The Fourth Dimension

So, I think I've come up with a way to make a neat, efficient representation of a function of complex numbers. The problem with doing so, of course, is that a complex number is kind of two-dimensional, at least if you are thinking in the reals (which most humans are; they are, of course, one-dimensional "over" the complex numbers themselves). For instance, to represent the complex numbers requires a plane, not just a line. A function would need to relate one plane to another, and just like representing a function between two one-dimensional real numbers requires two dimensions, representing a function between two two-dimensional complex numbers requires four. That's the problem: we don't have four dimensions, right?

Actually, we do.
The answer is that not all of the dimensions can be spatial. That's obvious, at least; there aren't four dimensions. But there are other types of dimensions, and I can think of one that it's easy for humans to perceive with the same sense they would use ordinarily to look at a graph of real numbers: color.

One can represent the complex numbers as a sphere of sorts, the projective sphere, where the "north pole" represents zero and the "south pole" represents infinite, or vice-versa. The technique for doing this is not hard: you take the sphere on which you want to project the complex numbers and place it somewhere in space relative to the complex plane, possible with the plane passing through its center though not necessarily. Then you pick the point on the sphere furthest from the plane (or pick one of the two, arbitrarily, if the sphere is bisected by the plane), and draw a line from that point to each point on the complex plane. The point where that line crosses the sphere is identified with the number it hits on the plane; the point you used to draw lines from is identified with infinite.

This is useful for my model because I've always kind of thought of color as a sphere. In Microsoft Word at least, when you're determining color you can choose either the Red/Green/Blue scale or the Hue/Saturation/Luminescence scale. I feel like the latter coordinate system would work very well in a sphere: hue would be the longitude, so to speak, commonly denoted in math as theta, and would vary about the "equator" between the two poles. Latitude, so to speak, would be luminescence, so that one pole would be white and the other black. Saturation is a measure of how grey or intense the color is, so in my model the greyscale line would run down the sphere's axis, and colors would get more intense as they moved away from the middle of the sphere. For me this represents the fact that colors all converge around black and white: it's harder to be a very intense color if you are very, very light or very, very dark than if you're in the middle somewhere.

These are, then, three dimensions of color, that also happen to line up with a sphere. We only need two, but after all the surface of that sphere is more intense than the interior anyway. So my idea is to take a projection of the complex plane onto a sphere, probably the standardized one where the unit circle is the diameter, and associate each point on that sphere's surface with a point on the surface of my color sphere. Then take a complex function, and color each point on the sphere by where it is moved to. So, for instance, if the point (1,0) [better known as "1"] becomes (0,1) [better known as "i"], and if (1,0) is colored red and (0,1) is colored yellow, then the point (1,0) would be colored yellow on the representation of this function. The coloring for the "infinity" point, which by the way is the same infinity whether it is arrived at on the positive real axis, the negative imaginary axis, or any other ray from the origin, would depend on the value of the function as the limit of the function approached infinity. That might not always be the same for each approach to infinity, as it often isn't for real numbers (exponential functions the easiest example since plus and minus infinity are being identified with each other here), so then one color would just abut another color with a sharp boundary line.

I think that this would be a cool way to represent functions of complex numbers, I think it would be reasonably easy to get accustomed to, and I think it would work. I don't know if anyone else has ever thought of it, but I kind of hope they haven't so I can patent it and make a lot of money.