In the Fall of 1996  I decided to do an experiment to show that if one starts from eight randomly chosen numbers, then one could find many striking numerical relationships between those numbers and various mathematical constants.  The relationships that I searched for are approximate and the level of accuracy that I allowed was rather generous.   I will explain in the commentary below why I did this experiment and how I chose the eight numbers.   Here are the eight numbers:

 A = 2361,      B = 4341,      C = 6441,       D = 5389,      E = 9747,     F = 8761,     G = 9411 ,     H = 4414

Most of the numerical  relationships which are given below will just involve simple ratios of these numbers .  There are 56 such ratios (not counting A/A,  B/B, etc.)   To explain the notation,  if a and b are positive numbers,  then  a ~ b  will mean that a and b are approximately equal.  The error will always be expressed as a percentage,  namely  the difference between a and b expressed as a percentage of the number b.  For example,  one might write  199~200. The difference is 1 and so the error is .5%  because 1 is .5% of 200.

Here are some of the most striking numerical relationships which I found.

1.  Relationships Involving the Number  e.

           The number e is one of the most important mathematical constants in modern mathematics.  For various ways to define this number, see e.   For a more elementary and historical introduction,  you might read a delightful book entitled  e: The Story of a Number by A. Maor  (published by Princeton University Press, 1994).

 Here is a relationship involving just the number  e  itself:      C/ ~   e                   (.4%)

 Starting from the above relationship involving e = e/1 ,  here is a set of relationships involving a simple sequence of numbers.

C/A   ~   e /1                            (  .4% )
F/C   ~    e /                              (.07% )
F/E   ~    e /3                            (  .7% )
C/G  ~    e /4                            (  .4% )
A/B  ~    e /5                            (.04%)
H/E  ~    e /6                             (.04%)

These numerical relationships could also be written in their inverted forms. That is,   A/C..~..1/e.,...C/F.~.2/e.,........,..E/H.~.6/e  . The numbers involved in these relationships form a six-term arithmetic progression : 1/e,  2/e ,  3/e,  4/e,  , 5/e,  6/e .  The difference between consecutive numbers in this sequence is 1/e . The percentage errors will be about the same as those above.
  Another arithmetic progression is the sequence of three numbers   e/3,  2e/3,   e . The difference between consecutive numbers is e/3 . The two numbers e/3  and  e  already occur in the above set of relationships (approximated by F/E and C/A). The number in between is approximated by E/D   :

E/D  ~    2e /3                            (.2%)

Here is a set of relationships involving another sequence of numbers formed from e and various squareroots.    It starts with  the number   e/2.=.e./\/ and ends with   e/3.=.e./\/9.   These relationships could also be presented in their inverted form.

F/C    ~    e /\/4        (.07%)
D/H   ~       e /\/5         ( .4% )
E/F        e /\/6    .. ( .3% )
E/G   ~     e /\/7         ( .8% )
G/E   ~    e /\/8          ( .5% )
F/E    ~   e /\/9         ( .7% )

2.   Relationships Involving the Number pi.

     I have not been able to find long sequences of numerical relationships involving pi which are as nice as those given above for the number e.   Nevertheless,  here are some relationships where the error is less than  .1%.

D/A...~...pi / \/6    +   1 .................(.002%)

C/A...~...pi / 3\/6   +   1.................(  .03% )

B/A...~...pi / 3\/5............................( .08% )

B/A...~...ln(2pi) .............................( .04%)

E/B...~...ln(3pi)..............................( .09%)

Two of the ratios are approximately equal:    C/H  ~  G/C    with error  .13% .  These ratios are both approximately equal to the cube root of pi.

C/H ~ 3\/pi                               ( .4%)

G/C ~ 3\/pi                              (.24%)

3.   Arithmetic Relationships Between the Eight Numbers.

          The numbers A, B, . . . , H  are all four-digit numbers. There are 9000 different four- digit numbers. Nevertheless, there are a number of simple arithmetic relationships involving these eight numbers which are fairly accurate.

The number 5391 is exactly halfway between B and C.  The number D=5389 is quite close. Expressing this mathematically, we can write:

D    ~ (B+C)/2                     (.04%)

The number 4401 is exactly halfway between A and C.  The number H=4414 is fairly close.   Mathematically,  we can write:

H    ~ (A+C)/                    ( .3%)

The above two relationships involve the arithmetic mean of two numbers. The next relationship involves a geometric mean of two numbers.

The product of G and H is an eight-digit number. Its squareroot is 6445.165. . . .  The number C=6441 is rather close.  Mathematically, we can write:

~ \/(GH)          ...........(.065%)

Here are a couple of relationships just involving addition:

~ B+H                              (.07%)

~ B+D                              (.17%)

All of these relationships can be stated in terms of ratios involving A, B, . . . , H.  For example, D.~.(B+C)/2 can be rewritten as B/D.+.C/D.~.2 and C.~.\/(GH) can be rewritten as G/C.~.C/H.  The percentage error for the first example is unchanged, but for the second example the percentage error increases to .13%.

