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picture using a projection, we sometime include information to help retain some of what was lost. The additional information might be shading or color, as in a painting or photograph. It might be a number, as in a topographic map to illustrate height. Or there might be no label at all, in which case the two-dimensional characterization simply offers less information.
    Without both our eyes, which work together to let us reconstruct three dimensions, everything we see would be projections. Depth perception is tougher when you close one of your eyes. A single eye constructs a two-dimensional projection of three-dimensional reality. You need two eyes to reproduce three dimensions.
    I am nearsighted in one eye and farsighted in the other, so I don’t properly combine the images from both eyes unless I’m wearing glasses—which is rarely the case. Although I was told I should have trouble reconstructing three dimensions, I don’t usually notice any problem: things still look three-dimensional to me. That is because I rely on shading and perspective (and my familiarity with the world) to reconstruct three-dimensional images.
    But one day in the desert, a friend and I were trying to reach a distant cliff. My friend kept telling me that we could walk directly there, and I couldn’t understand why he was insisting that we should walk straight through a piece of rock. It turned out the rock that I thought projected directly from the cliff, so that it would completely block our way, was in fact located much closer to us, in front of thecliff. The rock I had thought would bar our path wasn’t actually attached to the cliff at all. This misunderstanding occurred because we were near the cliff around noon, when there were no shadows, and I had no way to construct the third dimension that would have told me how the distant cliffs and rocks were lined up. I wasn’t really conscious of my compensating strategy of using shading and perspective until then, when it failed.
    Painting and drawing have always required artists to reduce what they see to projected images. Medieval art did this in the simplest manner. Figure 11 shows a mosaic image of a city as a two-dimensional projection. This mosaic doesn’t tell us anything about a third dimension; there are no labels or indications of its existence.
    Since medieval times, painters have developed ways to make projections that partially redress painting’s loss of a dimension. One approach that opposes the medieval flattening of space is the method used by the cubists in the twentieth century. A cubist painting (for example Picasso’s Portrait of Dora Maar , Figure 12) presents several projections simultaneously, each from a different angle, and thereby conveys the subject’s three-dimensionality.

Figure 11. A two-dimensional medieval mosaic.

Figure 12. Portrait of Dora Maar, a cubist painting by Picasso.

Figure 13. Dali’s Crucifixion (Corpus Hypercubus) .

    Most Western painters since the Renaissance, however, have used perspective and shading to create the illusion of a third dimension. One of the essential skills in painting is the ability to reduce a three-dimensional world to a two-dimensional representation that allows the observer to reverse the process and reconstitute the initial three-dimensional scene or object. We are acculturated to know how to decode the images, even though not all of the three-dimensional information is there.
    Artists have even tried representing higher-dimensional objects on two-dimensional surfaces. For example, Salvador Dali’s Crucifixion (Corpus Hypercubus) (see Figure 13) shows the cross as an opened-up hypercube. A hypercube consists of eight cubes attached in four-dimensional space. These are the cubes he has drawn. I’ve shown a few projections of a hypercube in Figure 14.

Figure 14. Projections of a hypercube.
    I have already mentioned a physics example: quasicrystals, which look like the projection of a higher-dimensional crystal into