New subject: [math-fun] More than you wanted to know about rotations of 4-dimensional space

4 Nov 2020

      I've been reading the Wikipedia article about rotations in
four dimensional space, and it has some interesting differences
from any other dimension.

The group of all rotations of R^n is the "special orthogonal 
group" denoted as SO(n). It's a Lie group of dimension
equal to (n-1) + (n-1) + ... + 1  =  n((n-1)/2.

SO(1) and SO(2) are respectively the group of order 2 and 
the circle group. 

For n ≥ 3, n ≠ 4:
-----------------
* If n is odd, n = 3, 5, 7, ...: SO(n) is a simple group*.  

* If n is even, n = 6, 8, 10, ...: SO(n)/{-I,I} is simple. 
Here {-I, I} is the group generated by the central inversion -I, 
and this group of order 2 is the center of SO(n) (the normal 
subgroup of the elements that commute with the entire group).
-----------------

But for SO(4) the situation is unique.

Let S^3 denote the unit sphere in 4-space, i.e., the unit quaternions.

Then any rotation of R^4 (a map 

   h : R^4 —> R^4 

that preserves all distances and orientation) can be described as

   x  —>  p x q

for some p, q in S^3, where x is any element of R^4 thought of
as a quaternion. Given h, the ordered pair (p, q) is almost
unique: the only other pair also describing h is (-p, -q).

(Because quaternions are associative, no parentheses are needed
in x —> p x q.)

An "isoclinic" rotation of R^4 is one where all nonzero points 
rotate by the same angle.  It turns out that all isoclinic
rotations can be described by either

   x —> p x

(a left isoclinic rotation), or

   x —> x q

(a right isoclinic rotation). The left isoclinic rotations
form a group, isomorphic with S^3, as do also the right isoclinic
rotations.  And each of these groups is a *normal subgroup* of SO(4).

Just like SO(n) with n even, n ≥ 6, the center of SO(4) is {-I, I}.

But S(4)/{-I, I} is *not* a simple group. It is isomorphic to the
cartesian product SO(3)×SO(3), and this is not simple because
SO(3)×{I} and {I}×SO(3) are each a nontrivial closed normal subgroup.

More at <https://en.wikipedia.org/wiki/Rotations_in_4-dimensional_Euclidean_space>.

—Dan
—————
* SO(n) is a group and a locally Euclidean topological space such that
multiplication and inversion are continuous: i.e., it is a Lie group.
Saying that a Lie group is "simple" means it has no nontrivial closed 
normal subgroup.

[math-fun] More than you wanted to know about rotations of 4-dimensional space

Dan Asimov

Adam P. Goucher

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