Pascal's Triangle Images mod N
| Pascal's Triangle mod 2 through 11 | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| |||||||||||
| 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | ||
Let's accelerate the process and expose the patterns by:
- coloring each non-zero number, i.e. 1 = blue, 2 = red, 3 = green, ...
- programming a computer to generate many rows of these colored images.
Above we have generated images of Pascal's triangle mod 2 through mod 11 with 128 rows in each image! You can download these images and enlarge them to examine the patterns in greater detail.
Mathematicians use the patterns in Pascal's triangle in probability, statistics, and general counting problems. Computer scientists and mathematicians use modular arithmetic to allow computers to do arithmetic with extremely large integers, as well as in the construction of highly secure encryption applications.
If you would like to take this to the next level, then you will want to investigate cellular automata for which you will find many references online and in the mathematical literature.