I just watched that interesting video: https://www.youtube.com/watch?v=tmNXKqeUtJM
In 2 dimensions a cloud of particles rotating in a plane is flat by definition since it's in 2 dimensions.
But in 3 dimensions even though the rotation of the cloud is given by one plane, particles can go around far up and down from that plane. As the particles bump into each others all the up and down motion tends to cancel out, its energy lost in crashing and clumping. Yet the whole mass must continue spinning inexorably because in our universe the total amount of spinning in any isolated system always stays the same. So over time through collisions and crashes the cloud loses its loft and flattens into a spinning roughly 2 dimensional disk shape, like a stellar system or a spiral galaxy.
However in 4 spatial dimensions the math works out such that there can be two separate and complementary planes of rotation. Which is both really really hard for our 3D thinking brains to picture, and also means there is no up and down direction in which particles lose energy by collisions. So a cloud of particles can continue being just that, a cloud.
So I'm wondering what does that all mean? Why can there be "two separate and complementary planes of rotation" in 4D? And I noticed the guy in the video used the word "can", does that mean that there are other possibilities? And what about in 5D or 6D?