Fig. 2 PEACE
I apologize for the terrible quality of the graphics below. They were scanned in from 10-year-old printouts; some were then compressed. But the visual effect of the modern computer-generating tools available to me right now are not as good. All criticisms accepted, but I am concentrating on the merit of the concepts involved, not on the technicalities of its presentation.
GRAPHIC REPRESENTATIONS OF THE DISTRIBUTION OF THE PRIME NUMBERS or WHY 11 AND 20 ARE AS ETERNAL AS PI (WELL, ALMOST) by Brendan J. Welch University of Massachusetts at Lowell Lowell, Mass. 01854 BRENDAN_WELCH@UML.EDU © 1986 May & 1996 April One way of finding the prime numbers is to use the ancient Sieve of Eratosthenes: a list is made of all the counting numbers, up to the highest desired or practical number; then all the multiples of 2 are eliminated, starting at 4; then all the multiples of 3 are eliminated, starting at 6; the process is repeated until all possible eliminations have taken place (which is arrived at with the multiples of the square root of the highest counting number used). A modern computerized version of the Sieve has already become one of the traditional ways of evaluating and comparing computers, because it is a lengthy, cpu-intensive process. As a casual observer, I thought that it would be interesting to list the primes graphically, for a variety of reasons. Although it probably has been done before, it seemed to me that the way in which the numbers die out as you march toward infinity would make a unique pattern which could be used to communicate with other civilizations. I had the preconception that such a picture would look quite ragged or straggly as you marched out toward infinity; I had the humorous ideas that maybe with some stretch of the imagination, it would come out looking something like the gold-plated picture plaque (shown in Figure 1) used by Carl Sagan on the Pioneer 10, or perhaps as a cosmic peace symbol (Figure 2). I say humorous, because actually it would be quite upsetting if some "local" earthly symbol would have an eternal, mathematical, galactic equivalent according to the Laws of Einstein that "physics is physics" no matter where you reside in the Universe, or if God and quantum mechanics had played some giant trick on us. ?????? placeholder here for plaque (Fig.1) currently at end of document to save your time. Figure 2 (below), explained at Figure 7
In any case, it would be nice to have some universally applicable "thing", whether it be the number pi, or the binary counting system, or the angle that the oxygen atoms make with the hydrogen atom in a molecule of water. One difficulty, on whatever planet or galaxy you reside, and whatever the size of your eyeball (or other optical sensor), is that presumably it is impractical to represent the number line from 1 to infinity. Certainly one approach toward compressing the picture, such as on the printed page, would be to stop your scan of the number line, as you go from whatever constitutes left-to-right on your planet, and place the next group of numbers "below" the first group, just as we in the English language have agreed to have our written documents extend perhaps 80 characters across, and then place the next part of our sentence below and all the way to the left of where we just cut it off. This is, of course, the same way a television camera scans a scene. The inhabitants of different planets, or even of the same planet, would probably disagree, maybe violently, about what number other than 80 would constitute a "natural" or "pleasing" or "eternal" or "universal" way. My approach, then, was to try several different numbers, and see which ones proved tractable, natural, mathematically revealing, or even just plain pretty or funny. This was all done, of course, on a computer. Some observations: ---it proved surprisingly easy to program, using Basic. ---you can represent a lot of numbers, even on a cheap screen; my first try was for 640 x 200 = 128,000. ---the biggest surprise was that there are a heck of a lot of primes; the math books say it goes as x/ln x, but I will just say that the bottom line of the screen looks almost as bright as the top line. An impromptu rule of thumb is that something like 10% of the counting numbers are primes; this is roughly true over a very wide range, falling off to about 5% by the time you count to a billion. ---because the average home computer today has as much computing and graphic power as the large computers of a few years ago, maybe I can learn something the professionals don't already know, or at least have not popularized among themselves. ---if you have an imagination, or like to look at clouds and see pictures in them, you can have a ball (or, they will send the men in white jackets to cart you away); I have seen such things as a baby boy (the child Jesus?), a heart, and an anchor by squinting at just a few hundred numbers when they have been displayed appropriately. Rorschach, beware. In the pictures below, each one has an associated cutoff number of "xmax" points per line. If xmax = 100, for example, then the first line contains the numbers 0 thru 99, the next line contains 100 thru 199, and we plot as many lines as can fit on the screen of the particular computer (either 200 or 800). A dot means a number which is prime. Thus the upper left corner contains the early primes 1, 2, 3, 5, 7, 11, 13, 17, etc. (except that I think I left out the first one, 1, to satisfy some purists who said 1 is not a prime number (?)).BIGGEST, xmax=512, using Matlab instead of Basic. =Fig. 3a (above).
Above are all the primes up to the number 384,000. Notice the diagonal striations, which I think are due to the many multiples of the smaller primes.
