Playing with Pascal’s triangle and angles hidden in cubes

In April 2018 I saw a great Numberphile video with Federico Ardila:

The project that we did after seeing that video is here:

Federico Ardila’s Combinatorics aand Higher Dimensions video is incredible

This week my younger son was learning about coordinates in 3 dimensions in his precalculus book and I though it would be fun to revisit some of the ideas about cubes from Ardila’s video.

We started by looking at cubes in 0 to 4 dimensions and discussing how we could see Pascal’s triangle hiding in the cubes:

In the last video we got a little hung up on the 4-dimensonal cube, so for the next part of the project we looked at the coordinates of the vertices of the various cubes to see if that could help us see Pascal’s triangle in the 4d cube.

Next we moved on to looking at the angles made by the long diagonal in the various cubes. This exercise was particularly nice since my younger son has been learning a little trig and my older son has been learning a bit of linear algebra.

For the final part of the project we looked at the 4-d cube. Here are zometool shape isn’t really helping us see the long diagonal. My younger son did a really great job seeing the pattern in the right triangles with the long diagonal. He also noticed the amazing fact that there is a 30-60-90 triangle hiding in a hypercube!

My older son was also able to find the same angle using ideas from linear algebra:

Definitely a fun project. It is fun to introduce coordinates not just for 3 dimensions, but for all dimensions at the same time. There’s also an enormous amount of fun math hiding in the seemingly simple idea of n-dimensional cubes, which makes this project sort of doubly fun!

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