# How Contact by Carl Sagan Ends

She learns from the aliens that there is a similar message hidden within one of the key transcendental numbersof mathematics, though they admit that they have not yet deciphered it themselves. The theological implications are intriguing (the whole universe must be constructed very carefully if one of its essential constants is in fact a coded message.) Then, at the very end of the book, she sets her computer program -- the same one which first noticed the message from Vega -- to looking at the numerical expansions of Pi (in various different bases).

She find a long string of 1's and 0's late in the expansion of Pi in base 11. It's length is a product of two primes, indicating a two dimensional array. So, she plots it on her computer screen (each digit representing a pixel) and sees a perfect circle. The constant which describes the ratio of a circle's circumference to its diameter itself contains a picture of a circle!

That's where the book ends.

Actually, as it turns out there is a theorem which almost guarantees that Sagan's "fiction" about Pi is true. In particular, I have been referred to Theorem 146 in the book "An Introduction to the Theory of Numbers" by Hardy and Wright which proves that the set of numbers that do not contain every arbitrary finite sequence in their decimal expansion has measure zero. (In other words, if you "randomly" pick a number, you can expect its decimal expansion to contain every finite sequence including the Gettysberg Address and the next e-mail message that you will write written out in ASCII.) There is no guarantee that this will be true for the number Pi...but there is also no reason to doubt that it is true.

Of course, the fact that Elie found this sequence that looks like a circle is really rather remarkable. The problem with Theorem 146 is that although every sequence appears in the decimal expansions, there is of course no way to find any given sequence. (Or, as visitor "Nils Tycho" points out, and as Sagan puts it in the story itself, the surprise is not that it appears that it appears "so early" in the sequence.)

Additional Comments from Mike Hennebry: `In Carl Sagan's novel Contact, he treats pi as if it were a physical constant and thus adjustable by Anyone with the wherewithall to adjust physics. The problem is that pi is not a physical constant, it is a mathematical constant, the ratio of the circumeference of a Euclidean circle to its diameter. It is not adjustable because its definition makes no reference to reality. Furthermore, there is no distinct physical constant that would be meaningful in any universe even vaguely resembling ours. In curved spaces, there is no constant ratio of circumferences to their corresponding diameters. The small circle limit, where no singularity is involved, is precisely pi, hence the qualification "distinct".'

I think that this is exactly the point, Mike. Remember that Sagan was an outspoken atheist, but the book is very much about religion as well. I think that Sagan was trying to find something that would give even a skeptic like himself that numinous feeling of amazement that goes beyond being impressed with an alien being's advanced technology. We can all imagine scientific advancements that could alter the physical universe, but to alter a constant derivable from Euclidean geometry itself seems, well, god-like! As "Nils Tycho" points out: "That is what makes the conclusion so spectacular."