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J. Volkmar Schmidt and Hael Yggs collaborated on Magic PiWorld, an art program that defines picture elements according to the digits of pi.

 

What's normal

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 = n=3,32,33...1/(n 2n).

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 of 2,3 (that's the 10100th binary digit) -- which happens to be 0.

More on randomness and normality

 

Hi-fi arithmetic

"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) Jean 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.

More on the mathematics of fast Fourier transforms

More on NERSC's library of high-precision software

Did You Ever Wonder Web Site

Ernest Orlando Lawrence Berkeley National Laboratory