## Patrick Honner’s Pi day exercise in 4d – Part 3

We are spending the week working through Patrick Honner’s Pi day exercise in 4 dimensions. The first two parts of our project are

Playing with 4 dimensional shapes using Zometool

Introducing Patrick HOnner’s Pi day idea in 4 dimensions

Also, since I didn’t want to really dive into the “surface volume” and “hyper volume” calculations, this website was critical for today’s project:

Regular Convex Four-Dimensional Polytopes

The main idea today is to calculate “$\pi$” for the first three 4-dimensional platonic solids – the 5-cell, the 8-cell (aka the hypercube), and the 16-cell. A fun twist is that the 5-cell and the 16-cell have some 3d projections that are quite similar, but give quite different values for “$\pi$

So, we started with a super quick review of the 4d formula for $\pi^2$ and then took a look at the 5-cell. Although we didn’t go through the calculation, I liked my son’s guess that the hyper-volume of a 4-dimensional pyramid would be given by (1/4) * volume of base * height.

Next we looked at the 8-cell, or hypercube. Luckily this shape has really easy “surface volumes” and “hyper-volumes.” That allowed us to calculate “$\pi^2$ exactly without too much difficulty – plus we got a little bit of exponent review 🙂

The last shape we looked at today was the 16-cell. This is the most difficult shape to understand, and understanding it is made even more confusing because we have a couple different 3-dimensional projects and they don’t look anything like each other! Also, as noted above, one of them looks a lot like the 5-cell.

It was fun to think about the “spherical-ness” of this shape prior to doing the calculation.

We are really having a lot of fun with this project. Tomorrow we’ll probably focus on the hyperdiamond because it is such a cool shape. Then we’ll talk about the 120-cell and the 600-cell for the grand finale 🙂