Zero - Charles Seife [12]
Figure 7: The mystical monochord
This philosophy—the interchangeability of music, math, and nature—led to the earliest Pythagorean model of the planets. Pythagoras argued that the earth sat at the center of the universe, and the sun, moon, planets, and stars revolved around the earth, each pinned inside a sphere (Figure 8). The ratios of the sizes of the spheres were nice and orderly, and as the spheres moved, they made music. The outermost planets, Jupiter and Saturn, moved the fastest and made the highest-pitched notes. The innermost ones, like the moon, made lower notes. Taken all together, the moving planets made a “harmony of the spheres,” and the heavens are a beautiful mathematical orchestra. This is what Pythagoras meant when he insisted, “All is number.”
Figure 8: The Greek universe
Because ratios were the keys to understanding nature, the Pythagoreans and later Greek mathematicians spent much of their energy investigating their properties. Eventually, they categorized proportions into 10 different classes, with names like the harmonic mean. One of these means yielded the most “beautiful” number in the world: the golden ratio.
Achieving this blissful mean is a matter of dividing a line in a special way: divide it in two so that the ratio of the small part to the large part is the same as the ratio of the large part to the whole (see appendix B). In words, it doesn’t seem particularly special, but figures imbued with this golden ratio seem to be the most beautiful objects. Even today, artists and architects intuitively know that objects that have this ratio of length to width are the most aesthetically pleasing, and the ratio governs the proportions of many works of art and architecture. Some historians and mathematicians argue that the Parthenon, the majestic Athenian temple, was built so that the golden ratio is visible in every aspect of its construction. Even nature seems to have the golden ratio in its design plans. Compare the ratios of the size of any two succeeding chambers of the nautilus, or take the ratio of clockwise grooves to counterclockwise grooves in the pineapple, and you will see that these ratios approach the golden ratio (Figure 9).
The pentagram became the most sacred symbol of the Pythagorean brotherhood because the lines of the star are divided in this special way—the pentagram is chock-full of the golden ratio—and for Pythagoras, the golden ratio was the king of numbers. The golden ratio was favored by artists and nature alike and seemed to prove the Pythagorean assertion that music, beauty, architecture, nature, and the very construction of the cosmos were all intertwined and inseparable. To the Pythagorean mind, ratios controlled the universe, and what was true for the Pythagoreans soon became true for the entire West. The supernatural link between aesthetics, ratios, and the universe became one of the central and long-lasting tenets of Western civilization. As late as Shakespeare’s time, scientists talked about the revolution of orbs of different proportions and discussed the heavenly music that reverberated throughout the cosmos.
Figure 9: The Parthenon, the chambered nautilus, and the golden ratio
Zero had no place within the Pythagorean framework. The equivalence of numbers and shapes made the ancient Greeks the masters of geometry, yet it had a serious drawback. It precluded anyone from treating zero as a number. What shape, after all, could zero be?
It is easy to visualize a square with width two and height two, but what is a square with width zero and height zero? It’s hard to imagine something with no width and no height—with no substance at all—being a square. This meant that multiplication by zero didn’t make any sense either. Multiplying two numbers was equivalent to taking an area of a rectangle, but what could the area