This applet graphically presents the Pascal triangle modulo an integer
number p, i.e. it computes the binomial coefficients
mod p and then plots them. The zero (mod p) values are depicted by gray
dots, the non-zero values- by black dots. I shall call the resulting figure
the "Pascal gasket of order p". If you try out my applet, you will see
that if p=2 then the resulting picture is the Sierpinski gasket. It is
a fractal of dimension less than 2, so its area (two-dimensional Hausdorff
measure) is zero. This shows that in a certain sense almost all binomial
coefficients are even. Actually (D.Singmaster, Notes on
binomial coefficients III-- Any integer divides almost all binomial coefficients,
J. London Math. Soc. 8 (1974), 555-560) same is true for
every integer p>1. This means in particular that the area of every Pascal
gasket is zero, though you can't quite see it from the picture, say, for
p = 15000!
For more information see: Ian Stewart, Four Encounters
with Sierpinski's Gasket. The Mathematical Intelligencer 17 (1995),
52-64 or Andrew Granville The Arithmetic Properties of Binomial Coefficients.
How to use the applet
If you have a slow system and choose a big number of rows, it might take
for a while for the program to build the image. Be patient: the program
writes "Calculating ... please wait!" in the status line while it is preparing
the image. It should type "Done." when the image is ready (of course you
will see the image also). If "Calculating..." disappears, but "Done" does
not come up, it probably means that your computer can not handle the number
of rows you entered. Try to decrease it. (I will try to make it more straightforward
in a future release). You need to use the scrollbars to navigate through
the image. If you scroll it very fast, the image may get corrupted. If
you scroll it back and forward again, it will restore.
Enter the value of the modulus p you want. This may be any integer between
2 and 15000. You might get very interesting pictures for quite big values
of p. Try, 1001, for example. You will be surprised!
Enter the number of rows. This is the number of rows in the triangles that
the program will compute and plot for you. I limited it by 650 since that
seemed to be the limit for my computer (486DX4-100, 16MB).
Click on "SHOW" and enjoy the image. Be patient: Java the way it is now
is pretty slow!
Last updated on 07 March,
Copyright © 1996 by Sergey Butkevich