4-Dimensional Cross-Polytope

When bad news gets me down, I often get insomnia. I wake up in the middle of the night, start thinking about how we’re all doomed, and can’t easily stop. To break out of these doom loops, I do elaborate visualization exercises. They don’t really put me to sleep, they just calm me down. Then later I can fall asleep.

Here’s what I’ve been doing this week. I visualize this shape made of two interpenetrating tetrahedra, called the ‘stella octangula’ or ‘stellated octahedron’. Then I imagine the yellow tetrahedron moving ‘up’ into the 4th dimension while the red one moves ‘down’.

At some point the distance between each corner of the yellow tetrahedron to the 3 nearest corners of the red one equals the distance between any 2 corners of the yellow tetrahedron. Then I’ve got a 4d shape called the ‘cross-polytope’. All its faces are regular tetrahedra.

There are easier ways to think about the cross-polytope, which is one of the six 4-dimensional regular polytopes. So the real challenge is to visualize how this way of getting it leads to the same result.

My go-to way to think about the cross-polytope is to imagine the 4 coordinate axes in 4-dimensional space and put two dots on each axis, one unit away from the origin in each direction:

These are the vertices of a cross-polytope. It’s the 4d analogue of an octahedron. Just as the octahedron has equilateral triangles as faces, this guy has regular tetrahedra as faces. Can you see those tetrahedra in this picture, and count them? Don’t worry—if you’re too busy now, you can do it when you’re lying in bed at 3 am thinking about global warming and the decline of democracy. Start by visualizing this picture:

But the hard part is to visualize the cross-polytope this way and then rotate it in your mind so you see it as made of two dual tetrahedra, one red and one yellow, and an edge connecting each vertex of the red one to the 3 nearest vertices of the yellow one. That’s been keeping me busy all week.

By the way, everything I just said has a 3d version! The 3d analogue of the cross-polytope is a regular octahedron. The corners of a regular octahedron are

But here’s a flattened picture of an octahedron:

See the two interpenetrating equilateral triangles? If you move one up, and move one down, they can become two opposite faces of a regular octahedron.

Actually this sort of trick works in any dimension. Take two regular n-simplexes, dual to each other and interpenetrating; then move one ‘up’ into the (n+1)st dimension and the other ‘down’. At some point their vertices will be the vertices of an (n+1)-dimensional cross-polytope. In 3 dimensions this is easy for me to visualize, while in 4 dimensions I can just barely visualize it… though it’s getting easier every night.