Earlier this year, I wrote three parts of a four-part series on how to talk to aliens. Here comes the final part.

My main thesis has been that if we want to communicate with complex self-aware systems about which we cannot make any assumptions (AKA aliens), then we have to strip our message of all arbitrary cultural and historical attributes, leaving only fundamental mathematical notions, such as integers. In part one, I examined some of the unfounded assumptions we’ve implicitly made about extra-terrestials in our past attempts to communicate. In part two, I explained how using continued fractions as a form of notation allows us to depict any number as a series of integers, and I explain why this is a far less arbitrary method than relying on base 2 or base 10, or bases tout court.

Now that we have a universal method for sharing numbers with aliens, which numbers should we send them? I promised to propose two such numbers, but delivered only the first, Khinchin’s constant, in part three.

This post is brought to you by the fine structure constant, or the letter alpha. Unlike Khinchin’s constant, which emerges as a fundamental property of numbers and thus exists entirely within the realm of mathematics, alpha defines a fundamental property of our universe.

In the broadest sense, alpha is the ratio of the strength of the electromagnetic force to the strength of strong force — the two strongest forces of the four fundamental forces in the universe.I’ll gladly outsource the gory details: Here it is defined as “the ratio of the speed of the electron orbiting the nucleus of a hydrogen atom to the speed of light”. Wikipedia’s updated definition has it as the ratio between “(i) the energy needed to bring two electrons from infinity to a distance of s against their electrostatic repulsion, and (ii) the energy of a single photon of wavelength 2.pi.s.” Alpha was “discovered” by physicists in 1916, and can currently be measured to an accuracy of 10 decimal places, at .007297352568(24), or 1/137.0359991(5).

But why send alpha, instead of another well-known physical constant, like the speed of light, c? The constant c is measured in terms of distance over time, so the actual number depends on the units we use for distance and time. These units are arbitary. The number 299792458, for example, defines c in terms of meters per second. Sending that number to aliens (or 186282.397…, which is c in terms of miles per second) imparts no information, because aliens are not privy to our measurement conventions. We might as well send them a random number.

Alpha is different. It does not have units of measure, (The term of art is that it is dimensionless.) The constant is a pure number, like pi or e. Unlike pi or e, however, alpha has resisted derivation from mathematical first principles.Not surprisingly, many people have tried to derive alpha mathematically. The physicist Arthur Eddington thought he could prove alpha was exactly 1/136, later that it was exactly 1/137. The phycisist James Gilson got a more accurate result, though the latest empirical data places alpha over one standard deviation away, making it unlikely he is right. Naturally, alpha also inspires the odd religious nut. Alpha is to early 21st-century humans still a fact of nature to be measured empirically — a given, an exogenous value not predicted by theory; and that, to physicists, is like catnip. Richard Feynman called alpha “one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man.”

John Baez, a renowned mathematical physicist, has listed 26 exogenous dimensionless constants that define our specific universe. Of these the fine structure constant is the most famous. But fame doesn’t fly with aliens. Why not choose any of the others?

We certainly could, but there is something else about alpha that makes it useful for our purposes: There is tentative evidence that suggests the constant may in fact change over time and space. A paper in Physical Review Letters in 2001 suggested that the constant may have been just a little smaller six billion years ago, based on the spectroscopic analysis of quasars. The authors give an broader overview of the state of the research in a Scientific American article earlier this year. The article, definitely worth reading, points to a paper that argues the value of alpha may also change depending on whether you are safely within the gravitational tug of a galaxy or out in intergalactic space. Inside a galaxy, alpha may remain more stable, goes the theory. Alpha may change across the universe due to the universe’s inherent “lumpiness”.

For our purposes, here are the salient facts about alpha: 1) Our level of technological advancement determines the precision with which we know alpha, and 2) It’s possible that alpha changes over time and distance. This allows us to transmit two pieces of information if we send alpha to faraway aliens: 1) Our level of technological advancement, and 2) the value of alpha here and now, which may act as something of a location marker or a data point, especially if their local value turns out to be slightly different.

We can achieve this result by sending both the highest likely value and the lowest likely value for alpha that we currently feel confident of — 0.007297352544 and 0.007297352592 — converted into a continued fraction, of course. This helpful site turns those numbers into the corresponding integer series

[0, 137, 27, 1, 3, 1, 1, 35, 2] and [0, 137, 27, 1, 3, 1, 1, 11, 3].Do reread part two of this series if you want a primer on continued fractions.

Given that our aliens will have deciphered part three‘s Khinchin’s constant (which depends on continued fractions to be meaningful), they will not be able to mistake these two sequences for anything other than two numbers that are extremely close to alpha. The aliens may have derived alpha exactly, or have measured it far more precisely; they could be aware that it changes across space, or they could be terrible at precise measurements. In all situations, our integer sequences corresponding to the upper and lower bounds for alpha as we currently know them will come in handy. I’m sure of it.

That concludes this series. I’ve found that often, the popularity of my blog posts is inversely proportional to the fun I have writing them. I’m not sure if that’s a constant, though.