While staying late at school on a Friday, I decided to put off some work I had planned to do before the weekend and played around with the holiday candy I received as a gift from the Parent’s Association.
Here’s a picture of Pascal’s Triangle where each black cell represents an even element and where each white cell represents an odd element. Pascal’s Triangle begins to look a lot like the Sierpinski Triangle!
It’s especially interesting to see where the rows of all odd elements are.
Below are representations of Pascal’s Triangle highlighting different multiples (3 through 8). All of them except for the multiples of 7 seem to have the potential to show some fractal-like properties.
Google SketchUp is a very fun piece of software to play around with. At my school, it’s used in computer classes for the middle school grades. I personally don’t know much beyond the basics, but I used SketchUp here to create another visualization of Pascal’s Triangle, this time in 3D:
The height of each tower represents the corresponding number in Pascal’s Triangle. The scale is 1 to 1 foot so the tallest tower among the first eleven rows is 252 feet high!
In order to represent more rows of Pascal’s Triangle using shades of gray, I have to scale the elements in some way. Using linear transformations (division, subtraction) doesn’t work too well since (a) the values in the triangle increase much faster and (b) it creates huge swaths of white or near-white which isn’t visually appealing.
Since the largest value on each row increases somewhat exponentially, I decided to scale every number in the triangle using a base 2 logarithm.
Using each scaled value to pick a shade of gray creates a much smoother transition, and 32 rows are shown before we get to a pure black.
I thought the borders of each square were somewhat distracting so I made another image where they are much more subtle:
Pascal’s Triangle has a recurring role in almost any class I’ve ever taught or will teach. Recently I’ve been thinking about different ways to represent the triangle for no other reason than the possibility that it might be artistically interesting.
Here’s my first attempt at a visualization. The exact shade of gray in each square represents the corresponding value in the triangle.
I was struck by how I could only adequately represent the first eleven rows of the triangle in this manner. Computer monitors typically have 256 shades of gray, and the largest number in the eleventh row is 252.