Examples of non-commutative rings in nature

Question

So this may sound kinda vague but my point is: mathematics can be applied in several ways to our understanding of nature and often even things that seems to be totally abstract can be, more or less rigorously and more or less naturally, successufully applied. Said that (abstract) algebra too applies to the world around us, so I was wondering if there is some practical examples of structure in our every day life that resembles the one of non commutative rings. Any examples or reference will be greatly appreciated!

Answer 1

The Hamilton quaternions are noncommutative and are routinely used in computer vision to model rotations. That's as close to nature as you can get!

Answer 2

When I think "noncommutative ring", the canonical examples I think of are the $n \times n$ matrices. And phenomena that can be described by matrices are everywhere, including not only "nature" (whatever that means) but also all other aspects of "everyday life".