# Patrick Honner’s Pi Day exercise in 4d part 5: The 120- and 600-cells

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

The first four 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

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

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

So, today we are looking at the 120-cell and the 600-cell. We have studied the 120-cell previously:

A Stellated 120-cell made from our Zometool Set

We’ve also played around with the game Hypernom and experienced the 120-cell first hand!

Using Hypernom to get kids talking about math

However, since we do not currently have a model of the 120-cell anywhere around the house, before starting the project tonight we took a look at two of Henry Segerman’s movies:

(1) Half of a 120-cell

and

(2) Toroidal Half 120-cell

With Segerman’s videos as background, we now calculated what “$\pi$” would be for the 120-cell:

We approached the 600-cell the same way – starting with a video from Henry Segerman:

After that quick introduction we discussed the shape and calculated the value of “$\pi$” for it. Turned out that it was the most 4-d spherical of the shapes that we looked at. That was a fun fact, and thinking about that fact caused my son to ask what a 4-d sphere actually looked like!

Well, I couldn’t end this week-long project with the question about the 4-dimensional sphere hanging in the air. We talked about the shape for about 5 min and then took the dog for a walk 🙂

As we walked up the street at the end of the walk my son turned to me and said:

“Wait a minute, all of the spheres would have to differ by infinitesimal amounts . . . . but, oh, there are infinitely many of them so I guess that’s ok.”

The project couldn’t have ended on a better note! Thanks to Patrick Honner for the great Pi day exercise which inspired this project. Thanks also to Henry Segerman for his videos about the 120- and 600- cells. I hope to own a few more of his 3d prints soon!