The Pascal Fractal at the Other End of the

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					  The Pascal Fractal at the
  Other End of the Rainbow
                        John Holte
              Gustavus Adolphus College

 NCS-MAA Spring Meeting at Minnesota State University
              Mankato, 22 April 2006


It is well known that Pascal’s triangle modulo a prime p has a self-similar
pattern, and the nonzero residues may be viewed as forming a fractal that
Mandelbrot calls a Sierpiński gasket. The nonzero residues correspond to
binomial coefficients whose “order of divisibility by p” is zero. What if we
look at the binomial coefficients in the initial pn-by-pn Pascal’s diamond,
marking those binomial coefficients exactly divisible by pdn? Then dn is the
order of divisibility, and d, the “degree” of divisibility, must be between 0
and 1. As n→∞, these “prefractals” for d = 1 (maximum p-divisibility)
approach a fractal of the same dimension as the Sierpiński gasket, the d = 0
case. Furthermore, it turns out that these dimensions for d = 0 and d = 1
correspond to the endpoints of a rainbow—a spectrum of fractal dimensions
f(d).