbers precisely: In base 10, for example, 6/2 is precisely 3 and 1/4 is precisely 0.25. But 1/7 is 0.142857142857… the 142857 repeating forever, the actual number never quite managing to get nailed. In base 7, meanwhile, the number equivalent to 1/7 in base 10 can be precisely represented as 0.1, but 1/4 becomes 0.151515… ad infinitum.

Base 2 merits some special consideration. The received wisdom is that binary notation is somehow more natural, because it is the simplest system, requiring the fewest symbols (2) without having to resort to pebbles or prison wall notches as a counting tool. Base 2 is special, it is argued or assumed, because its ones and zeros (“bits”) lend themself perfectly to representing the trues and falses of Boolean logic, which in turn can be physically embodied in the presence (“on”) or absence (“off”) of electromagnetic charges in our computers’ transistors and circuits.

That amounts to attaching too much importance to historical accident, however. Alternative logic systems do exist. In ternary logic, for example, base 3’s zeros, ones and twos map to “unknown”, “true” and “false”. And base 3 is more efficient than base 2 at representing numbers, as this great article points outHow to measure this efficiency? Take a number, any number. Depending on your choice of base, you will require different amounts of digits to write out the number. For example, the number 66 requires two digits in base 10; the base 2 equivalent of that number, 1000010, requires seven digits; the base 3 equivalent, 2110, requires four digits. Clearly, the higher the base, the fewer digits you need. Using a higher base means having to differentiate between more number symbols, which uses up bandwidth or computing resources — though it also means needing fewer digits, which saves resources. You can measure the total resources needed by multiplying the base a number is written in by its length in digits. In our case, 66 requires 10 x 2 = 20 “units” in base 10, 2 x 7 = 14 units in base 2, and 3 x 4 = 12 units in base 3. It turns out that for almost all numbers, using base 3 provides us with the notation that is the most economical in terms of these “units”. (The same argument applies to using ternary logic instead of Boolean logic. Often, fewer steps are needed to obtain the same result.). Computers using “trits” instead of bits were developed in the 1950s, notably by the Russians, but these efforts never caught on. According to the article, which you still haven’t read (it’s worth printing out the PDF and unplugging for this one), the likely reason why base 2 became the Microsoft of information theory notation is that in those days we didn’t have the technology to make transistors that could reliably represent three states. We could do two states, and that was good enough, so people like Claude Shannon ran with the idea and we never looked back. The cost of switching would now be too great. (Or would it? When it comes to data transmission, the Swedes are all over this.)

If aliens assume anything, why shouldn’t they assume that since base 3 is the most efficient notation, this must obviously be the base to use for interstellar communication? Since radio transmissions use phase shifts variable signal strength to encode “on” and “off” bits, it would be trivial to encode multiple states. Why didn’t we do so in our data transmissions? I’m guessing at more anthropocentrism, and because we forgot that sometimes simplicity is not the same as efficiencyWay off topic: When quantum computing happens, and it will, there is every reason to consider using the opportunity to swith to ternary logic, and using qutrits instead of qubits. This paper [PDF] and this paper make the case. This one [PDF] even has pretty pictures!.

But I want to hold off on the aliens for a while longer. I’m trying to make a case here for jetissoning bases in favor of a more rigorous approach to describing the numbers that lie between whole numbers. That’s because no number system is precise when it comes to representing arbitrary rational numbers, and downright hopeless when it comes to irrational numbers — and I think it is some of those numbers that we should consider beaming to aliens. (But I’m getting ahead of myself.)

The solution to the division problem is ingenious, and therefore first discovered by the Greeks. I’d come across continued fractions before, in high school, but their deeper significance completely passed me by until Roger Penrose, of course, rubbed my nose in them in his The Road to Reality (Chapter 3)You might remember my commitment to blog each chapter of that book. It’s not feasible, and frankly boring to impose my half-baked solutions to his problems on you. Perhaps a wiki is in order. Later. Instead, there’ll be posts like this one, inspired by the eye-opening (to me) stuff from the book..

It turns out that every number on the number line is representable by a unique continued fraction that looks like one of these:

It’s not as bad as it looks, so bear with me. You can make these continued fractions yourself: Use a calculator to divide 13 by 11. The answer is 1.181818… Subtract the bit to the left of the decimal point (it’s a 1) and put it aside. Divide the remainder into 1, and you get 5.5. Again subtract the bit to the left (the 5), then again divide the remainder into 1, and you get 2, this time with no remainder (if your calculator is any good). The numbers you took away from the running total, 1, 5 and 2, are the terms along the left edge in the continued fraction, above, which uniquely describes 13/11. This is the “basic” way of constructing a continued fraction — there are other ways [PDF].

