Walking Paths with Complex Numbers

https://dannythecoder.wordpress.com/2016/12/02/walking-paths-with-complex-numbers/

by dannybirch • 10 years ago

86 21 10 years ago

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deckar01• 10 years ago

Here is the pseudo-code for matrix rotation for comparison:

  def rotate(vector, angle):
    rotation_matrix = [
      [cos(angle), -sin(angle)],
      [sin(angle), cos(angle)]
    ]
    return rotation_matrix * vector
  
  position = (0, 0)
  velocity = (0, 1)
  for instruction in [x.strip() for x in input.split(",")]:
    direction = instruction[0]
    distance = int(instruction[1:])
    angle = 90 if direction == 'L' else -90
    velocity = rotate(velocity, angle)
    position += velocity * distance
  print position

andars4• 10 years ago

Since you only work with 90 degrees, you don't even need the full machinery. In my solution I used (a,b) -> (-b,a) and its dual.

ajkjk5• 10 years ago

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.)

ThrustVectoring2• 10 years ago

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.

ajkjk5• 10 years ago

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.

posterboy3• 10 years ago

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.

ajkjk5• 10 years ago

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.

andars4• 10 years ago

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.

ajkjk5• 10 years ago

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.

posterboy3• 10 years ago

I still don't get the significance of complex numbers, to me they seem like vectors with certain operations and I still suppose there is no deeper revelation, that sqrt(-1) represents, and every example could have a geometric interpretation, even if this isn't motivated by the example. Likewise, the only explanation for e^(pi*i)-1=0 I know is the conversion to polar coordinates.

Since you need to keep track of your current orientation this slightly complicates things a bit
If at each step we keep the imaginary multiplication term from the previous step we can effectively encode the rotation in our expression

Why, you could just keep track of the rotation as an angle, which would likewise be a result of the previous step. In binary there wouldn't even be much of a difference: Look at a 2-bit value either as a modulo 4 ring of type pi/2, or as a modulo 2 vector of the two complex components. That's my first thought, at least. Although, the real and imaginary parts would always have opposite values, so only one bit is needed. I wonder if complexity analysis would show a difference.

ThrustVectoring2• 10 years ago

The best non-geometric interpretation of complex numbers is algebraic closure, IMO. Basically, mathematicians invent new kinds of numbers when they want to do things and have the result of the operation always be a number. Adding and multiplying natural numbers gets you a natural number. Subtraction on integers yields integers. Division for rational numbers. Infinite series for real numbers. And factoring all polynomials is the raison d'etre for complex numbers.

pdkl95• 10 years ago

(first, I second ThrustVectoring's comment about algebraic closure. That's a key part of understanding complex numbers)

I recommend reading this[1] article. I found it very enlightening about why complex numbers are useful.

[1] https://acko.net/blog/how-to-fold-a-julia-fractal/

edit: Warning - [1] requires WebGL.

posterboy3• 10 years ago

e^(pi*i)+1=0

fixed, otherwise e^pi would be trivially wrong

jacobolus2• 10 years ago

A complex number is a “complex” consisting of a scalar part (“real” or “parallel” number) and a bivector part (“imaginary” or “perpendicular” number) complected together into one single object.

IMO the best way to think of it is as the quotient or ratio between two arbitrary vectors T = v/u in a Euclidean vector space. Or equivalently, as a type of mathematical operator which you can apply (by multiplication on the left) to one vector u to produce another vector v, i.e. Tu = (v/u)u = v. This operator T will scale and rotate any other vector in the plane spanned by your two vectors u and v in the same way.

http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf http://geocalc.clas.asu.edu/pdf/GrassmannsVision.pdf

jstewartmobile• 10 years ago

Good stuff! Even though this may not be the best application for complex numbers, it is a very simple example for a practical use. For some person out there, this may even serve as a "gateway drug" to complex analysis.

rosstex• 10 years ago

https://acko.net/blog/how-to-fold-a-julia-fractal/

Walking Paths with Complex Numbers

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