[ajkjk]

Erg. It's true that complex numbers model a rotation operator in 2d geometry. But I think it's harmful to say that it's better to think about 2d geometry in terms of complex numbers rather than, well, 2d geometry (vectors and rotation operators/matrices/geometric algebras). This kind of thinking doesn't extend to higher dimensions, obfuscates the actual discovery under the 'magic' of complex numbers, and requires knowing 'one weird trick' to figure out yourself.

(edit: added reference to geometric algebras, which are cleaner than rotation matrices for intuitively modeling this.)

[andars]

I'd say if it depends on how you think about complex numbers. If your understanding of the imaginary unit is limited to "that thing that when you square it you get -1", I agree with you. But if you think of the imaginary unit as the 2-blade in the geometric algebra of R^2, I would argue it is one of the best ways to think about 2D geometry.

[ajkjk]

My point is that almost no one thinks about it that way, and that's the problem. People are content using complex numbers and saying 'it magically' works instead of dealing with the actual structure they want to be using.

[andars]

In that case, I agree with you wholeheartedly.

[ThrustVectoring]

i is a rotation operator in R3 as well - it's just a 3-vector instead of a 1-vector, and textbooks usually call these "quaternions".

If you do want to think of geometry in terms of linear algebra structures, you probably want to be studying Geometric Algebra.

[ajkjk]

ah, I'm aware of those things and have studied them extensively --

My point is, I think it's harmful to say "hey, you - you're trying to solve a problem in 2d space? Have you considered using complex numbers?" When the correct thing to do is "have you considered using the algebra of N-d spaces?" (that is, Clifford/geometric algebra), and leave all the magical complex numbers out of it.

[arh68]

What would that look like, code-wise? Is there a good library for this stuff?

It's easy to reach for complex numbers when they're built-in. Do you have to use Maple/Mathematica for this stuff, or can you do it easily in Python/JS/&c ?

[andars]

If you are interested in this stuff, I have heard that the book Geometric Algebra for Computer Science is good.

You can find the first chapter as a pdf download here: http://geometricalgebra.org/downloads/ga4cs_chapter1.pdf. It contains some code examples, and the complete source code is here: http://geometricalgebra.org/sandbox.html

[jacobolus]

I made a list recently: https://news.ycombinator.com/item?id=12941658

The GACS book is indeed great. The “conformal model” defined there is very pleasant to work with. If you want to learn about it without buying a textbook, there’s also this PhD thesis: http://www2.eng.cam.ac.uk/~rjw57/pdf/r_wareham_pdh_thesis.pd...

[ajkjk]

Well, implementing rotation matrices in code isn't hard, and they should be supported in every mathematical library already.

Even more lightweight than matrices is just implementing the transformation on vector objects yourself: a 90-degree rotation left is R(x,y) = (-y, x), and (-R)(x,y) = -(R(x,y)). In higher dimensions this remains true for any 2-dimension subplane, so R_zx(x,y,z) = (z,y,-x).

Rotation matrices / other formulations become necessary when you're dealing with being able to rotate through any angle.

Though, doing rotations of the matrix 'y = Rx' form suffers mathematical problems like numerical instability and bad behavior. Quaternion rotations or Geometric Algebra's rotors (which are equivalent for N=3, and both take the form y = RxR^-1) work a lot better, and also eliminate a lot of the redundancy in the (skew-symmetric) rotation matrices formulation (4 variables instead of 9, for 3d).

[posterboy]

I think it does extend to higher dimension. I have to read up on hypercomplex numbers, Clifford algebra and geometric algebra. Quaternions are known to facilitate arbitrary rotations in 3D. I suppose, complex numbers are more powerful in analysis rather than discrete mathematics, if there is a difference at all.

[ajkjk]

Quaternions model 3d rotations, and Clifford algebras in general model rotations in arbitrary higher dimensions, but it's definitely non-trivial to go from "complex numbers model 2d" to all of that machinery.

What I mean is that the intuition around complex numbers doesn't extend. Certainly not the "i is the square root of -1"-level understanding of complex numbers.

Clifford algebras are better thought of as structures built over the higher-dimension vectors spaces, with little relationship to algebras/fields. They have much more in common with the matrix representation than the algebraic one. The algebraic perspective only supports complex, quaternion, and octonion numbers before breaking down.