a, 5). "If we extend Pascal's triangle to infinitely many rows, and reduce the scale of our picture in half each time that we double the number of rows, then the resulting design is called self-similar -- that is, our picture can be reproduced by taking an subtriangle and magnifying it," Granville notes.The pattern becomes more evident if the numbers are put in cells and the cells colored according to whether the number is 1 or 0 (Peterson's, 5).Similar, though more complicated designs appear if one replaces each number of the triangle with the remainder after dividing that number by 3. So, I get: 1 1 1 1 2 1 1 0 0 1 1 1 0 1 1 1 2 1 1 2 1 1 0 0 2 0 0 1 This time, one would need three different colors to reveal the patterns of triangles embedded in the array. One can also try other prime numbers as the divisor (or modulus), again writing down only the remainders in each position (Freedman, 5). Actually, there's a simpler way to try this out. With the help of Jonathan Borwein of Simon Fraser University in Burnaby, British Columbia, and his colleagues, Granville has created a "Pascal's Triangle Interface" on the web. One can specify the number of rows (up to 100), the modulus (from 2 to 16), and the image size to get a colorful rendering of the requested form.It's a neat way to explore the fractal side of Pascal's triangle. Here's one example that I tried out, using 5 as the modulus (Petetson's, 5). ...