There is only one continued fraction of this kindBy “kind” I mean continued fraction where all the numerators are 1. for each number because if you changed any of the denominators you would (obviously) get a different number. Another way of saying this is that every real number (whole, rational, irrational, transcendental) has a one-to-one correspondence to a specific sequence of whole numbers. In the case of the number 13/11, this sequence is short and finite: 1,5,2. In the case of the square root of 3, the sequence is 1,1,2,1,2,1,2… with the 1 and 2 alternating ad infinitum.

You might be wondering why this is any better than the repeating “181818…” we got in the decimal notation of 13/11. I have two reasons. First, the whole numbers that make up these sequences do not change depending on the base we use to write them. The sequence is base-neutral. Second, these sequences are more elegant: They are always finite when it comes to describing rational numbers exactly, and they are far better at divulging patterns in irrational numbers than any number system you care to use.

Here are some examples showing off the elegance of continued fractions when describing some irrational numbers — the square root of 41, and Phi, the Golden Mean.

In decimal notation, and in any other base, the numbers following the decimal point convey no order; they might as well be random. Not so when it comes to the representation of these numbers via continued fractions; for the square root of 41, the sequence is [6,2,2,12,2,2,12,2,2,12,2,2,12,…] ad infinitum. The pattern is clear.

What about that special class of “really” irrational numbers, the transcendental numbers, such as pi and eTranscendental numbers are irrational numbers that cannot be the solution to a polynomial equation. There are in fact far more transcendental numbers than any other kind of number — pi and e are just the most famous ones.?

For e, there is pattern visible, though it is one that changes in a set way with each repetition. As for pi, there is no discernible pattern via this kind of continued fraction, though much has been [Swedish PDF] made of the sequence of numbers that corresponds to it.

In sum, what we now have is a means of representing any real number in a base-independent manner using only sequences of whole numbers. This is good for communicating with aliens: No bases means no opportunity for anthropomorphising, while whole numbers are about as close to made-for-radio blips as you can get.

So, now that we can unambiguously send real numbers to aliens, which numbers shall we send them? Stay tuned for part III.

Before talking to aliens, it would be helpful if we had some insight into what it is like to be one. Since we don’t, and won’t, we will have to do the next best thing — identify all our anthropocentric assumptions about intelligent life in the universe and then ruthlessly eradicate these from the messages we send them.

To that end, we need to ask ourselves: What common ground could there be between all forms of intelligence in he universe? What is the minimal definition of the term intelligence in this regard? And is there a medium for communication that is shared by all these intelligences?

Jules Verne implicitly had a go at some answers when he conjured up a scheme for communicating with moon dwellers, the “Selenites,” in From the Earth to the Moon:

Thus, a few days ago, a German geometrician proposed to send a scientific expedition to the steppes of Siberia. There, on those vast plains, they were to describe enormous geometric figures, drawn in characters of reflecting luminosity, among which was the proposition regarding the ‘square of the hypothenuse,’ commonly called the ‘Ass’s Bridge’ by the French. “Every intelligent being,” said the geometrician, “must understand the scientific meaning of that figure. The Selenites, do they exist, will respond by a similar figure; and, a communication being thus once established, it will be easy to form an alphabet which shall enable us to converse with the inhabitants of the moon.” So spoke the German geometrician; but his project was never put into practice, and up to the present day there is no bond in existence between the Earth and her satellite. (Found via the University of Zimbabwe)

Here a geometric proof of the Pythagorean theorem is proposed as something so fundamental that it must be one of the first discoveries made by a budding civilization’s mathematicians. The Greeks indeed discovered it first here on Earth, but still: How fundamental is it exactlyIn hyperbolic space, the angles of a triangle add up to less than 180 degrees. In elliptical space, they had up to more than 180 degrees — as is the case if you draw a triangle on a sphere, for example. We don’t currently know what kind of space the universe is made of. On a very small scale, the human scale, space certainly looks Euclidian, but that would also hold true in hyperbolic or elliptical universes. My own gut feeling is that space is hyperbolic. But aliens might know for sure.? The theorem only holds true in Euclidian space, which we think of as “normal” space, but not in hyperbolic or elliptical space. We’re biased that way, however. We’re instinctive flat-Earthers — we prefer to shoehorn the elliptical plane we call Earth into flat, Euclidian maps, replete with massive Greenlands. We exude pro-Euclidian sentiment in everything we do.

We shouldn’t expect aliens to know about this proclivity of ours. Actually, they probably wouldn’t even notice proofs transmitted at wavelengths suited specifically for our eyes — aliens have no reason to suspect we’d be broadcasting at those entirely arbitrary wavelengths. Or else they might be in a phase of their development where geometric proofs are unfashionable, not to be trusted, much as was the case for generations of our own mathematicians in the era between Descartes and Riemann — and they’d even be right in this caseIn Verne’s time, aliens might (correctly) have interpreted our depiction of the Pythagorean theorem across Siberia as a statement of our (unjustified) belief in a Euclidian universe; or else they might (incorrectly) have concluded that we think the theorem holds true on an elliptical plane (Siberia). Either way, they’d think we’re stupid..

