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The Irrationality of Pi
Young Bucky Fuller seems to have been haunted by p. Discovering that the ratio of a sphere's circumference to its diameter is an irrational number commonly referred to by the Greek letter p was perhaps the most disturbing of all geometry's strange lessons. Just as he could not imagine where its accomplice in deception, the infinite straight line, stopped, Bucky could not let go of a vision of thousands upon thousands of digits trailing after 3.14159..., spilling out the classroom window and stretching absurdly to the next town and then farther, never allowed to stop.
He has told the story countless times, each rendition sounding like the first, with an air of revelation. If he happens to be standing by the ocean, so much the better; he's bound to talk about the foarning waves.
"Look at them all"! he marvels, "beautiful, beautiful bubbles, every one of them!" Bucky recalls that growing up near the ocean gave him plenty of time to think about the structuring of these bubbles. Looking back at the wake trailing behind his boat, or standing knee-deep in breaking waves at the shore, he saw the water continuously being laced with white foam. Its whiteness was created by vast numbers of tiny air bubbles, each one suddenly formed and emerging at the water's surface. "How many bubbles am I looking at"? Bucky would ask himself, "fantastic numbers, of course." He could not help wondering: if each and every one of those spheres involves p, to how many digits does nature carry out the irrational p in making one of those bubbles, before discovering that it can't be completed? At what point does nature stop and make a "fake bubble"? And how would the decisions be made? In meetings of the chemistry and mathematics departments? No, the young Bucky concluded, I don't think nature's making any fake bubbles; I don't think nature is using p.
It is a decidedly amusing image: nature gathering department heads, nature's consternation over fudging the numbers, getting away with imperfect bubblesand it is through such deliberate personification that the ideas become memorable. The story of a young man looking out from his ship carries a deeper message. Fuller uses this particular eventeven assigning a time and place, 1917 on a U.S. Navy shipas a moment of revelation, the threshold of his conscious search for nature's coordinate system. The scene is bait, to draw us in, make us as curious about nature's structuring as that young sailor was. It works. If he had started on a heady discourse about mathematical concepts versus natural structures, his audience might have walked away.
It's important to realize the nature of his rebellion: not to challenge the theoretical numerical ratio between the circumference and diameter of the ideal sphere, but rather to challenge that sphere to materialize. Irrational numbers don't belong in tangible experiences. It is a question of sorting out the demonstrable from the impossible and then developing models based on the former. Essentially, Bucky is choosing not to play with p, posing the question "why shouldn't mathematics deal with experience"?
"Nature Isn't Using Pi"
Nature can have no perfect spheres because she has no continuous surfaces. The mathematician's sphere calls for all points on its surface to be exactly equidistant from the center. This "sphere," explains Fuller, has no holes. It is an absolutely impermeable container sealing off a section of Universe, a perpetual energy-conserving machine defying all laws of nature. The illusion of a physical continuum in any spherical system is due to the limitations of the human senses.
On some level of resolution, all physical "solids" and surfaces break down into discrete particles. A magnifying glass uncovers the tiny dots of different colors that make up the "pure" blue sky in a magazine photograph, and a microscopeif it could be set up in the seawould reveal that nature's bubbles are likewise fragmented, consisting of untold numbers of discrete molecules located approximately equidistant from an approximate center. If such a phenomenon could be precisely measured, we would find that the ratio of a bubble's circumference and diameter is some number very close to the elusive p, but the point, says Bucky, is that the bubble differs from the Greek ideal. A physical entity is necessarily demonstrable and finite, while an irrational number such as p is just the opposite. A real sphere consists of a large but finite number of interconnected individual energy events. Moreover, the middle of each of the implied chordal (straight-line) connections between events is slightly closer to the sphere's center than the ends, thereby violating the mathematical definition, and insuring some small departure from the ratio p. Any cross-section of this sphere will be, not a circle, but a many-many-sided polygon. Likewise, the most perfect-looking circle, carefully drawn with a sharpened pencil and accurate compass, will appear fuzzy and fragmented through a good magnifying glass. The most precise ball bearing is imperfect: a good approximation of p, but always imperfect, with bizarre mountains and valleys dramatically visible after 5000 times magnification. In Chapter 15 we shall examine the specifics of one alternative model for the mathematician's impossible sphere.
"Fake bubbles" led to further contemplation about p. Nature is always associating in simple whole rational numbers, thought Bucky: H20, never HpO. Irrational numbers do not show up in chemical combinations of atoms and molecules. Clearly, nature employs "one coordinate, omnirational, mensuration system." (410.011)
None of this is new information; any one of us could have thought about it for a while and understood that of course p does not play a role in the making of bubbles. What happens is that turbulence introduces air into the water and the air is so light that it floats toward the surface to escape. It's an easy problem of getting the most air in each pocket with the least surface area of water pushing in to collapse the bubble. A spherical space allows the most volume per unit of surface area (as we shall learn in later chapters), providing the most efficient enclosure for that air. (Nature is exquisitely efficient.) But, even if we did contemplate this mystery at the seashore, we probably didn't bother to take our children aside and explain it to them before they became perplexed and frustrated by the never-ending p in mathematics class.
If nature's lack of employment of p isn't news, why does Bucky persist? Why use his own considerablebut still finiteenergy to lecture tirelessly on the subject for half a century? The answer lies in the fact that our education and popular awareness fall short of the mark; Fuller felt that society did not sufficiently emphasize this discrepancy between theoretical games and real structures. We introduce "solids" to school children long before they learn the real story about energy-event reality. If humanity is to feel comfortable with science, argues Fuller, education must present accurate models in the first place. The message behind the bubble's story is: why not tell the truth from the beginning? He calls for widespread recognition of discrete energy events, a reality that cannot be perceived by human senses.
Unfortunately, the issue is not that easily resolved. We shall not ever completely escape the brain-teasing presence of irrational numbers. For example, the construction of the simplest polygons, with carefully measured unit-length edges, frequently produces irrational diagonals. The distance between nonadjacent vertices in a regular pentagon is the irrational number known as the "golden section," which we shall see again in Chapter 11, and the diagonal of a square is incontrovertibly Ö2. However, in every actual construction, a finite approximation can be determined according to the specific limitations of available measuring instruments. That numerical fractions must terminate is a consequence of investigating verifiable experience rather than theoretical relationships. Just as architects do not work with irrational-length two-by-fours, measurements obtained through "operational procedure" are necessarily real. Finally, Bucky's dismissal of these troublesome values calls attention to the granular constitution of physical reality, as described in the following section.
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