MadSci Network: Physics

Re: How many rotational axes would there be in a 4- or 5-dimensional object?

Date: Mon Sep 25 09:18:16 2006
Posted By: Joe Fitzsimons, Grad student, Quantum and Nanotechnology Theory Group, Department of Materials, Oxford University
Area of science: Physics
ID: 1159015136.Ph


Good question!

Basically rotations are things you do to a vector, and can be described by
square matrices. If we have a two dimensional matrix, then any operator on
it must be a 2x2 matric. All the entries of these matrices are real numbers. 

For the operation to be a rotation, we require that the vector doesn't
change length, and that all vectors orthogonal vectors are mapped on to
orthogonal vectors. Basically this means that the vectors which are at
right angles to each other must still be at right angles after the roataion.

The columns in the matrix correspond to the vectors that the basis vectors
(like [1 0 0]' [0 1 0]' and [0 0 1]') will be mapped to. 

In n dimensions, the restriction that we cannot change the length of a
vector means that we can only choose n-1 entries in the first column of the
nxn matrix.

Since the next column has two restrictions (1. it must maintain the length
of the vector, and 2. that it must be at right angles to the first column),
we can only choose n-2 entries freely.

For column m we have m restrictions, and so can only choose (n-m) entries. 

So we now add up all the free entries in the matrix, and that gives us the
number of rotations we need! So we have (n-1)(n-2)(n-3)...(1) = n(n-1)/2.

For 1 dimension this is 0, for 2d this is 1, for 3d it is 3, so for 4d it
is 6 and for 5d it is 10.

I should point out that in 3d you can actually get any rotation you want
from rotations about only two axes. You can do this by rotating about the X
axis, then the Y axis and then the X axis again. These are called Euler
angles, and I've added a reference for them at the bottom if you want to
learn more.

Hope this helps!


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