Thread by @johncarlosbaez, A rotation in 4 dimensions is almost the same as a pair [...]

A rotation in 4 dimensions is almost the same as a pair of rotations in 3 dimensions. This is a special fact about 3- and 4-dimensional space that doesn't generalize. It has big implications for physics and topology. Can we understand it intuitively?

(1/n)

Probably not but let's try.

For any rotation in 3d, you can find a line fixed by this rotation. The 2d plane at right angles to this line is mapped to itself.

This line is unique except for the identity rotation (no rotation at all), which fixes *every* line.

(2/n)

For any rotation in 4d, you can find a 2d plane mapped to itself by this rotation. At right angles to this plane is another 2d plane, and this too is mapped to itself.

Each of these planes is rotated by some angle. The angles can be different. They can be anything.

(3/n)

A very special sort of rotation in 4d has both its 2d planes rotated by the same angle. Let me call these rotations "self-dual".

Another very special sort has its 2d planes rotated by opposite angles. Let me call these rotations "anti-self-dual".

(4/n)

Amazingly, self-dual rotations in 4 dimensions form a GROUP!

In other words: if you do one self-dual rotation and then another, the result is a self-dual rotation again.

Not obvious.

Also, any self-dual rotation can be undone by doing some other self-dual rotation.

(5/n)

Similarly, anti-self-dual rotations in 4 dimensions form a group. And this is isomorphic to the group of self-dual rotations.

What is this group like?

Another amazing fact: it's *almost* the group of rotations of THREE-dimensional space.

(6/n)

The group of self-dual (or anti-self-dual) rotations in 4d space is also known as SU(2).

This is not quite the same as the group of rotations in 3d space: it's a "double cover". That is, two elements of SU(2) correspond to each rotation in 3d space.

(7/n)

If I were really good, I could take a self-dual rotation in 4d space, and *see* how it gives a rotation in 3d space. I could describe how this works, without using equations. And you could see why two different self-dual rotations in 4d give the same rotation in 3d.

(8/n)

But I haven't gotten there yet. So far, I only know how to show these things using equations. (They're very pretty if you use quaternions.)

Another amazing fact: you can get *any* rotation in 4d by doing first a self-dual rotation and then an anti-self-dual one.

(9/n)

And another amazing fact: self-dual rotations and anti-self-dual rotations *commute*.

In other words: when you build an arbitrary rotation in 4d by doing a self-dual rotation and an anti-self-dual rotation, it doesn't matter which order you do them in.

(10/n)

All these amazing facts are all summarized in a few equations.

In 3d the group of rotations is called SO(3). In 4d it's called SO(4). The group SU(2) has two special elements called ±1. And we have

SO(3) = SU(2)/±1
SO(4) = (SU(2)×SU(2))/±(1,1)

(11/n)

The equation

SO(3) = SU(2)/±1

says SU(2) is a double cover of the 3d rotation group: every element of SO(3) comes from exactly two elements of SU(2), and both 1 and -1 in SU(2) give the identity rotation (no rotation at all) in 3d.

(12/n)

The equation

SO(4) = (SU(2)×SU(2))/±(1,1)

says every rotation in 4d can be gotten by doing a self-dual rotation and an anti-self-dual rotation, each described by an element of SU(2). It says the self-dual and anti-self-dual rotations commute. And it says a bit more!

(13/n)

The equation

SO(4) = (SU(2)×SU(2))/±(1,1)

also says that each rotation in 4d can be gotten *in two ways* by doing a self-dual rotation and an anti-self-dual rotation. So SU(2)×SU(2) is a double cover of SO(4).

(14/n)

I could prove all this stuff to you using quaternions, but this series is getting too long. What I really want to do now is get better at visualizing this stuff and explaining it more clearly. Maybe you can help me out!

Learn more here:

https://en.wikipedia.org/wiki/Rotations_in_4-dimensional_Euclidean_space

(15/n, n=15)

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