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?

  • $\begingroup$ A link to the code of the simulation ? anyone, please. $\endgroup$ – Helder Velez Feb 11 '15 at 6:50

The "math works it out" indeed. I try to write it as accessible as I can.

  • A 0-domensional Euclidean space is just a point.
  • The 1-dimensional is a line.
  • The 2-dimensional is a plane.
  • The 3-dimensional is the space as we know it.
  • This can be continued to 4-5-6 whatever dimensions.

On a plane you can draw lines and point, but not planes. In space you can have many planes, lines and points. So you can embed Euclidean subspace that has fewer dimensions than the Euclidean space you embed it into.

Also if you draw two lines on a sheet of paper they must intersect at a point (even parallel lines considered to intersect infinitely away). In three dimensions a plane and a line intersects at least in a point. Two planes intersect at least along a line. $m$ dimensional and and $n$ dimension Euclidean subspace embedded into a $d$ dimensional Euclidean space intersect in a subspace that has at least $m+n-d$ dimensions.

  • So line and line in a plane: $1+1-2 = 0$ - at least a point.
  • So plane and plane in space: $2+2-3 = 1$ - at least a line.
  • Two lines in space: $1+1-3 = -1$ - they don't need to intersect at all.

This wasn't a rigorous description but you can work it out using n dimensional vectors too.

Now back to physics: if you have two rotating disks around the same point they will inevitably intersect along a line. If they are two disks of particles the collisions will smooth out the differences in the angular momentums and you'll have a single rotating disk after several revolutions.

In 4 dimensions two planes must intersect only at a single point. So you can have two disks that are revolving around the same point but the particles in it doesn't interfere with each other: the disks can remain stable for long time. This means the cloud won't collapse into disks.

But this assumes that rotation in 4D is a circular motion along a plane. If angular momentum is also a vector in 4D than some unimaginable spatial rotation is possible around it. Planar rotation is only possible if the angular momentum is defined by two vectors - a bivector. I'm not sure which is the correct.


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