COMMENTARY:  The eight four-digit numbers A=2361, B=4341 , . . . , H=4414 were formed by dropping the three-digit prefixes from the first eight telephone numbers listed under the name "Hoagland" in the 1996 Seattle Telephone Directory. In effect, I used the telephone directory as a random-number generator.   In The Monuments of Mars, Richard Hoagland presents a set of numerical relationships involving nine angles which were identified by Erol Torun in connection with the so-called D&M Pyramid in the Cydonia region of Mars.  He suggests that those numerical relationships are unusual and that their existence is evidence that the D&M Pyramid is an artificial object in which those relationships were part of an intentional design.  When I first saw those relationships in Hoagland's book, my immediate, instinctive reaction was that there was nothing at all unusual about them and that it would not be hard to find an equally striking set of relationships starting from a random set of angles.  Mr. Hoagland was scheduled to come to Seattle in mid-October, 1996 to give a public lecture.  I decided to try to convince him that he was mistaken by doing a simple  "control experiment" in which I used the angles 23.61o, 43.41o, . . . ,  44.14o  (formed from the above numbers) in place of angles from the D&M Pyramid, and then considered ratios of the angles, values of the trigonometric functions, and radians, allowing myself the same margin of error in the approximations as Hoagland and Torun did.  I devoted about 10-12 hours to this experiment and found a large set of numerical relationships which seemed to me to be even more striking than those presented in Hoagland's book.  Before Hoagland's lecture, I presented him with a summary of some of the results of my experiment. He said that he would study it, and respond at a later date.  In fact, he never did respond even after I sent several reminders to him.  Eventually (in 1998), I even challenged him to a debate on this issue, but he adamantly refused. The points that I would have raised in such a debate are described on another page - The D&M Pyramid on Mars .
            The relationships presented above have a large overlap with those which I presented to Mr. Hoagland in 1996.  However, here I have restricted the relationships to ones which just involve simple ratios of the eight numbers A, . . . , H.   (Note: Interpreting those numbers as angles with the decimal point placed as above has no effect on these ratios.)  In the course of writing this page, I found quite a few new relationships not included in the original set, and also decided to leave others out.  In particular, I have left out various relationships involving some of the crop circles in England and the pyramids in Egypt, and also various isolated relationships involving mathematical constants, some of which were quite accurate but did not fall into nice sequences.
            There is one important point that should be made concerning this experiment.  I started with a set of eight four-digit numbers. I decided in advance that I would look for relationships involving the 56 ratios (and also values of trigonometric functions and radians when I originally did this experiment in 1996). I also decided in advance on the level of accuracy that I would require, namely that it would be similar to the level of accuracy that Hoagland and Torun allowed.  I also decided in advance that the relationships should involve well-known mathematical constants and that they should fall into some nice, striking patterns.  These were the guidelines that I chose in advance.  All of this was motivated by my intention to imitate what Richard Hoagland presents in his book.  Now, if I had also specified in advance that I would search for the specific pattern  1/e,   2/e,  3/e,  4/e,  5/e,  6/e, and then actually found it, I would have been extremely surprised.  The probability of finding this specific arithmetic progression in relationships with the 56 ratios is small. (The guidelines for the percentage errors are not really precise enough to even try to calculate this probability.  Even if we made those guidelines more precise, it would be extremely difficult to arrive at a reliable estimate of this probability.)  However, I did not decide in advance to look for that specific arithmetic progression. My guideline was to search for any pattern whatsoever that seemed unexpectedly striking. Even though any such pattern by itself might be very improbable, the number of possibilities for such patterns is very large, and therefore the probability of finding at least one such striking pattern is reasonably high, perhaps close to 1.  For this reason, when I found a pattern as nice as the arithmetic progression 1/e,..2/e,.......,.6/e,  it did not seem so remarkable to me.
           On my page Secrets of a Certain Pentagon, I discuss another experiment of a similar kind. There I start with a certain specific pentagon and considered relationships involving areas and lengths associated with that pentagon.. In doing that experiment, I decided to search for relationships which have a much higher level of accuracy. I expected to find striking relationships which were also highly accurate just because I had more numbers to work with. Instead of considering just 56 ratios as in the Telephone Number Experiment described on this page, I had literally hundreds of quantities that I could extract from that pentagon.  I restricted myself to relationships with a percentage error of at most .1% and managed to find about a hundred such relationships.
           As I was preparing this page, I discovered to my amusement that one can interpret these numbers in terms of the geometry of a tetrahedron inscribed in a sphere: Tetrahedral Geometry in the Seattle Telephone Directory.  The constant e'=(\/3/2)pi  shows up. (The tetrahedral significance of this number was pointed out in The "Message of Cydonia": First Communication from an Extraterrestrial Civilization?)  Even Richard Hoagland's favorite angle of 19.5o shows up.   Of course, this is just a further demonstration of the ubiquity of coincidences whenever one deals with a large enough collection of numbers.

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