Exploring different L^p versions of the torus

A few weeks ago we did a fun project on L^p spheres after watching Kelsey Houston-Edwards’s video on different ways of measuring distance:

Sharing Kelsey Houston-Edwards’s Pi video with kids

Playing around a little with our 3d printing software last night made me want to try a similar project with a torus in various L^p metrics. I made 5 different ones and set the printer to print them overnight . . . and the print failed. Boo 😦

So, I’m re-printing them to use for a project this afternoon, but for now the project with my younger son just used the shapes on the computer.

Here’s what he thought about the usual torus and the torus in L^1

Next we moved on to looking at the torus in the L^3 and L^5 metrics:

Finally, we looked at some of the shapes when p was not an integer. We looked at p = 0.75, 1.5, and 1.05.

Using the computer program was a nice way to save the project after the print failed. I’m really hoping that the 2nd time is a charm with the print and we can explore the 3d printed shapes this afternoon!