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."