# Studies in the Geometry of Color

## inspired by log real-imaginary and x-y color maps

The lightness and hue of a single pixel can be used to represent respectively the log[magnitude] and direction of a vector, or the log[amplitude] and phase of a complex number. This allows a single pixel to encode any two floating point numbers between -Infinity and +Infinity, or equivalently any single number in the complex plane. A saturation parameter e.g. between 0 and 1 can also be represented by that same pixel, but since this washes out color differences its utility may be more limited. Applications of this fact so far examined by us include:

In the process some ideas to be described below have emerged for 4-color decorating (look out!), investigating the food preferences of your ancestors, and a more colorful Rubik's Cube®.

A cyclic array of hues based on the three types of color sensors in the human eye looks like...

red, orange, yellow, chartreuse, green, seagreen, cyan, turquoise, blue, indigo, magenta, pink, red, etc.

If butterflies have four types of color sensor, how might their hue cycle differ? Would butterfly computer stores specialize in RGBU instead of RGB monitors? Did you hear about the recent (~May 2005) science news about butterfly usage of their ultraviolet sensors for global navigation?

Then there is the subject of color decorating. Are there elements here for scientific underpinnings of taste in contemporary fashion? From the plot above, you can see that...

The line between red and cyan makes right angles with that between chartreuse and indigo.

The line between blue and yellow is perpendicular to that between pink and seagreen.

The line between green and magenta is likewise orthogonal to that between turquoise and orange.

Each of the sentences above shows a "spanning group" of four colors, while the (approximate) vertical columns show "spanning groups" of three. Of the spanning triads, I vote for indigo, seagreen, and orange (below left). Should this be taken into account when, for example, choosing clothes, decorating a room, or deciding what colors to use in plotting graphs?

Why do each of the orthogonal 4-color sets below look like a theme one might use to decorate a playroom? You may want to bookmark this page if just reading the sentences above makes you cheerful! On the other hand, would a business ensemble in such colors (right below) prevent folks at the office from being able to concentrate on their work?

How about color themes restricted to only one quadrant (e.g. below left)? Might these colors help with someone's sense of focus, or for some might they instead exacerbate feelings of claustrophobia? Which of these single quadrant collections do you like best? Do the colors in one quadrant seem, to you, much more varied and if so why? For example if you're especially good at recognizing color differences in the orange-yellow quadrant (third from left), might that indicate that the ability to detect ripening fruit was an important source of selection pressure in your ancestors' environment?

If visual segregation in the orange-yellow quadrant is useful to fruit-eaters on land, might visual segregation in other quadrants be important to metazoans occupying other niche types, like leaf-eaters, carnivores, and cetaceans? How would you design experiments to check this out? Do you think that the quadrant patterns above match up with the four seasons, and if so how? What else do these various color sets bring to mind?

For each of the basic colors (hues) in the images above, a wide range of lightnesses and saturations are possible as illustrated in the boxes for the basic color orange at right. Another way to put it is that basic colors in patterns above represent only a subset of the colors shown in the color circle at the very top of this page, and these in turn constitute only the outside surface of an RGB color cube that in principle contains all possible colors your computer can display. The contents of that cube are illustrated in the animation below, of surfaces of varying chromaticity (HSL saturation). This parameter "satL" is the parameter mentioned in the opening paragraph above, that washes out color as it goes to zero along the grayscale body-diagonal of the RGB cube. In this and the images below that, the black lines are iso-contours of complex number amplitude and phase in the log color scheme.

In addition to the RGB mapping (far right), two other standard ways to parameterize color space are illustrated below. Note in particular that the HSL scheme (left) maps all of these "complex/coordinate" colors onto a single layer of its hue and lightness manifold, leaving its color saturation or chromaticity manifold (satL) available as a third free parameter that one can store (along with amplitude and phase) in a single RGB image.

The plots above suggest an interesting way to color a Rubik's Cube with "seamless color". For example both of the RGB cubes below are the same, although the one on the right has been scrambled. Solving the scrambled one might require a bit more of one's color imagination than does solving the usual 6-color version of the cube, since each sub-cube face (or sticker) has nearest cousins in all directions. It would be tougher still if we let colors change continuously across each sticker so that center rotations become distinguishable.

Any one up for trying to name each of the colors on these 6×3×3=54 stickers? Note that none are pure red, green, blue, cyan, magenta, yellow, white, or black even though the 8 vertices point in those directions. Minor modification of Mathematica's interactive Rubik Cube model allows one to set up contests with a virtual cube of this sort.

Is it worthwhile trying to market physical cubes with this color pattern? It's possible that a replacement sticker set would be adequate to put almost any cube into this RGB form.

This page is http://www.umsl.edu/~philf/colors.html. Although there are many contributors, the person responsible for errors is P. Fraundorf. This site is hosted by the Department of Physics and Astronomy (and Center for NanoScience) at UM-StL. Mindquilts site page requests ~2000/day approaching a million per year. Requests for a "stat-counter linked subset of pages" since 4/7/2005: .