The mystical symbol of the Pythagorean cult was, naturally, a number-shape: the pentagram, a five-pointed star. This simple figure is a glimpse at the infinite. Nestled within the lines of the star is a pentagon. Connecting the corners of that pentagon with lines generates a small, upside-down, five-pointed star, which is exactly the same as the original star in its proportions. This star, in turn, contains an even smaller pentagon, which contains a tinier star with its tiny pentagon, and so forth (Figure 6). As interesting as this was, to the Pythagoreans the most important property of the pentagram was not in this self-replication but was hidden within the lines of the star. They contained a number-shape that was the ultimate symbol of the Pythagorean view of the universe: the golden ratio.
Figure 5: Square and triangular numbers
Figure 6: The pentagram
The importance of the golden ratio comes from a Pythagorean discovery that is now barely remembered. In modern schools, children learn of Pythagoras for his famed theorem: the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. However, this was in fact ancient news. It was known more than 1,000 years before Pythagoras’s time. In ancient Greece, Pythagoras was remembered for a different invention: the musical scale.
One day, according to legend, Pythagoras was toying with a monochord, a box with a string on it (Figure 7). By moving a sliding bridge up and down the monochord, Pythagoras changed the notes that the device played. He quickly discovered that strings have a peculiar, yet predictable, behavior. When you pluck the string without the bridge, you get a clear note, the tone known as the fundamental. Putting the bridge on the monochord so it touches the string changes the notes that are played. When you place the bridge exactly in the middle of the monochord, touching the center of the string, each half of the string plays the same note: a tone exactly one octave higher than the string’s fundamental. Shifting the bridge slightly might divide the string so that one side has three-fifths of the string and the other has two-fifths; in this case Pythagoras noticed that plucking the string segments creates two notes that form a perfect fifth, which is said to be the most powerful and evocative musical relationship. Different ratios gave different tones that could soothe or disturb. (The discordant tritone, for instance, was dubbed the “devil in music” and was rejected by medieval musicians.) Oddly enough, when Pythagoras put the bridge at a place that did not divide the string into a simple ratio, the plucked notes did not meld well. The sound was usually dissonant and sometimes even worse. Often the tone wobbled like a drunkard up and down the scale.
To Pythagoras, playing music was a mathematical act. Like squares and triangles, lines were number-shapes, so dividing a string into two parts was the same as taking a ratio of two numbers. The harmony of the monochord was the harmony of mathematics—and the harmony of the universe. Pythagoras concluded that ratios govern not only music but also all other types of beauty. To the Pythagoreans, ratios and proportions controlled musical beauty, physical beauty, and mathematical beauty. Understanding nature was as simple as understanding the mathematics of proportions.
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 of a rectangle with zero height or zero width be?
Nowadays the great unsolved problems in mathematics are stated in terms of conjectures that mathematicians are unable to prove. In ancient Greece, however, number-shapes inspired a different way of thinking. The famous unsolved problems of the day were geometric: With only a straightedge and compasses, could you make a square equal in area to a given circle? Could you use those tools to trisect an angle?* Geometric constructions and shapes were the same thing. Zero was a number that didn’t seem to make any geometric sense, so to include it, the Greeks would have had to revamp their entire way of doing mathematics. They chose not to.