# Patrick Honner’s Pi Day exercise in 4d day 4: The Hyperdiamond

This is the 4th (of 5) in series of 4 dimensional explorations inspired by Patrick Honner’s Pi Day exercise:

The first three parts in the series are here:

Playing with 4 dimensional shapes using Zometool

Introducing Patrick Honner’s Pi day idea in 4 dimensions

Patrick Honner’s Pi day exercise in 4d: part 3

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

Today we looked at the 24 cell – aka the Hyperdiamond. We’ve looked at the shape previously after seeing this great video from Matt Parker:

Using Matt Parker’s Platonic Solid video with kids

Unfortunately my older son didn’t remember the previous exercise – ha! Oh well, luckily we started with a quick review of the hyperdiamond and the rhombic dodecahedron:

After that quick talk about the 24-cell we returned to our whiteboard to talk about the value of “$\latex \pi$” for this shape. The “surface volume” and the “hyper volume” for the 24-cell turn out to be fairly simple numbers, and that but of luck gives us an easy value of “$\pi$ for this shape!

So, one last project tomorrow. We’ll look at the 120-cell and the 600-cell. Can’t wait!