Pascal's Triangle and Tetrahedrons

Here's Pascal's Triangle as we left it yesterday. Comparing to the one I posted yesterday, it looks like we made a few mistakes in the bottom two rows. Oops! I think the infinity signs on the right are there to show that the triangle can keep going on and on forever.

Today we started with square numbers, so called because they're the numbers of dots you can arrange into squares of increasing sizes (see upper right corner of chalkboard). There are also "triangular numbers," which are the numbers of dots you can arrange into equilateral triangles of increasing sizes (see middle right chalkboard).

The diagonal of Pascal's triangle that contains the numbers 1, 3, 6, 10, 15, etc. consists of the triangular numbers.

Pascal's triangle also contains "tetrahedral numbers." To make sense of this we had to know what a tetrahedron is. First we tried to make one out of ourselves:

Since our building pieces (the kids) were different heights, this didn't exactly work. I started to build a tetrahedron using pencils, and then the kids brought out the zometools. I had never played with these before. Now I want buckets of zometools. So much fun!

We all built tetrahedra, myself included.

Lydia built a tetrahedron tree.

Saul made a very large tetrahedron and filled it with the little white zomespheres. This is where the "tetrahedral numbers" come in - by counting up the number of spheres it takes to make tetrahedra of increasing sizes.

He very kindly let Noah put on the last one:

The tetrahedral numbers are 1, 4, 10, ... which just happen to be the next diagonal in Pascal's Triangle. We have the natural numbers, the triangular numbers, and the tetrahedral numbers:

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

1 8 28 56 70 56 28 8 1

1 9 36 84 126 126 84 36 9 1

1 10 45 120 210 252 210 120 45 10 1

1 11 55 165 330 462 462 300 165 55 11

When Saul counted up all the zomespheres he stacked inside his tetrahedron, he got 165, which is indeed one of the tetrahedral numbers. No zomespheres were lost in the counting of this tetrahedral number.