Pascal's Triangle

In the beginning, there was an infinitely long row of zeroes. And somewhere in the midst of these zeroes there was a lonely 1.

Row 0 (with invisibles) cropped

To this long row was applied a certain rule:

Rule cropped

The figure then looked like this.

Rows 0-1 (with invisibles) cropped

That wasn’t exciting enough, so the rule was applied to the new row that had just been generated.

Rows 0-2 (with invisibles) cropped

Looking better. Now the rule again to the newest row:

Rows 0-3 (with invisibles) cropped

At this point, all those zeroes are getting in the way. So let’s make them invisible.

Rows 0-3 cropped

There, that’s much easier to see. Remember, we haven’t gotten rid of the zeroes; we’ve just hidden them so we can focus on the interesting part.

Continuing the pattern for a few more rows – with each number in the new row being the sum of the two numbers above it – we get:

Rows 0-7 cropped

If we continue this on to infinity, we get a structure known as Pascal’s Triangle.

Rows 0-10 and beyond cropped

This curious construction has some very remarkable properties, as discovered by the French mathematician Blaise Pascal (for whom the triangle is named). Let’s start with the basics…