Moduli of Pascal's Triangle

An exploration of some of the properties of Pascal’s triangle.

Pascal’s triangle has multiple properties, one of which being that the numbers that it comprises present certain structures. The Macromedia Flash animation shown here outlines some of these structures by calculating Pascal’s triangle and taking the modulus of each entry of the triangle by a user-selected number. Each number thus obtained is represented by a grayscale pixel whose amount of white reflects how close the number is to zero—that is, a smaller number is whiter than a high number.

Macromedia Flash player required.

The grayscale interval in RGB spanning up to 256, the user is allowed to select numbers from 2 to 256. The grayscale interval is divided into this user-selected number to allow for maximum visibility of all colours.

Notice that pattern 2 is Sierpinski’s triangle, and that all other patterns obtained from prime numbers are a “scaled up” version of Sierpinski’s triangle, whose smaller triangles present interesting patterns themselves. Also notice that the height of the smaller triangle in a pattern obtained from a prime number is that exact prime number, and that a bigger blank triangle is obtained at the square of that particular prime number. Also, patterns obtained from numbers which are not prime are “interference” patterns of the prime numbers which make up the selected number.

Plaintext source of this program, released under the GPL.