Fig. 3e (above), xmax=510
Multiples of 2,3,5, and 17 cause the noticeable banding. Fig. 3g (above), xmax=616
Fig. 3h (above), xmax=639
Tilt your head 90 degrees toward your right shoulder, until I straighten out this next one. Squint.BULL, xmax=256
Several people can see a bull above, horns between 100 and 150. xmax=256
Ah, life is not simple. How can we compress more information? If we were sending a radio signal consisting of a little burst of energy representing each binary bit of each prime number, we would be wasting a lot of significant digits (and time, and energy) on essentially repetitive information. It would be easier just to send successively the differences between successive numbers. So the next thing I did was something which can be assailed as being arbitrary, artifical, or culture-dependent. Presuming that these alien beings are as oriented to two dimensions as we are, I grouped the differences between the primes in groups of four; within each group, the sequence of plotting is up, left, down, right; thus the sequence of prime numbers is wrapped around in a rectangular ball. What is surprising is the unusual picture which resulted. If any alien civilization thinks at all like we (I?) do, they certainly would recognize this picture as being singular, eternal, and mathematically beautiful. Figure 4 (below), rectangular derivs
But I was not satisfied to believe that a raster type of left-to-right scan was the only way to communicate across time, space, and biological types. Another way of compressing data from a single, infinitely extended, straight number line would be to wind the number line around in a spiral. Furthermore, we could communicate another "eternal" idea, that of radian measure. Consider the well-known mathematical spiral, r = theta. If we place a dot at each counting number, where a value of 1 = 1 radian = 57.925 degrees, then we get a picture like this (it has been compressed radially by a factor of 16, both to fit in more numbers, and to show its structure of radiating spiral arms more distinctly; also, it is upsidedown from the traditional textbook picture because +y is downwards on my computer screen): Figure 5 (below) spiral of ALL counting numbers (i.e. the integers) 44 arms compressed by a factor of 16
Even more surprising is the picture which is left after the Sieve program has removed the various multiples! What could be more appropriate to communicate across the galaxies than a picture of the most "prime" or "eternal" galaxy? Figure 6 (my favorite) the spiral "galaxy" compressed by factor of 128
The empty white bands were even more impressive when I first saw them, as a road of black, on a green-screen monitor, giving the galaxy its distinctive form. I thought it might be due to the fact that there are no even numbers which are primes, but in our continuing round of surprises, this is not the case. In order to understand what has happened, it is necessary to look back at Figure 5, containing all the counting numbers in spiral form. Note that after every count of 6, we get a circular slippage of 16.225 degrees, due to the difference between the 6.28 (=2*pi) radians of once around the circle and the 6 whole numbers we can fit into one rotation (6.28 - 6 = .28 rad = 16.225 degrees). Also note that after 22 trips around the circle, we have almost perfectly come back to where we started, since 22 X 16.225 = 356.95 or almost 360 degrees. But it turns out that there are 44 radiating spiral arms, not 22. By hindsight, this is explainable as follows. When you get halfway thru your 22 trips, the slippage puts another number halfway between your expected 22 arms; partly because of the radial compression of the picture, partly because of the fairly wide separation of the existing arms, and partly because of the lines which the combination of eye and mind tend to follow, another spiral arm is generated approximately 180 degrees away. Note that these 44 spiral arms have "nothing to do" with the single, tightly wound spiral arm generated by the original formula, r = theta. You have generated a pattern which has no points actually connected, and there are many ways the eye can look at such a pattern and generate a mental or visual connection. As you walk out any one of the radial spiral arms, the numbers you encounter there are spaced 44 away from the previous number; your eye will also tend to follow another set of lines, which seem to run almost circularly, whose number spacing is 25. When the Sieve program erases the multiples of 2, it immediately removes every alternating arm, leaving a fairly symmetrical picture with 22 arms. As the multiples of 3, 5, 7 etc. remove other points, (you might even say removes them "randomly"), some arms may seem to be hit a little harder than others, but the basic structure of 22 arms is still visible. The next surprise, perhaps the biggest of the whole experiment, is that when the multiples of 11 are removed, the job is completed of totally removing 2 more arms. Sure it is easy to explain after the fact; the factors of 44 are 2,2, and 11. The result is a picture of a "galaxy" with 20 arms; they are spaced every 16.225 degrees; the "missing" arms give the galaxy the look of a barred spiral, a classic astronomical category of galaxy. But don't give up yet, folks. Do you remember up above, where we fed the spacing between the primes into a rectangularly-based display? Let's do the same thing again, and feed those same numbers into the spiral display program. The resulting picture, shown below, except with no radial compression, gives another "eternal" picture, unwittingly adopted by the U.S. Dept. of Transportation as a logo. Maybe Eratosthenes could be credited with contributing the letter lambda to the Greek alphabet. And, I have found that there does exist some sort of cosmic version of the "peace symbol" of Figure 2. Figure 7 (below), lambda
There is a corollary of Murphy's Law which says that anything can be made to work (or in this case, can be explained) if you kludge at it long enough. There are only 6 possible radials at this magnification of the spiral (your eye is not connecting the same radial or circular lines as it did above); prime numbers are always odd; thus their differences are always even; many of the early primes have differences of 2, 4, or 6 (or of 8, 10, or 12, etc.). Thus there will be 3 spiral arms, containing even numbers, whose bright points are lighted up by many (redundant) contributions of rather small numbers.
Figure 1 below is the plaque aboard the Pioneer 10 spacecraft. © NASA. (Moved here to the end of the document to prevent slow loading at beginning.)