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  Beyond Pythagoras

  Today, mathematicians use different terms to describe various kinds of numbers that the Pythagoreans either did not distinguish, or did not recognize. The natural numbers are those they knew. These are derived by counting visible objects: 1, 2, 3, 4 . . . , and the Greeks used letters or pebbles (Latin calculi) to represent them. Much later, zero was added as a way of signifying the absence of objects: 0, 1, 2, 3 . . . The invention of zero, like that of the wheel, was one of the turning points in the history of civilization. It radically simplified the process of calculation, since a set of nine symbols could be rearranged to make numbers of any size by attributing a value to their position in the composite (without zero it would not have been possible to distinguish between 45, 405, and 40,050). The number series is extended in either direction from zero to yield the complete set of integers, both negative and positive: –3, –2, –1, 0, 1, 2, 3 . . .16

  Zero, by the way, came from India (where it was placed after 9) and transmitted by the Arabs into medieval Europe (where it was placed before 1).17 In nature there is no zero.18 Some writers on number symbolism therefore regard it as an interloper, whose introduction as a placeholder led to the loss of awareness of the symbolic properties of number, and especially of Unity (displacing it from its position at the beginning of the number series), creating a framework for the development of atheism. Robert Lawlor writes, for example:

  The “western” rationalistic mentality negated the ancient and revered spiritual concept of Unity, for with the adoption of zero, Unity loses its first position and becomes merely a quantity among other quantities . . . With zero we have at the beginning of modern mathematics a number concept which is philosophically misleading and one which creates a separation between our system of numerical symbols and the structure of the natural world. On the other hand, with the notion of Unity which governs ancient mathematics, there is no such dichotomy.19

  Personally, I am not so sure. Zero could also be taken as the ground of being, and a symbol for the return to one. Perhaps the mistake lay not in introducing zero, but failing to read it symbolically.

  Integers can be divided as well as multiplied by each other, enabling us to add between each whole integer a series of fractional numbers, or ratios: ½, ¼, and so on. These are called “rational” numbers (from the same root as “ratio,” referring to the relationship of one number with another). In theory, the quantity of these is indefinite, since we can go on subdividing any one of them as much as we like. Each fraction can also be written in decimal notation, where each space from the point represents a leap of magnitude, in this case tenfold. When this is done, every fraction forms a closed or repeating pattern of decimals: 0.25, 6.3333 recurring, and so on. These regular patterns reflect the discrete nature of every rational number. (A similar pattern appears in any rational base: binary, trinary, and so on.)

  But this means that, although there is an indefinite number of possible fractions between each integer, they are still discrete. In other words, there are gaps between them, no matter how many smaller fractions we decide to create. For that reason, in between every rational integer and fraction of an integer there is actually room for an indefinite quantity of other numbers, namely all those fractions that do not form a closed or repeating pattern. These are called the irrational numbers. (Rational and irrational numbers together are called “real numbers.”)

  If talk of “irrational” numbers sounds abstruse, there are nevertheless many practical implications in everyday life. The solar year, for example, like the lunar month, is not a rational multiple of a day. If it were precisely 365.25 days, as we were told in school, then astronomical events related to the apparent position of the sun would repeat precisely over that period, and the year and the day would be back in step at exactly the same point. “So nothing ever repeats exactly. Calendars therefore have to make compromises, and it is the history of those compromises in different cultures that has led to a plethora of calendar systems.”20

  Irrationals cannot be used to count things, but are encountered in relationships between things that can be counted. So if the sides of a cube are one unit long (1:1:1), the diagonal of the cube will be √3, and of each square face √2. Both are irrational numbers, much used in the construction of Gothic cathedrals. Most famously, the circumference of a circle is irrational compared to its diameter. The name we give to the number of times you can divide the diameter into the circumference is the Greek letter π (pi). Like all irrational numbers, when expressed as a decimal, π goes on forever (signified by the three dots at the end): 3.141592 . . .21 Another famous irrational number is Φ or phi, the so-called “golden ratio” that describes a particularly harmonious relationship we will look at more closely in a moment.22

  It is often claimed that because the Pythagoreans at first believed all numbers to be natural, their discovery of “incommensurables” or irrational numbers was a shock to them, and at first had to be kept secret; in fact, there is a legend that they drowned one of their brotherhood to keep it quiet. For it meant that the world could not be explained entirely on the basis of whole numbers such as those in the sacred Tetraktys. But the “crisis of the irrational”—if there was one (and we have little or no real evidence for this)—was only temporary. Irrationals may not have been expressible as whole numbers or even as fractions, but this was because they were the result of using discontinuous numerals to describe continuous spatial or qualitative relationships. Such indefinable quantities were, however, easy to locate geometrically—that is, in two or three dimensions—and the ancient Greeks did their mathematics pictorially in any case. A higher-order mathematics (such as the one developed by Eudoxus and rediscovered in the modern period) incorporated them without difficulty.

