Let me give fair warning: We are about to do some physics, and we will have to use a little mathematics in the process. This section of the Features of Light and Darkness help file isn't so much for people who know physics as for people who might have known a little of it once and can remember it with prompting. It does take a little physics to understand how the app works, though, and that in turn takes a little mathematics. So here goes.

Recall that there are many wavelengths of light, some that we can see and some that we cannot. Light of any one visible wavelength will appear to have its own particular color, different from any other. What humans usually think of as a single color or kind of light is most often composed of many different wavelengths of light mixed together in varying proportions. Simple optical gadgets like a prism or a piece of diffraction grating can separate the mixture of colors: If you pass ordinary white light through a prism, what comes out is a long streak of colored light, that is red at one end and violet at the other. Such a streak of light is called a spectrum (the plural is spectra), or we speak of the spectrum of a source of light. An ordinary rainbow is a spectrum that occurs in nature: Droplets of water, acting like myriads of tiny prisms, break up sunlight into the familiar curved, colored band.

If you pass light from some other source through a prism, you may see a streak of light in which the brightness obviously varies from place to place, or perhaps just a few spots of different colors. If you have a prism, try it on a beam of light from a fluorescent light or a light-emitting diode (LED). These examples show that the intensity of light in a spectrum may vary with wavelength.

When Features of Light and Darkness deals with light, whether as a light source or as light reflected back at you from a colored pigment, it has a whole spectrum in mind. You can see those spectra if you press the buttons for light sources and pigments. They are stylized: The shape of the curve that forms the upper edge of each spectrum indicates how bright the light is for a particular wavelength -- and that is what counts -- but the area under the curve has been filled in with a colored rainbow to indicate the colors of light at various wavelengths. The range of wavelengths shown is from 380 nanometers at the blue ends of the spectra through 700 nanometers at the red ends.

The spectra for light sources indicate the actual relative brightnesses of different wavelengths of light. Thus the spectrum "Neutral White" has all wavelengths equally intense, while the spectrum "Display White" shows the proportions of different wavelengths that a typical liquid-crystal display (LCD) emits when it is set to create white light.

The spectra for pigments indicate what proportion of light of any given wavelength reflects back from a thick layer of the pigment. That may be a little different from what you might have expected, because pigments are most often described by how much light they transmit, not how much light they reflect, and Features of Light and Darkness makes no attempt to deal with transmitted light. You can think of the pigments in Features of Light and Darkness as representing thick acrylic or oil paint daubed clumsily on a surface. Thus if you were to shine "Display White" light on pigment "Simple Green", what would reflect back would be a lot of green light and a lot less of redder and bluer wavelengths, so the layer of pigment would look green.

To be pedantic, the pigment spectra in Features of Light and Darkness are the coefficients of reflection of the pigments in question, as functions of wavelength. The comment in the last paragraph about pigments often being described by the light they transmit means that the spectra given for pigments are more often coefficients of absorption, showing how much light is absorbed by a layer of pigment of a certain thickness. Features of Light and Darkness does not use that latter kind of spectra.

There are four special spectra among the pigments of Features of Light and Darkness, that are not coefficients of reflection; namely, "Cone Cell -- Blue", "Cone Cell -- Green", "Cone Cell -- Red", and "Rod Cell". These spectra represent the sensitivity of the different kinds of light-detecting cells in the retinas of your eyes. Thus the rod cells of your eye are most sensitive to a wavelength near the border between blue and green light: They see that wavelength of light best. Such spectra might technically be called coefficients of spectral response or coefficients of spectral sensitivity, again as functions of wavelength. I put them there pretty much just for reference.

Spectra can be added and multiplied. The result is a whole new spectrum. For addition, the intensity of the new spectrum at any given wavelength is the sum of the intensities of the two spectra that were added, at that same wavelength. For multiplication, the intensity of the new spectrum at any given wavelength is the product of the intensities of the two spectra that were multiplied, at that same wavelength. When you combine lights, their spectra add. When light reflects off a layer of pigment, the spectra that results is the product of the spectrum of the light itself and the spectrum that represents the coefficients of reflection of the pigment.

Sometimes it is important to know the total area under the curve that represents a spectrum. The technical name for that area is the definite integral of the spectrum, and the word "integral" often causes trembling and panic among people who have bad memories of studying calculus, but have no fear -- all we need here is the idea; there will be no equations and we won't have a quiz on it tomorrow, or even next week. The integral of a spectrum of light represents the total amount of light, or perhaps the total light energy, that is present, but Features of Light and Darkness has an even more important use for integrals of spectra.

If we multiply two spectra and then take the integral of the result, we have obtained what mathematicians call the convolution of the two spectra. Spectral convolutions are very important in Features of Light and Darkness, because they describe the response of the eye to light, and that is what Features of Light and Darkness is all about.

To get a number that represents the response of one kind of the eye's light-detecting cells when light falls on the retina, we do the following: We multiply the spectrum of the light in question by the spectrum that represents the coefficients of spectral sensitivity for that kind of light-detecting cell, then take the definite integral of the result. That is, we calculate the convolution of the spectrum of the light with the spectral sensitivity of the cell. The idea is really simple: The light is composed of a mixture of tiny amounts of light of a vast number of different wavelengths; for each wavelength, we calculate how much the light-sensitive cells respond to the amount of light of that wavelength that is present, and then we add up the results.

When we convolve the spectrum of light that hits a part of your retina with the spectral sensitivities of the three kinds of cone cells and of the rod cells, we get four numbers that represent the responses of those four different kinds of cells. Those numbers are the starting point for color vision. They are all the information your brain gets from your eye about the colors of light that you are seeing. All the complexity of the shape of the spectral curve, with a different intensity for every wavelength, is boiled down into just four numbers.

Most of the time, things are even simpler. It turns out that in bright light, the rod cells are turned off, or at least their response is overwhelmed by the response of the three kinds of cone cells. In bright light, the response of the rod cells is effectively zero, and your brain only knows three numbers about each color of light that it sees; namely, the response of the red, green and blue cone cells. In bright light, color vision is a three-number system.

In extremely dim light, on the other hand, the cone cells do not work well, and their response is effectively zero. Thus in extremely dim light, only the rod cells are working, and color vision is a one-number system. That means that in extremely dim light, you cannot see color. More strictly, in extremely dim light, you can only see one color, which your brain interprets as different shades of gray, according to the intensity of the light. The fact that only the rod cells are working in extremely dim light is the reason why "At night, all cats are gray."

Yet there is a range of brightness of light in which the cone cells are still working, but in which the light is so dim that they no longer overwhelm the rod cells. In this region of brightness, which corresponds roughly to what we humans call "twilight", color vision is a four-number system: The three kinds of cone cells and the rod cells are all capable of providing responses to the brain that are all noticeably different from zero, and those responses vary according to the spectrum of light that is falling upon the retina. Color vision is particularly interesting when that happens, as Features of Light and Darkness attempts to show.

I can't quite let myself make so many simple statements about such a complicated matter without adding a few caveats. As I have said elsewhere herein, biology is full of exceptions: Color vision is complicated and not well understood, and new information about it turns up regularly, so I am sure that the description of color vision that I have just given is not quite complete and not quite correct. Nevertheless, it will do to explain and present a lot of interesting material, as we shall see in Features of Light and Darkness itself, and as we shall discuss in other sections of this help document.

Okay, class, that is all for today. Remember, pop quiz tomorrow, so get busy studying those equations.

No, wait! STOP!! I am just kidding! No quiz! No equations! Just kidding! Come back!! ...