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truly imagine a sphere. From his new perspective, he recognizes the sphere as the shape made by gluing together the two-dimensional slices he witnessed. Even in his two-dimensionalworld, A. Square could have plotted the disks he sees as a function of time (as in Figure 6) to construct the sphere. But it wasn’t until his trip through a third dimension opened his eyes that he fully comprehended the sphere and its third spatial dimension.

Figure 6. If a sphere passes through a plane, a two-dimensional observer would see a disk. The sequence of disks that the observer sees over time comprises the sphere.
    By analogy, we know that if a hypersphere (a sphere with four spatial dimensions) were to pass through our universe, it would appear to us as a time sequence of three-dimensional spheres that increase, then decrease, in size. 3 Unfortunately, we don’t have the opportunity to journey through an extra dimension. We will never see a static hypersphere in its entirety. Nonetheless, we can make deductions about how objects look in spaces of different dimensions—even dimensions that we don’t see. We can confidently deduce that our perception of a hypersphere passing through three dimensions would look like a series of three-dimensional spheres.
    As another example, let’s imagine the construction of a hypercube —a generalization of a cube to more than three dimensions. A line segment of one dimension consists of two points connectedby a straight, one-dimensional line. We can generalize this in two dimensions to a square by putting one of these one-dimensional line segments above another and connecting them with two additional segments. We can generalize further in three dimensions to a cube, which we can construct by placing one two-dimensional square above the other and connecting them with four additional squares, one on each edge of the original squares (see Figure 7).

Figure 7. How we put together lower-dimensional objects to make higher-dimensional ones. We connect two points to make a line segment, two line segments to make a square, two squares to make a cube, and (not pictured since it’s too difficult to draw) two cubes to make a hypercube.
    We can generalize in four dimensions to a hypercube, and in five dimensions to something for which we don’t yet have a name. Even though we three-dimensional mortals have never seen these two objects, we can generalize the procedure that worked in lower dimensions. To construct a hypercube (also known as a tesseract), put one cube above the other, and connect them by adding six additional cubes, connecting the faces of the two original cubes. This construction is an abstraction and difficult to draw, but that doesn’t make the hypercube any less real.
    In high school, I spent a summer at math camp (which was far more entertaining than you might think), where we were shown a film version of Flatland . * At the end, the narrator, in a delightful British accent, tried futilely to point to the third dimension that was inaccessible to Flatlanders, saying, “Upward, not Northward.” Unfortunately, we have the same frustration if we try to point to a fourth spatialdimension, a passage. But just as Flatlanders didn’t see or travel through the third dimension, even though it existed in Abbott’s story, our not having yet seen another dimension doesn’t mean there is none. So although we haven’t yet observed or traveled through such a dimension, the subtext throughout Warped Passages will be, “Not Northward, but Forward along a passage.” Who knows what exists that we haven’t yet seen?
    Three from Two
    For the rest of this chapter, rather than thinking about spaces that have more than three dimensions, I will talk about how, with our limited visual capacity, we go about thinking and drawing three dimensions using two-dimensional images. Understanding how we perform this translation from two-dimensional images to three-dimensional reality will be useful later on when