^{-20}to 2

^{-40}). So, thirty bits to prove our intelligence. What should they be? Here are some ideas. A mathematical constant, for example, pi. But pi itself got me thinking: will an alien civilization actually use 3.1416 as their version of pi? Perhaps not: 2pi=6.2832 is also a very reasonable constant. It is the period of sine and cosine, the ratio of the radius to the circumference. sqrt(pi)=1.7725 appears in various formulae for the volumes of higher-dimensional spheres and in probabilities, for example in the normalizing term of the error function. Real numbers greater than one pose a problem because it's not clear how to encode the decimal point. Maybe we can broadcast just the portion after the decimal point, or the inverse (1/x) of the constant. Two more constants "of the ancients" are the sqrt(2)=1.414 (and 1/sqrt(2)=0.7071) and the golden ratio phi=0.618. They nicely fall between zero and one. There are plenty of other mathematical constants that aren't pi to choose from. e=2.71828 is the next most famous, and it doesn't have the problems of 2e or sqrt(e) being plausible constants as well. 1/e=0.3679 is an important constant in probability -- the limiting probability of success of repeating a gazillion times an experiment that has a one-in-a-gazillion chance of succeeding: a very apt constant to be sending through space. By choosing the constant, we as a civilization can show off our level of mathematics. pi says we've advanced as far as Archimedes: basic geometry. The sqrt constants say we know algebra. e says we know calculus. Yay us. (I'm generalizing of course, after all, what is the difference between "basic geometry" and "algebra". Uh...) There is another constant that also says we know calculus, and calculating it to high accuracy is quite difficult. This is the Euler-Mascheroni constant γ = 0.577, or the "left-over" from subtracting log from the harmonic series. This constant is pretty obscure and difficult to calculate that we're tooting our own horn in broadcasting it. It's difficult to calculate: using the naive formula would require billions of terms for 30 bits of accuracy. But Euler himself calculated it to 16 decimal places (53 bits), obviously using some other formula. Euler was a calculating machine, though! Going further into obscure mathematical constants, how about Khinchin's Constant == 2.685 or the Feigenbaum Constant == 4.669 ? Brun's constant = 1.902 in fact is only known to today 9 digits (29 bits) (see link). Human mathematics has taken a detour through the Riemann Zeta function to get those nine digits, so it's pretty high math. The sum of the reciprocals of the midpoints of the twin primes, i.e., 1/4 + 1/6 + 1/12 + 1/18 + 1/30 + 1/42 + 1/60 + ... does converge to a nicely transmittable value between zero and one (by the Harmonic-Mean Arithmetic-Mean inequality), but I don't know if the same detour for Brun's constant can get us 30 bits. Instead of broadcasting a fractional constant, we can broadcast an integer. Sadly there won't much agreement. The best integer is 78557, which is conjectured to be the answer to the Sierpinski problem, but it's thought to be at least a decade until the number is proved. Or we could go with the Mersenne exponent of month (currently 24063583) which is the pinnacle of human computing. Unfortunately these numbers are 17 and than 24 bits respectively -- quite small. The largest Mersenne prime is likely to grow, a lot, in the course of the centuries of our broadcast. In the realm of constants we haven't calculated yet, how about the probability of a random Turing machine halting? Unfortunately, current theory states that expected to never know even one bit of that number. Instead of mathematical constants, how about a physical constant? The glaring example of course is the fine structure constant α (alpha) = 0.007297. It has a nice bit of zero padding at the front so that the aliens will be able to figure out where in the loop our message begins. Kind of like pi, sqrt(alpha) is also a plausible value. Through much cleverness by the experimental physicists, we now (barely) know the constant to a sufficient number of bits. A tiny possible fly in the ointment is that we are not sure that the constant is even constant! If the constant changes over the billions of years the message flies through space, they'll be some pretty confused aliens on the receiving end. Finally, here's an idea that's not so complicated. The problem with all these constants mentioned above is that each year one needs to "look up" whether to turn on the bit. How about something calculated from the year itself? Counting to 8 in binary takes up 24 bits: let's try to transmit the sequence 000001010011100101110111. This requires only some simple instructions mumble mumble divide the year by 24 and keep the remainder mumble mumble. Even better yet, Chinese zodiac mumble mumble! The reason an easy-to-calculate sequence is good is because we don't want to tie up a radio transmitter for centuries broadcasting. Instead the population of entire planet can participate, but we can't expect the entire population to have the binary value of (say) the fine structure constant handy. But a simple rule, like count to 8 in binary, people can memorize. On on-bit years, do something; on off-bit years, don't do it. For example, leave the backyard light on for the aliens. It doesn't even have to be something done continuously through the year. A crazy idea might be to only hold the World Cup on years with on-bits. The world emanations of TV being watched would make a recognizable pattern. Maybe people can vote: (a) 000001010011100101110111 (counting to 8) (b) 00000001110111100011110101 (fine structure) (c) 100001101001010101110011111100 (1/Brun) (d) 0101111000101101010110001101100010110100 (1/e) Not that I'll care about your votes when I become dictator of the world and institute alien broadcasting on the populace...

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