I saw an incredible tweet from John Carlos Baez last week:

It's fun to think about higher-dimensional cubes. But the great mathematician Hilbert took it further! He studied an INFINITE-DIMENSIONAL cube: the "Hilbert cube".

A point in the Hilbert cube is an infinite list of numbers between 0 and 1.

Here’s the picture in case the tweet isn’t embedding all that well for you:

I thought exploring some of these shapes would make a great project for kids, so I began by asking my son (in 7th grade) for his thoughts on the shapes he was seeing:

Next we built a few of the cubes from our Zometool set and talked about some of the shapes. First, though, I asked my son to give his definition of what an n-dimensional cube was:

Finally, we played around by a different version of a 4-dimensional cube – “Hypercube B” by Bathsheba Grossman. This amazing version of the hypercube makes amazing shadows and my son was able to find a projection that was a little closer to the projection of the 4d cube in Baez’s tweet:

Also, here’s a video I made a while back showing some other (almost freaky) 2d projections from our Zometool model of Hypercube B:

Definitely a fun project – thanks to John Carlos Baez for sharing some of his ideas about higher dimensional cubes on twitter!