On this topic of 4-dimensional rotations, I thoroughly recommend: 'On Quaternions and Octonions' by J. H. Conway and D. A. Smith. Amongst other things, it includes a classification of all finite subgroups of SO(4). -- APG
Sent: Wednesday, November 04, 2020 at 6:47 PM From: "Dan Asimov" <dasimov@earthlink.net> To: "math-fun" <math-fun@mailman.xmission.com> Subject: [math-fun] More than you wanted to know about rotations of 4-dimensional space
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.
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