Look closely and you will see that the area of the wall on which the projector is projecting is brighter than an area of the same wall that is out of the projector's reach, despite the fact that the projector is supposed to be projecting perfect blackness. First of all, turn off the room lights, turn on the projector, and then send an image from your computer to the projector that is completely black everywhere. There are a few ways that you can convince yourself of this concept. In this way, a projector can throw some white light on a wall and convince you that it is black. Returning to the case of the image cast on the wall by the projector, in order for our brains to perceive a part of the image as black, it just has to be less bright than all the other parts of the image. By basing our color perception on relative color differences rather than absolutes, our brains are able to link object color to intrinsic properties of the object (such as ripeness), rather than assume that object color only tells us the color content of the light source. If humans were only able to evaluate colors according to their absolute spectral content, then we would bizarrely conclude that ripe bananas become unripe every time they go in the shade. Therefore, a ripe banana sitting in direct sunlight has an absolute color of yellow, while the same ripe banana has an absolute color of yellowish green when in the shade on a clear sunny day. For instance, objects that are in shade on a sunny day are more blue than objects in direct sunlight, because they are being illuminated by the whitish blue sky rather than the white sunlight. In contrast, the absolute color of the bananas does change as the light source changes. The relative color difference between the two bananas does not change, even if the light source does change. A ripe banana placed next to an unripe banana is always more yellow and less green than the unripe banana, no matter what type of light is shining on it. The tendency of our brains and eyes to evaluate a color based on its relation to nearby colors is actually beneficial. Public Domain Image, source: Christopher S. They look like different colors because of the way our brains perceive color based on the color of surrounding areas. The blue colors immediately surrounding both black dots are the exact same color. You can convince yourself that the areas right around the two black dots are the same color blue by holding a thick piece of paper up to this image with two holes cut out right over the black dots, allowing you to see the colors directly around the dots without seeing the rest of the image. They look different to humans because of the way our brains perceive color based on the color of surrounding areas.
Directly surrounding both black dots are patches of blue that are the exact same color. Our human eyes and brains are designed to evaluate a color based on how it looks relative to the colors of the surrounding objects, rather than based on the absolute spectral content of the color. Note that the "black" squares in the checkerboard on the right are actually brighter than the regions of the wall outside of the projected image. Regions of the projected image that are supposed to be black are actually a dim white color, which our brains interpret as black if the image contrast is high enough. Since black is actually just the absence of light and not something that can literally be projected, the illustration on the right shows what we actually see. Note that the dark regions in the projected image would be darker than the regions of the wall outside of the projected image. On the left is an illustration of what a black-and-white checkerboard image would look like projected on the wall if a projector could literally project black.