A way of arranging the colors. Used in art. The color wheel looks like this: (represented using abbreviations for the colors)

         Y
     YG / \ YO
    G  /   \  O
   BG /     \ RO
     B-------R
      BV   RV
         V

Y = Yellow     YO = Yellow-Orange
R = Red        RO = Red-Orange
B = Blue
               RV = Red-Violet
O = Orange     BV = Blue-Violet
V = Violet
G = Green      BG = Blue-Green
               YG = Yellow-Green

The color wheel traditionally has yellow at it's top, and the other primary colors (red and blue) spaced equidistantly. Between the primary colors are the colors you can create by mixing paints of the primary colors, including the secondary colors and the intermediate colors.

Also, you can use the color wheel to idenitify "special" pairs of colors. Colors opposite each other on the color wheel are called contrasting colors and usually make grey when they are mixed. For example, violet and yellow are contrasting colors. Colors next to each other on the color wheel are called analogous colors. Red-violet, red, and red-orange would be an example.

Note that the color wheel does not show tints and shades. Also, colors like red-violet are named with the primary color first.

Unasked for addendum:
Wog's colorwheel is for pigment.
When working with light (as in theatre, for example) the primary colors are green, blue, and red, as opposed to yellow, blue, and red.
The color wheel for photography is slightly different as well.

The primary colors are yellow, magenta and cyan, and their opposites are:
yellow - blue
magenta - green
cyan - red

For more explanation about how these colors and their opposites relate to one another and how tinkering with them can affect the outcome of your photographs, see on color photography and artificial light.
What wog says about not having shades or tints on a colour wheel is not quite true. Sometimes a colour wheel is a filled wheel, not just a rim.

The usual technique is to have white at the centre, fading through progressively stronger tints until you reach a totally saturated colour, then moving through darker shades as you reach the black rim.

However, this does not show all colours, as colour is three dimensional: the saturation (difference of the colour from an equally bright shade of grey) is always 100% on this wheel. Also, all points on the outside are the same colour: #000000, or black.

The correct colour-shape should (I think) be a sphere: white at the top, saturated at the equator, and black at the bottom. Running from pole to pole is a monochrome bar, and running from the core of the sphere to the outside growing more and more vivid.

Or, it could be visualised as a cube of dimensions RGB. Or of hue-saturation-luminance. Or in many other shapes.

Of course, octarine is the the 4th dimension, into a hyper-cube or -sphere... of which the colours we see around us are only pale reflections of octarine-colours. But this analogy could also be used to describe alpha: how transparent a colour is. But transparency is not the same for all colours... and there are colours outside of our sight (like UV and IR that don't fit into a nice colour wheel.

A color wheel is more than just a reference for art and decorating, it is also a vital component in non-CRT video projectors. For that application, a color wheel is s solid disk with an outer rim made up of red, green, and blue (RGB) color filters. Sometimes the color sectors are like the blocks on a paper color wheel, and sometimes the filters are arranged in a swirl pattern, as if streaked by the motion of the wheel.

It works by spinning very quickly in front of the projector’s lamp, breaking the white light into its primary colors, shining them in rapid sequence at the image-creating device, be it LCD (transmissive), LCoS (reflective), or DLP(also reflective). The imager is simultaneously sequencing through the red, green, and blue portions of the image in time to the red, green, and blue light falling upon it so that the proper colors are directed at the screen in the proper sequence to create a full-color image through persistence of vision.

A better, but more expensive solution is to use a dedicated image chip for each color. Three-chip projectors display flicker-free images with a brightness up to three times greater than a similar 1-chip system using a color wheel.

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