Using the prime numbers to communicate with other civilizations

  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 (?)).

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.

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.)

The plaque aboard the Pioneer 10 spacecraft. © NASA.