# What if Flatland wasn’t a plane

I had a couple of requests to bring the squircles to a New Year’s Eve party:

I threw a couple of other 3d prints into the bag, too, because I thought the kids there would like more than just squircles. The 4d shapes generated a lot of conversation. Bathsheba Grossman’s “Hypercube B” in particular:

Those conversations got me thinking about common ways of understanding higher dimensional shapes if you live in a lower dimensional world. The sphere passing through a plane generating a series of larger and then smaller circles is one common way of explaining how a 2d being could understand what a sphere looks like. So, today we discussed what a sphere passing through two other 2D shapes would look like.

We started by talking about the problem of a sphere passing through the plane and then briefly talked about what we thought a sphere passing through a “v” shaped piece of paper (a bent plane) would look like:

Next we went to Mathematica to see the shapes for ourselves. The first set of shapes we explored was the sphere passing through a “V”. I should have published these videos with the computer in much higher quality – sorry that the computer text is basically impossible to read.

For the sphere passing through the “V” we see shapes that look nothing at all like normal slices of spheres:

The next set of shapes we looked at was a sphere passing through a cone. In this video the sphere is centered directly above the point of the cone:

Finally we looked at how the shapes from the previous video change if the sphere is not centered on the point of the cone. Here you get some pretty strange shapes. I think if you encountered these shapes in the wild it would be really hard to imagine that they would assemble into a sphere!

So, a fun project to start the new year. Turns out there are lots and lots of different “slices” of spheres passing through flat, 2d shapes!