In linear algebra, an orthogonal matrix is an n x n matrix for which its transpose is also its inverse; when mutiplied by its own transpose, an orthogonal matrix yields the identity matrix. The identity matrix basically amounts to the higher-dimensional equivalent of the number one.
Now, every orthogonal matrix I have come up is symmetric; it equals its own transpose. (It might be possible for a non-symmetric matrix to be orthogonal, I don’t know.) But this means that the orthogonal matrix multiplied by its transpose is the same as the orthogonal matrix multiplied by itself. And since when we multiply any of these orthogonal matrices by their transposes we get the identity matrix, what we really have in a symmetric orthogonal matrix is a higher-dimensional square root of the number one.
In one dimension, a number one has only one square root (√1 = 1). But in higher dimensions, there are multiple possible square roots. I wrote a program in Excel that lets me find symmetric orthogonal matrices in 3×3, 4×4, and 5×5 systems. You are welcome to try it out, it is here, and it includes the notes and calculations I made along the way. You will see I also tried plotting out some of these matrices, just to see what they looked like physically.
(The program is automatic; just fill in numbers in the fields in yellow, heeding any instructions in the blue-green fields. The rest takes care of itself.)
I have yet to find a practical use for any of this, but there is something elegant about a matrix that multiplies away into identity. If you know what orthogonal matrices are good for, or if you have any further thoughts on this, then please post a comment.