We have on several occasions made real attempts to talk to aliens. In the early 70s we attached a plaque with an engraved diagrammatic message to the Pioneer 10 and 11 space probes, which we then sent on flyby missions to Jupiter and Saturn and on into outer space.

Pioneer 10 is travelling to Aldebaran, a giant aging star around 65 light years away. As of 2005, the probe is about 12 light hours from us — one 50,000th of the way there.

In the late 70s we launched the Voyager 1 and 2 probes on a similar, upgraded mission. They contain a more ambitious attempt at communication — a gold-plated phonograph record! With a supplied needle, no less.

Both attempts unintentionally reveal some less flattering aspects of humanity, though fortunately these should be way above aliens’ potential heads. For example, the naked woman on the Pioneer plaques has no genitalia. Apparently the plaque’s designer, astronomer Carl Sagan, left them out rather than risk rejection of the entire project. The fact that there were naked people at all on these plaques nevertheless led to angry letters accusing NASA of peddling smut to the stars. More embarrassing, at least to me, is that one of the two greetings on the Voyager records is by an ex-Nazi stormtrooper, Kurt Waldheim. How on Earth — literally — did we manage thatThe other greeting is by Jimmy Carter.?

I rather doubt aliens will ever “hear” the encoded sounds on the record. If they find a Voyager spacecraft they will conclude it was built by an intelligence, but in the absence of ears, atmosphere or human brain circuitry, the bits on that record will be as revealing to aliens as a jpeg of Monica Bellucci is to a blind personCue yet another reference to Thomas Nagel and his essay, What is it like to be a bat?. And how should aliens tell if they are seeing the plaque the right side up, given they have never met a human before? It’s just a load of Pollocks to them.

In 1974 we carefully aimed the Arecibo radio telescope’s transmitter at M13, a globular star cluster 25,000 lightyears away, and sent it a three minute message containing exactly 1,679 bits (around 0.2 kilobytes). The message, travelling at the speed of light, will unfortunately miss M13 completely, as the cluster will by then have moved out of the signal’s path, seeing as our galaxy rotates.

This problem aside, using radio was not an arbitrary choice. Radio transmissions offer a faster medium than plaques and a less anthropocentric medium than records or light. Radio is as fast as light, since both are just electromagnetic radiation made up of photons, but radio frequencies are far lower than light frequencies, so radio photons require much less energy to produce.

Which exact frequency to use for radio transmissions (and thus also for listening)? In this universe, one part of the radio spectrum — at around 1420 Megahertz — has far less background noise than other parts, so anybody who would want to maximize their signal-to-noise ratio would use it. We use it when we listen for aliens with SETI. So did we use it to send the Arecibo message? Not exactly:

It’s interesting to note that in 1974 the Arecibo message was transmitted at 2380 MHz, a frequency well above the “water hole” band. In Earth’s first “Active SETI” attempt we didn’t transmit at a well known and preferred frequency of either 1420 or 1665 MHz. Furthermore, 2380 MHz is the second harmonic of no particularly special frequency. The Arecibo transmitter was designed for S-band planetary radar experiments and SETI used it because it was available.

Luckily, it technically relatively easy for aliens to listen to many frequencies simultaneously, as this fuller discussion on making guesses about the medium makes clear.

The one thing the makers of the Arecibo message did get right, in my opinion, was to use the basic properties of whole numbers to encode it. Because the sequence of prime numbers is the same regardless of where you are in this universe, the prime factors of 1,679 — the number of bits sent in the message — will be 23 and 73 everywhere. Aligning the bits sequentially on a 23 x 79 grid produces the patterns that make up the message. That’s really clever, and hints at the kind of message I think we should be sending.

Do the patterns that comprise the Arecibo message make any sense to aliens? As there will not be any opportunity to start a dialogue, aliens won’t be able to ask for clarifications. I think mixing binary counting systems with graphical representations is therefore really just a way of not imparting any information at all. Aliens might think we look like the blobs representing binary representations of our DNA molecules. Maybe they look like the binary representations of our DNA molecules. Maybe our representative pinheaded human provides them with an unsolveable binary counting puzzle which they just can’t crack.

And why oh why do we have to count to 10 in our binary counting lesson at the start of the message? Haven’t we learned anything about anthropocentrism? Much better, I think, to drop using bases altogether when talking to aliens, and just focus on winking at them unambiguously. How? I have a plan.