Getting Started

A graph of a real valued function of a real variable fits very nicely on a piece of paper or a computer screen: position along a horizontal x-axis represents the variable and position along a vertical y-axis represents the value. When we move to complex numbers, the (x,y) coordinates of a point in the plane represent the real and imaginary parts of the complex variable. We need to plot at that point something representing the real and imaginary parts of the value of the function. Since we perceive color as a three dimensional quantity, two of those dimensions can be used to represent a complex number.

Color televisions and computer displays make color images by stimulating red, green and blue elements on the screen. The color may be described by indicating the amount of red, green and blue on a scale of 0 to 1, with 0 indicating that the element is not to glow at all, and 1 indicating a maximum stimulation. This is an additive RGB system, which produces black when none of the elements is stimulated and white when all are stimulated maximally. Yellow is produced by equal stimulation of red and green but no blue. A small amount of red combined with an even smaller amount of green produces a brown color.

The creation of colors by combining various colored inks on a white piece of paper is quite different. Here the observed color is the result of reflected light, and each color of pigment placed on the paper absorbs certain colors of light. This is a subtractive CMY system, and uses cyan, magenta, and yellow pigments, which are complementary to the additive red, green, and blue of the computer screen. These colors may also be measured on a scale of 0 to 1, and are related to the computer's red, green, and blue by the formulae: Cyan = 1 - Red; Magenta = 1 - Green; and Yellow = 1 - Blue. Using no pigment leaves the paper white, while maximum pigment of all three colors produces black.

Although images may be reduced to RGB components to display on the computer, and to CMY for printing, another parametrization is more appropriate to establish a correspondence with the complex numbers. In the Hue, Saturation, Value, or HSV color scheme, the parameter Hue is an angle measuring position around the color wheel. Red has a Hue of 0 or 360 degrees; green has Hue 120 degrees, and blue has Hue 240 degrees. Yellow has Hue 60 degrees, half way between red and green. A Hue of 30 degrees corresponds to orange, etc. A 'pure' red or green or orange can be mixed with white to create a pastel shade. As more and more white is mixed in, the original Hue becomes indiscernible. The parameter Saturation measures how far a color is from a white or gray toward a 'pure' color. When Saturation is zero, the color is white, gray or black. When Saturation is one, the color is 'pure' with no white mixed in. 'Pure' colors are composed of at most two of the primary colors red, green, and blue. The parameter Value is also measured on a scale of 0 to 1. Decreasing Value is like turning down the brightness control on a television set or computer monitor. For every Hue and Saturation, Value zero gives black.

It is easy to set up a correspondence between the argument of a complex number and the Hue of a color since both are measured as angles. We use contour lines in Saturation and Value to represent the modulus of the complex value. The contours distinguish moduli between 0 and 1, and between 1 and e, and e and e², etc. Moduli between 0 and 1 (or any of these ranges) are essentially wrapped around a rectangle in saturation-value space with one corner at saturation = 1 and value = 1, and the opposite corner at saturation = 0.3 and value = 0.6.