Playing with 4 dimensional shapes using Zometool

We had a 2 hour school delay due to snow this morning – perfect time to . . . .

My goal for this week with the boys is to work thorough Patrick Honner’s Pi day exercise in 4 dimensions. I already did the 3d version with my older son:

Patrick Honner’s Pi Day Exercise

and worked through the 4d version on my own last Monday:

Patrick Honner’s Pi Day Exercise in 4d

The path to getting the boys to understand the 4d project goes something like this:

(1) The 4d versions of volume and surface area for a sphere involve $\pi^2$ rather than $\pi$, so we have to figure out how to modify the formulas in Patrick Honner’s 3d exercise to take this 4d feature into account.

(2) We have to understand some of the simple 4d shapes – I’m looking at the 6 4d platonic solids. Our Zometool set helps us understand those shapes.

By luck, we’ve already done a project on the one Platonic solid that exists in 4d that doesn’t exist in 3d:

Using Matt Parker’s Platonic Solid Video with Kids

(3) Finally, we have to come to understand the “volume” and “surface area” formulas in 4 dimensions. That is the part that I worked through last Monday after finding this awesome website:

Polytopes

Today’s project was mainly about point (2) above.

We started by looking at the 5-cell:

The the 8-cell, or Hypercube:

And finally the 16-cell which is a little more difficult to build and visualize. Luckily we found this helpful website:

The 16-Cell and some relatives

It was nice to have the surprise snow day today to play with these shapes!