Pi shared fairly
Mathematicians edge closer to proving that all numbers get an equal slice of pi.
1 August 2001 
ERICA KLARREICH

 
Pi has been causing chaos in mathematical circles for years. 
 
 

Carl Sagan's science fiction classic Contact ends with its heroine searching
for a cosmic message hidden in the digits of the number pi. Two mathematicians
have now taken the first step towards proving that pi contains not a single
message but every conceivable message, meaningful or not1.

David Bailey of Lawrence Berkeley National Laboratory in California and
Richard Crandall of Reed College in Portland, Oregon, present evidence that
pi's decimal expansion contains every string of whole numbers. They also
suggest that all strings of the same length appear in pi with the same
frequency: 87,435 appears as often as 30,752, and 451 as often as 862, a
property known as normality. 

Pi is the ratio of a circle's circumference to its diameter. Mathematicians
have known for more than two centuries that the number is an infinite,
non-repeating decimal. 

Whether that decimal is random or ordered is one of the hardest questions
in mathematics, says Stan Wagon, a mathematician at Macalester College,
in St Paul, Minnesota. "It is a breakthrough that these researchers have
anything to say about the matter at all," he says.

Bailey and Crandall showed that the normality of pi will follow if
mathematicians can prove a conjecture in a completely different field,
chaos theory.

"We haven't proved the normality of pi, but we've laid out a road map,"
says Bailey. Following that path to the end may be difficult, but he hopes
to prove at least a simplified version of the chaos conjecture within a
few years. 

Pi slinging

The quest to conquer pi's infinite expanse has led some mathematicians into
fierce calculating competitions. The current record, achieved with the help
of supercomputers, is 500 billion digits.

The new work arises out of a surprising formula for pi that Bailey and his
colleagues discovered in 1996, which allows mathematicians to compute any
digit of pi without knowing the previous digits.

To link pi's normality to chaos theory, Bailey and Crandall construct a
sequence of numbers between zero and one out of pieces of pi's expansion.
Their process goes something like this: 

The decimal expansion of pi starts as the familiar 3.1415926535.... Consider
the sequence 0.314, 0.141, 0.415, 0.159, 0.926, 0.265, 0.653, 0.535...
obtained from consecutive groups of three digits of pi. If pi's digits
are random, the sequence should jump about randomly between zero and one.

Bailey and Crandall work not with pi's decimal expansion but with its
binary expansion. This expresses what comes after pi's decimal point as
an infinite string of zeros and ones, the language of computers. 

Using the binary expansion, they build a sequence similar to the one above.
They then use the 1996 formula for pi to write down a simple expression for
each number in the sequence. Finally, they prove rigorously that if the
sequence bounces about chaotically between zero and one, pi is normal. 

The new work brings the question of pi's normality in contact with a much
broader field of mathematics, says Jeffrey Lagarias, a mathematician at
AT&T Labs in Florham Park, New Jersey. "It also indicates connections with
areas not previously thought to be related," he says.

"I'm not aware of any other link between chaos theory and number theory,"
Bailey agrees. "One field arises from computational physics, and the other
is the purest of pure mathematics."

If pi is indeed normal, looking for a message in its digits would be like
searching for meaning in Jorge Luis Borges' imaginary library of Babylon,
in which the books contain every possible combination of letters and
punctuation.

While there may be no cosmic message lurking in pi's digits, if they are
random they could be used to encrypt other messages as follows:

Convert a message into zeros and ones, choose a string of digits somewhere
in the decimal expansion of pi, and encode the message by adding the digits
of pi to the digits of the message string, one after another. Only a person
who knows the chosen starting point in pi's expansion will be able to decode
the message.  
   
References
Bailey, D. and Crandall, R.On the random character of fundamental constant
expansion. Experimental Mathematics, 10, 175 - 190, (2001). 
 

© Nature News Service / Macmillan Magazines Ltd 2001