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This is a follow-on from this earlier question, which asked for examples of simple rotation matrices.

I'm interested in rotation matrices whose entries are simple rational numbers, because these are easy to use in hand calculations, using nothing more than your brain and a pencil.

So, let's denote by $Q_n$ the set of rational numbers whose numerator and denominator are at most $n$. For small values of $n$ (up to around 15, maybe), I'd like to know how many different $3 \times 3$ rotation matrices there are with entries in $Q_n$, and I'd like to have some systematic process for generating these matrices.

The answers to the earlier question give some good techniques for generating nice simple rational matrices. But it's not clear (to me, anyway) that all the desired matrices can be generated by those techniques.

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  • Rotation matrices are matrices made out of orthogonal column vectors, of length $1$, with a determinant of $+1$. To enumerate them you would first have to enumerate rational points on a sphere, which is not a simple problem. – Karolis Juodelė Dec 15 '13 at 10:24
  • True. But I'm not asking for all rational points on the sphere, only the ones with small denominators. – bubba Dec 15 '13 at 10:38
  • Is enumerating rational points on a sphere sufficient? In other words, will this enumeration process generate all the rotation matrices I want, or only some subset of them. – bubba Dec 15 '13 at 10:40
  • Well, yes, what you need is actually enumeration of integers on a sphere (multiply both sides by $n!$). And, no, this is not sufficient. I suppose you should take $\vec{a}$ from enumeration of sphere, then build $\vec{b}$ using enumeration of circle, perpendicular to $\vec a$ and finally $\vec c = \vec a \times \vec b$ where $\vec a, \vec b, \vec c$ are columns of the rotation matrix. I'm not certain about any duplicates though. – Karolis Juodelė Dec 15 '13 at 11:48
  • Incidentally, since the numerators of the rational numbers are always going to be less than the denominators, you can simplify to 'denominator at most $n$'; the generic word that people generally use for a valuation like this is the height of the rational number. – Steven Stadnicki Dec 18 '13 at 20:28

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