Zero by Charles Seife Page A

our world is four-dimensional, as we shall see in a later chapter.) The clock on your dresser, the cup of coffee you drink in the morning, the book you’re reading right now—all these are three-dimensional objects. Now imagine that a giant hand comes down and squashes the book flat. Instead of being a three-dimensional object, the book is now a flat, floppy rectangle. It has lost a dimension; it has length and width, but no height. It is now two-dimensional. Now imagine that the book, turned sideways, is crushed once again by the giant hand. The book is no longer a rectangle. It is a line. Again, it has lost a dimension; it has neither height nor width, but it has length. It is a one-dimensional object. You can take away even this single dimension. Squashed along its length, the line becomes a point, an infinitesimal nothing with no length, no width, and no height. A point is a zero-dimensional object.
    In 1425, Brunelleschi placed just such a point in the center of a drawing of a famous Florentine building, the Baptistery. This zero-dimensional object, the vanishing point, is an infinitesimal dot on the canvas that represents a spot infinitely far away from the viewer (Figure 18). As objects recede into the distance in the painting, they get closer and closer to the vanishing point, getting more compressed as they get farther away from the viewer. Everything sufficiently distant—people, trees, buildings—is squashed into a zero-dimensional point and disappears. The zero in the center of the painting contains an infinity of space.
    This apparently contradictory object turned Brunelleschi’s drawing, almost magically, into such a good likeness of the three-dimensional Baptistery building that it was indistinguishable from the real thing. Indeed, when Brunelleschi used a mirror to compare the painting and the building, the reflected image matched the building’s geometry exactly. The vanishing point turned a two-dimensional drawing into a perfect simulation of a three-dimensional building.

    Figure 18: The vanishing point
    It is no coincidence that zero and infinity are linked in the vanishing point. Just as multiplying by zero causes the number line to collapse into a point, the vanishing point has caused most of the universe to sit in a tiny dot. This is a singularity, a concept that became very important later in the history of science—but at this early stage, mathematicians knew little more than the artists about the properties of zero. In fact, in the fifteenth century, artists were amateur mathematicians. Leonardo da Vinci wrote a guide to drawing in perspective. Another of his books, about painting, warns, “Let no one who is not a mathematician read my works.” These mathematician-artists perfected the technique of perspective and could soon depict arbitrary objects in three dimensions. No longer would artists be restricted to flat likenesses. Zero had transformed the art world.
    Zero was, quite literally, at the center of Brunelleschi’s painting. The church, too, dabbled with zero and the infinite, though church doctrine was still dependent on Aristotelian ideas. A contemporary of Brunelleschi, a German cardinal named Nicholas of Cusa, looked at infinity and promptly declared, “Terra non est centra mundi”: the earth is not the center of the universe. The church didn’t yet realize how dangerous, how revolutionary, that idea was.
    One of the old declarations of the medieval Aristotelian doctrine—as strong as the ban on the vacuum—was the statement that Earth was unique. It was at the universe’s very center. Earth’s special position at the center of the universe made it the only world capable of containing life, as Aristotle held that all objects sought out their proper place. Heavy objects, like rocks or people, belonged on the ground; light objects, like air, belonged in the heavens. Not only did this imply that the planets—in the