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interpreting lower-dimensional “pictures” of higher-dimensional worlds. Think of this section as a warm-up exercise for wrapping your mind around extra dimensions. It might be good to remember that you cope with dimensionality all the time in ordinary life. It really isn’t that unfamiliar.
    Often all we can see are parts of the surface of things, the surface being only the exterior. This exterior has two dimensions, even though it curves through three-dimensional space, because you only need two numbers to identify any point. We deduce that the surface isn’t three-dimensional because it has no thickness.
    When we look at pictures, movies, computer screens, or the figures in this book, we are generally looking at two-dimensional, not three-dimensional representations. But we can nonetheless deduce the three-dimensional reality that is being portrayed.
    We can use two-dimensional information to construct three dimensions. This involves suppressing information in making two-dimensional representations while trying to keep enough information to reproduce essential elements of the original object. Let’s now reflect on the methods we often use to reduce higher-dimensional objects to lower dimensions—slicing, projection, holography, and sometimesjust ignoring the dimension—and how we work backwards to deduce the three-dimensional objects they represent.
    The least complicated way of seeing beyond the surface is to make slices. Each slice is two-dimensional, but the combination of the slices forms a real three-dimensional object. For example, when you order ham at the deli, the three-dimensional lump of ham is readily exchanged for many two-dimensional slices. * By stacking all the slices you could reconstruct the full three-dimensional shape.
    This book is three-dimensional. However, its pages have only two dimensions. The union of the two-dimensional pages comprises the book. † We could illustrate this union of pages in many ways. One is shown in Figure 8, in which we view the book edge on. In this picture we’ve again played with dimensionality, since each line represents a page. So long as we all know that the lines represent two-dimensional pages, this illustration should be clear. Later on, we’ll use a similar shorthand when we depict objects in multidimensional worlds.

Figure 8. A three-dimensional book is made up of two-dimensional pages.
    Slicing is only one way to replace higher dimensions with lowerones. Projection , a technical term borrowed from geometry, is another. A projection gives a definite prescription for creating a lower-dimensional representation of an object. A shadow on a wall is an example of a two-dimensional projection of a three-dimensional object. Figure 9 illustrates how information is lost when we (or rabbits) make a projection. Points on the shadow are identified by only two coordinates, left-right or up-down along the wall. But the object that is projected also has a third spatial dimension that the projection doesn’t retain.

Figure 9. A projection carries less information than the higher-dimensional object.
    The simplest way to make a projection is to just ignore one dimension. For example, Figure 10 shows a cube in three dimensions being projected onto two dimensions. The projections can take many forms, the simplest of which is a square.
    To return to our earlier examples of the graphs of Ike and Athena, we might make a two-dimensional plot of Ike by neglecting his driving fast cars. And we might not really want to know the number of owls Athena raises, and might therefore make a four-dimensional rather than a five-dimensional plot. Disregarding Athena’s owls is a projection.

Figure 10. Projections of a cube. Notice that the projection can be a square, as we see in the middle diagram, but that projections can also take other shapes.
    A projection discards information from the original, higher-dimensional object (see Figure 9). However, when we make a lower-dimensional