  Irrational Beauty

  Let’s look more closely at these so-called irrationals, and especially at the famous “golden ratio” or “golden section.”

  When one thing is compared to another we call the relationship a ratio (it is this word from which we derive the word “rationality”). Slightly more sophisticated is a proportion, or analogy, which is based on the comparison of one ratio with another (A to B is like C to D), especially when there is a common term linking the two ratios that acts as a mediator between them (A to B is like B to C). The most elegant form of this proportionality is one in which there are just two terms, rather than three: A to B is like B to A + B. This is what is called the “golden” or “divine” ratio, also known as the “extreme and mean ratio,” designated Φ or phi.

  In this diagram, A/B = B/C, and if B is equal to 1, then A = Φ = 1.61804. . . .23 In other words: if the relation of the larger section to the whole line is proportionally the same as the relation of the small section to the larger, then that relation is Φ. The point dividing the line is called the golden mean.

  A rectangle is described as golden when the ratio between its sides is Φ. If you cut a square out of a golden rectangle, the remaining piece is also a golden rectangle.

  Both Leonardo da Vinci and Piet Mondrian used such rectangles frequently in their paintings, and the ratio itself can be found governing the lengths of sections in many Beethoven movements.

  Phi is called “Divine” because, like God, it contains within itself both identity and difference. Meditating on Φ and observing it in nature is thought to be a way of raising the human mind toward the divine unity. In his book Sacred Geometry, Robert Lawlor relates the golden ratio to both the Trinity and the Logos. Phi represents “Three that are Two that are One.” He sees this as an exact transcription into mathematics of the words: In the beginning was the Logos [Word], and the Logos was with God, and the Logos was God (John 1:1).

  In a sense, the Golden Proportion can be considered as supra-rational or transcendent. It is actually the first issue of Oneness, the only possible creative duality within Unity. It is the most intimate relationship, one might say, that proportional existence—the universe—can have with Unity, the primal or first division of O
ne. For this reason the ancients called it “golden,” the perfect division, and the Christians have related this proportional symbol to the Son of God.24

  Phi “is the perfect division of unity: it is creative, yet the entire proportional universe that results from it relates back to it and is literally contained within it, since no term of the original division steps, as it were, outside of a direct rapport with the initial division of Unity.”25 It is a division that creates difference rather than self-duplication. As a proportion, not strictly a number, phi can be taken to represent the experience of knowledge, or mediation, of analogy—the Logos in all things.

  Phi and the Natural Numbers

  Phi has a special relationship to the number 5. Not only is it roughly equivalent to 8 divided by 5 (= 1.6), but it is accurately derived from the square root of 5 by adding 1 and dividing by 2. It can also be calculated from the ratio of the lengths between adjacent and alternate tips of a five-pointed star. (Other lines within this figure are divided according the golden ratio: try to work out which ones.)

  Of course, it is even more closely related to Unity. If you divide Φ into 1, you get a number exactly 1 less than Φ: namely 0.61804 . . . , while if you square it, you get Φ + 1, or 2.61804. . . .

  The Fibonacci sequence also approximates to Φ. This sequence, which featured in the best-selling thriller The Da Vinci Code, is derived from the work of a mathematician and merchant known as Fibonacci, whose book Liber Abaci or “Book of Computation” persuaded Europe to adopt Arabic numerals after AD 1202.26 The sequence is composed of an open-ended series of numbers starting with zero and one, in which each consecutive set of two numbers added together produce the next: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, etc. The sequence has many remarkable mathematical properties, apart from the fact that the bigger the number, the closer it approximates to the golden ratio when divided by the number immediately preceding it. It can also be used geometrically to construct a so-called “golden spiral,” in which each turn follows the Fibonacci sequence and grows from within—a pattern that seems to recur in galaxies, whirlpools, shells, and plants.

  It has often been claimed that this “golden” proportion is one of the defining characteristics of objective beauty, since it is found throughout nature, including the relation of the various parts of the human body (such as the lengths of finger bones or the proportions of the face) to one another. The Greek sculptor Phidias, after whom the ratio is named, is said to have used the proportion in his designs for the Parthenon. The evidence for this is thin, however, and the claim that the ancient architects were familiar with it may have been exaggerated. Quite possibly the presence of the proportion in ancient works such as the Great Pyramid may have been the result of intuition as much as calculation. In the case of the golden ratio, taken up so enthusiastically by artists and architects and even composers after the twelfth century, we may have seen how it is possible not only to reestablish a tradition that the ancients knew well, but to develop it creatively in the light of other knowledge.

  Symmetry

  The same possibility of creative development applies to the tradition I want to mention briefly in the final section of this chapter.

  Symmetry, or “patterned self-similarity,” has always been treated as one of the fundamental principles of beauty. It is an underlying structural principle, around which a multitude of spontaneous or unpredictable variations can be woven. Such underlying structures are laid bare in mathematics, particularly a branch of mathematics called group theory. Mathematicians tell us that the most basic kind of symmetry is reflectional or bilateral, as seen in the two sides of the human body. A square has four axes of symmetry, because there are four ways to fold it making two equal halves. A circle has infinitely many such axes. Other more complex types of symmetry apply to rotating objects, helices (DNA is a non-repeating helix, thanks to its irrational radians), objects on different scales of magnitude (e.g., fractals), etc.

