Discussion problem from my math class
We had a pretty interesting discussion in my online math class this week. Here was the prompt from my teacher:
Module 13: Color Planet
People who write about the theory of color often envision the range of visible colors as a geometric "color solid." To explore this concept, imagine we live on a planet of which every cubic inch is a different color. The familiar "color wheel," containing the colors of the rainbow at their highest intensities, is a band around the equator. The North pole is pure white and the South pole is pure black. Let's say pure red is located at the point of 0º latitude and 0º longitude, with yellow at 120º W and blue at 120º E of this point (I am using the simplified red-yellow-blue color wheel we all learned about in grade school–you CMY and RGB sophisticates can keep your comments to yourselves for now!)
Describe what it would be like to travel around this planet. What colors are on the North-South axis through the center of the planet? What color do you think corresponds to the place you were born? What would happen, color-wise, if you moved West from that location? What about if you moved North? What happens if you dig straight down? Can you come up with any general rules relating movement on or within the planet to what color changes you'd experience?
If all visible colors are uniformly distrbuted on and within this planet, what is its shape? Is it spherical like the Earth, or not? Explain your answer.
We'll be talking about both the nature of color perception and the geometrical mapping of space in this topic, so try to check in often throughout this module–there will be many further questions for you to comment on as we explore and develop our understanding of this color planet.
And my response: (after the many other responses to the orginal prompt the question turned to finding specific colors based on Lat./Long.)
Well it's actually pretty simple to "convert" between latitude/longitude and a color value. I'm going to begin in the HSL color system (Hue, Saturation, Lightness) since our globe is already close to that. There are three components to HSL: Hue is the actual pure color from red to blue to green, etc. Saturation is how strong that color is from weak (being gray) to strong (being the full pure color). Lightness is how much white or black is in the color. Any color we can see can be described in the HSL color space.
The first problem is that that HSL has three components and we have only two on our globe. The longitude on our globe corresponds to the hue and the latitude corresponds to lightness. Saturation is not represented on the surface of our globe (notice there is no gray anywhere, everything is a color!) but I'll come back to it later.
Longitude/Hue
Lucky for us both longitude and hue are measured in degrees. Longitude 0° is the prime meridian and we measure 180° east (positive) and 180° west (negative) from there. Hue is measured from 0° to 360° (0° is red, 120° is green, 240° is blue, and 359°cycles back around toward red again). I suggest we just add 180° to the longitude so that it has a range of 0° to 360° instead of -180° to +180°. So,
Hue = Longitude + 180°
Latitude/Lightness
Latitude ranges from -90° (South Pole) to 0° (The Equator) to +90° (North Pole). Lightness is a percentage ranging from 0% (Black) to 50% (Neutral change to the color) to 100% (White). Converting from latitude to lightness is very similar to the formula to change between Celsius and Fahrenheit. To convert the range of 180 units to 100 units we multiply the latitude by 5/9 (5/9 is 100/180 simplified). Then we add 50 because the center points of the two scales are 50 units apart.
Lightness = Latitude * (5/9) + 50%
Elevation/Saturation
At this point we have all we need to define any point on the surface of our globe with a color. Basically saturation on the entire surface is 100%. Inside the core of the globe the color would get grayer, until we get to the core which would be middle gray (0% saturation, 50% lightness, undefined hue) But we'll just stay on the surface and 100% saturation.
HSL to RGB
Most people are more used to dealing with colors as ratios of three primary colors. On computers that'd be Red, Green, and Blue, the additive primaries, instead of Red, Yellow, Blue, the subtractive primaries. I'm going to use RGB because we have a standard system of giving them numerical values, where RYB are used in pigments and paints and are often subjectively mixed without measured amounts. HSL to RGB conversion is a little complicated unless you're a computer programmer but if you're curious you can find the formulas here. Otherwise, you can use an online convertor, like this one.
Problems and Issues to be worked out
The original problem describe a globe with Red, Yellow, and Blue primaries, and my methods don't address this. I haven't found a method online for converting between the two, nor have I found an accepted way that people use to give numerical values to colors in the Red, Yellow, Blue system. Saturation could also be added to this problem as well, but should the core of the globe be 0% saturation or the deepest part of the ocean? Would all of the surface be 100% saturation, or should that just be the top of Mt. Everest with all the other land getting "grayer" as we approach sea-level?
An Example
A spot here in San Francisco would be Latitude: 37.7733, Longitude: -122.4178, which using the formulas above would be 57° Hue and 71% Lightness which is this color (the map on the first page is already calibrated to my system, red, aka 0° hue, is already on the prime meridian over England). Imagine a gps device that showed colors instead of numbers. Crazy.
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