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The Visible Spectrum
The human eye can perceive electromagnetic waves with wavelengths from approximately 390nm (violet) to 750nm (red). The visible wavelengths of light are collectively referred to as the visible spectrum, and constitue a small part of the total electromagnetic spectrum. From longest to shortest wavelength, the colors are: red, orange, yellow, green, blue, and violet. A chart of the visible spectrum is just a cross section of rainbow that labels the colors with what wavelength of light they produce (see http://www.photo.net/photo/edscott/vis00010.htm). Other animals can see different frequency ranges, but otherwise their visible spectrum is fundamentally no different from ours.
The Color Wheel
You can probably remember seeing a color wheel somewhere back in elementary school - that picture with the primary colors and intermediates you would get if you mixed them together? Technically, if yours had red, yellow, and blue (or magenta, yellow, and cyan) as the primary colors, it was demonstrating subtractive colors - the kind of colors you get by mixing paints together. Light is additive, but the concept is the same; for light, the primary colors are red, green, and blue, and the intermediates are brighter than the primaries because there is more total light present in them. Check out http://home.wanadoo.nl/paulschils/06.00.html for a nice picture of an additive color wheel.
The visible spectrum is laid out in a line, but the color wheel is circular. You can get a fairly good idea of their relationship if you imagine bending the visible spectrum into a circle. The color wheel is useful for observing multiple wavelengths of light simultaneously (what do I get if I mix red and blue?). The color wheel is not representative of the real world - only our perception of it. For many species, our color wheel would be meaningless.
The primary colors are the basis for the color wheel. The whole point of the wheel is that all the other colors on the wheel are made up of varying amounts of just those three colors. However, the primary colors are entirely a product of our perception of light - they are meaningless outside of our eyes. Each primary color of light (red, green, and blue) corresponds to a type of cone in our eyes suited to observe that particular wavelength of light. Red cones are best at 564nm wavelengths, green cones at 534nm, and blue cones at 420nm. Upon seeing red light, the "red" cones are excited and send an appropriate signal to the brain. Yellow light (560nm) is between red and green in wavelength, and will excite both red and green cones - the brain interprets the combined red-green signal as yellow. Your brain cannot tell the difference between a pure single-frequency yellow light, and a combination green/red light that triggers the green and red cones in the same proportion and amplitude.
Computer monitors take advantage of this fact. If you look at a monitor with a magnifying glass, you will see tiny clusters of three dots; red, green, and blue. By adjusting the brightness of each dot, our eyes can be fooled into thinking they are seeing any color of the visible spectrum (and even a few that aren't in the spectrum - keep reading).
So if the primary colors are only a function of the cones in our eyes, could animals have different primary colors? Yes, and in fact they do. The red-sensitive cone is a fairly recent evolution in primates, and many mammals (including dogs and cats) do not have red sensing cones in their eyes. Bees, on the other hand, have extra primary colors, including one in the ultraviolet! The number of possible colors grows exponentially with the number of primary colors available because each color is perceived as a combination of the primaries.
What About Violet?
It is easy to see how mixing red (650nm) and green (535nm) light might make you see yellow (560nm) light, because yellow is somewhere in between red and green. A wavelength between red and green will trigger both the red and green cones, and can be mimicked by separately triggering both. Similarly, any frequency of light between green and blue can be recreated in our eyes by using a combination of pure green and blue light (remember that the light does not actually combine to form an intermediate frequency, it just looks that way to our eyes). But why would combining red light (700nm) with blue light (420nm) make our eyes perceive violet (390nm)?
Each of the three types of cones in our eyes is tuned to a particular wavelength of light (Red: 564nm, Green: 534nm, Blue: 420nm). These are the wavelengths of light each of these cones responds to most strongly, but they are not the only wavelengths they respond to. Green and blue cones have fairly centralized responses around their associated wavelength, but red cones actually respond faintly to wavelengths much shorter than normal red light. The red response declines steadily from 564nm to about 500nm, but then remains constant and even rises slightly below 450nm. This means that although blue light primarily triggers the blue cones, it also excites the red ones slightly.
When we see the blue light, we get a lot of response from the blue cones and a slight response from the red ones. Heading toward violet, the red response remains slight but steady, while the blue response falls off. In this way, the wavelengths below blue make our eyes see proportionately more red, even though the absolute value of red cone response is remaining the same! Green and blue cone response is much more restricted to their respective wavelengths; under violet light, green cones will respond very slightly, but not enough to be noticeable compared to the blue or red cone response.
It is impossible to see "pure" green or blue, because green and blue light will always trigger a slight red cone response. If the optical nerves were directly stimulated, it might be possible to see a color more green than is physically possible; under ordinary circumstances all green light will trigger some red cone response in addition to the main green cone one.
Now what if you have light composed of a lot of red with a little bit of blue - a color like magenta. This is not a color that can be achieved with a single wavelength of light! Take a look at the visible spectrum (http://www.photo.net/photo/edscott/vis00010.htm) - magenta isn't there. Any color between violet and red on the color wheel actually requires two wavelengths of light to create. Of course, white light (when all three types of cones are active) requires multiple wavelengths of light as well.
Also in the visible spectrum, you will notice that at shorter wavelengths than blue the colors turn more and more violet until we can't see them. But once you reach red, our vision just stays red - once our red cones are active and the blue and green cones are not, we have no way of distinguishing 675nm from 700nm red. The longer wavelengths will seem darker the further they get from the red cone's main frequency, but to us this is indistinguishable from just dim lighting.References: