Volkmar Schmidt and Hael Yggs collaborated on Magic
PiWorld, an art program that defines picture elements according
to the digits of pi.
The technical word for digit randomness is normality. In base 10,
for example, "any single digit of a normal number occurs one-tenth
of the time, any two-digit combination occurs one one-hundredth
of the time, and so on," David Bailey explains.
Bailey and Richard Crandall hypothesized that sequences of a particular
"chaotic-dynamical" kind "uniformly dance in the
interval between 0 and 1." If so, the normality of many fundamental
constants would follow.
The BBP digit-calculation formula yields just this kind of chaotic
sequence for pi, and there are similar formulas for many other constants.
While Bailey and Crandall have yet to prove their hypothesis for
pi, recently they proved it for the number written 2,3
Richard Stoneham published a different, little known proof for
the same number in 1973, but Bailey and Crandall proved normality
for a much wider class. And to show that, like pi, its digits can
be rapidly extracted, they calculated the googolth binary digit
(that's the 10100th binary digit) -- which happens to
"The 16-digit, 64-bit, floating point arithmetic common to
most computers is sufficient for almost all scientific applications,"
says David Bailey, "but a few crazy people need more. I'm one."
The PSLQ algorithm is one example of a calculation that needs high-precision
arithmetic. Modeling of the global climate is another; insufficient
precision means that the same program running on different computers
may come up with different answers.
One fruitful source of high-precision techniques has been fast
Fourier transforms, named after mathematician (and Egyptologist)
Baptiste Joseph Fourier. A Fourier transform translates a complex
wave into a spectrum of frequencies -- "something your ear
does routinely when listening to music," Bailey says.
Fourier transforms have applications far beyond studying waves.
"I got interested because they allow rapid multiplication of
very large numbers," says Bailey. "Indeed, they are useful
for calculating anything that can be formulated as a '250convolution.'"
Applying these and other techniques, Bailey and his collaborators
have developed an extensive library of high-precision software.