  Transcending the division between rational and irrational, the idea of symmetry can be applied to numbers, shapes, music, words, and ideas. The universe as a whole is highly symmetrical—even more so, it seems, as you wind time backward to the Big Bang. The physical distribution of matter and energy, the laws of conservation of energy and momentum, and even the invariance of physical laws across time and space, are all now seen as manifestations of the same principle, which is why so many popular science books have appeared with titles like Symmetry and the Beautiful Universe, Hidden Unity in Nature’s Laws, and even Why Beauty Is Truth.

  Once again, we find the ancient intuitions of the Great Tradition borne out in unexpected ways by modern science. The cosmic symmetries exposed by group theory reveal creation as the interplay of the One in the Many. In one of the most helpful of the popular expositions of modern physics, Stephen M. Barr writes,

  Symmetry contributes to the artistic unity of a work, to its balance, proportion, and wholeness. The connection between symmetry and unity is exceedingly important and applies also to symmetry in physics. If one part of a symmetrical pattern or structure is removed, typically its symmetry is spoiled. Cut off one arm of a starfish, and it will no longer have a five-fold but only a two-fold symmetry. Remove one pentagonal facet of a soccer ball, and instead of 120 symmetries only 5 will be left. Symmetry requires all the parts of a pattern to be present, and is therefore a unifying principle.27

  He goes on to show how the incompleteness of a symmetric pattern often leads to the discovery of a new subatomic particle. In fact all the four basic forces of nature (gravity, electromagnetism, the strong force, and the weak force) are based on and controlled by principles of symmetry, and the “ultimate theory” that is the Holy Grail of physics is currently assumed to lie in the direction of some grand unification of all these four systems of symmetry (as we will see later on).

  In a way it could be argued that arithmetic, geometry, astronomy, and music originated in the quest for the ultimate mandala: this being the oriental name we give to any symbol of wholeness exhibiting a variety of intensive symmetries, in which each part communicates with and corresponds with every other. A mandala is fascinating mainly for its ability to integrate multitudinous variety in a simple pattern. It illustrates one of the most important aspects of beauty: the convergence of extreme unity with extreme complexity. And in such patterns we can see not only the world, but ourselves reflected—or at least ourselves as we aspire to be, images simultaneously of creation and the Creator.

  1. See Joost-Gaugier 2006.

  2. Cited in Pickover 2005, 226. According to Paul Friedländer, “Plato’s physics of elements being transformed into each other and of regularly divisible atoms was incomprehensible as long as classical physics reigned supreme, i.e., from Newton to the recent past. Now, these chapters of the Timaeus have acquired a new meaning, and perhaps Plato may be looked upon as a predecessor of Rutherford and Bohr in the same sense that Demokritos was a predecessor of Galileo and Newton” (257).

  3. Cited in Barrow 2003, 102.

  4. Cited in Pickover 2005, 229.

  5. Schneider 2006, 81–82.

  6. See, e.g., Iamblichus 1988.

  7. Burkhardt 1987, 79.

  8. Cf. Lundy 2005.

  9. (1) “Literal” or historical-factual, (2) “allegorical” or doctrinal, (3) “tropological” or moral, and (4) “anagogical” or mystical, perhaps corresponding to the four types of causation.

  10. For further details see Sutton 2005, and Critchlow 1994. The solids themselves are not the most fundamental elements of nature, according to Plato, since each is composed of equilateral or right-angled triangles. Thus the whole of nature is reduced (or raised) to a triadic principle, through which it is implicitly resolved back into unity.

  11. As the Greeks also knew, there is only one perfect number between 1 and 10, between 10 and 100, between 100 and 1000, and between 1000 and 10,000 (6, 28, 496, and 8128 respectively). After that the pattern changes, and the next perfect number is in the tens of millions
.

  12. The writings of the Jewish Platonist Philo of Alexandria (d. AD 50) were preserved by Eusebius and were influential on the church fathers. For his arithmology see Philo 1981, 85.

  13. Meyer-Baer 1970, 80, 351.

  14. “The Tetraktys encapsulates the numbers one, two, three, and four as the fractional lengths of a vibrating string that produces the natural seven-tone musical scale” (Schneider 1994, 335–36). This will make more sense after we look at the nature of harmony in a later chapter.

  15. Ratzinger 1995a, 26.

  16. Among the integers, the prime numbers are fundamental. These are numbers that are only exactly divisible by themselves and one, and which can therefore be said to compose all other integers by multiplication. They are the building blocks of the world of numbers. The first number, 1, is usually not classed as a prime because multiplying by it does not produce another number. In a sense it is more fundamental than any prime, because it produces and is reflected in every other number.