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our three-dimensional world. Projections can also be used for practical, not just artistic purposes. Medicine contains many examples where three-dimensional objects are projected onto two dimensions. An X-ray always records a two-dimensional projection. CAT (computer-assisted tomography) scans combine multiple X-ray images to reconstruct a more informative three-dimensional representation. With X-rays taken from sufficiently many angles, one can use interpolation to piece together full three-dimensional images. An MRI (magneticresonance imaging) scan, on the other hand, reconstructs a three-dimensional object from slices.
    A holographic image is another way to record three dimensions on a two-dimensional surface. Although a holographic image is recorded on a lower-dimensional surface, it actually carries all the information of the original higher-dimensional space. You probably have an example of this technique in your wallet: the three-dimensional-looking image on your credit card is a hologram.
    A holographic image records relationships between light in different places, so that the full higher-dimensional image can be recovered. This principle is much the same as that used in a good stereo, which lets you hear where instruments were being played in relation to each other when they were recorded. With the information stored in a hologram, the eye can truly reconstruct the three-dimensional object it represents.
    These methods tell us how we might get more information from a lower-dimensional image. But maybe all you really need is less information. Sometimes you just don’t care about all three dimensions. For example, something might be so thin in the third dimension that nothing interesting happens in this direction: even though the ink on this paper is really three-dimensional, we lose nothing by thinking of it as two-dimensional. Unless we look at the page under a microscope, we simply don’t have the necessary resolution to see the ink’s thickness. A wire looks one-dimensional even though, on closer inspection, you can see it has a two-dimensional cross-section and therefore three dimensions in all.
    Effective Theories
    There is nothing wrong with ignoring an extra dimension that’s too small to be seen. Not only the visual effects, but also the physical effects of tiny, undetectable processes can usually be ignored. Scientists often average over or ignore (often unwittingly) physical processes that occur on immeasurably small scales when formulating their theories or setting up their calculations. Newton’s laws of motion work at the distances and speeds he could observe. He didn’t need thedetails of general relativity to make successful predictions. When biologists study a cell, they don’t need to know about quarks inside the proton.
    Selecting relevant information and suppressing details is the sort of pragmatic fudging everyone does every day. It’s a way of coping with too much information. For almost anything you see, hear, taste, smell, or touch, you have the choice between examining details by scrutinizing very closely, and looking at the “big picture” with its other priorities. Whether you are staring at a painting, tasting wine, reading philosophy, or planning your next trip, you automatically parcel your thoughts into the categories of interest—be they sizes or flavors or ideas—and the categories that you don’t find relevant at the time. When appropriate, you ignore some details so that you can focus on the issue of interest, and not obscure it with inessential details.
    This procedure of disregarding small-scale information should be familiar because it’s actually a conceptual leap people make all the time. Take New Yorkers, for example. New Yorkers living in the thick of the city see the details and variation within Manhattan. To them, downtown is funkier, older, with narrower, more crooked streets. Uptown has more real estate that was designed for human beings to actually live in, as