GEOMETRY

GEOMETRY . During the last two millennia bce, the period that produced most religious texts, geometry (lit., "earth measurement," from Greek gaia, ge, "the earth," and metrein, "to measure") was essentially a "geometrical algebra" with a focus on number. Problematic allusions to number and space, which abound in sacred texts, are presumably inspired by this early mathematical protoscience. During the much later development of Christianity and Islam, Euclidean geometry—based, so it seemed, on irrefutable deductive logic built from definitions, postulates, and theorems—became the rational paradigm for all sciences, including theology. The discovery of other geometries in the last two centuries has brought the realization that Euclidean geometry is merely a special case within a wider realm. Efforts to rid mathematics of its logical paradoxes have taught that perfect consistency and certainty are unattainable in rational thought. These developments, together with a new awareness of the complexity of physical space and a better understanding of how culture shapes perceptions, have dramatically altered philosophical dogmatism, making ancient and Eastern modes of thought more congenial to the modern West and contributing to the problems of contemporary religion.

Neolithic Cultures (6000–3500 bce)

In southeastern Europe and the Near East, Neolithic peoples decorated the surfaces of cult objects with geometric motifs—circles, ovals, parallel lines, chevrons, triangles, squares, meanders, and spirals. These abstract designs came to abound in the folk arts of most of the cultures of the globe, seemingly irrespective

of time, of the degree of civilization attained, or of concomitant skill in the realistic depiction of natural objects such as animals, human faces, leaves, and landscapes. Creation myths inherited from Mesopotamia and Egypt of the third and second millennia bce and later from Palestine, China, and Greece, as well as those recorded in modern times in the Americas, Africa, and Oceania show that the act of divine creation is universally conceived as an ordering, a shaping and selection that brings a world, a cosmos, into being. During the Neolithic period, the abstract geometrical motifs that ornament dress, vessels, walls, and other artifacts found in the earliest shrines and villages were expressions of an intuitive identification of order with the sacred and a consequent mobilization of aesthetic feeling in control of design.

In A History of Mathematics (1968), Carl B. Boyer observes that pottery, weaving, and basketry, from the time of their Neolithic origins, "show instances of congruences and symmetry, which are in essence parts of elementary geometry." To Boyer, "simple sequences in design"—such as translations, rotations, and reflections (see figure 1)—"suggest a sort of applied group theory, as well as propositions in geometry and arithmetic." Formal propositions did not appear, however, until the Greeks initiated them in the fifth and fourth centuries bce. Group theory was not developed until the last two centuries; only recently has it been extended to cover the geometric symmetries in space that were already a concern in Palestinian stoneware, for instance, as early as 10,000 bce. The oldest mathematical texts, dating from circa 1900 to 1600 bce, show that geometry developed historically as "the science of dimensional order," in close alliance with arithmetic and algebra, although "the 'spaciness' of space and the 'numerosity' of number are essentially different things" (Alfred North Whitehead, An Introduction to Mathematics, 1911).

"There is a direct correlation between complexity of weaving and sophistication of arithmetic understanding,"

Walter A. Fairservis notes in The Threshold of Civilization (1975). In a settlement like that at Çatal Hüyük in central Turkey, occupied before 6500 bce, the frame posts of the houses are filled in with sunbaked bricks made from molds, furnishing strong economic motivation for knowing precisely "how many bricks were necessary for each wall"; hence, "counting and notation were very much a part of the cultural scene." Far cruder artifacts—sequences of notches incised in bone, studied by Alexander Marshack (1972)—suggest "systems of lunar and other notation" that push the origins of arithmetic far back into Paleolithic times, twenty-five thousand and more years ago. Modern archaeologists and anthropologists are thus producing alternative theories to those of Herodotos (fifth century bce), who believed geometry began in Egypt, motivated by the necessity of reestablishing boundaries after the annual Nile floods, and by Aristotle (fourth

century bce), who also assumed an origin in Egypt, but because "there the priestly caste was allowed to be at leisure."

Ancient Egypt

Chief sources of knowledge of early Egyptian geometry are the Moscow Papyrus (c. 1890 bce) and the Rhind Papyrus (c. 1650 bce). The emphasis here is always on calculation, so that their geometry "turns out to have been mainly a branch of applied arithmetic" (Boyer, 1968). The concept of geometric similarity is applied to triangles, and there is a rudimentary trigonometry. There is a good approximation to π in the formula that computes the area of a circle by constructing a square on eight-ninths of its diameter (see figure 2). In addition, the Egyptians knew the formulas for elementary volumes and correctly calculated the volume of a truncated pyramid.

Modern scholars, however, are disappointed to find so little cause for the high estimation in which the Greeks later held Egyptian science. Respect for the organizational and engineering skills required for the building of palaces, canals, and pyramids, for example, is tempered by the realization that such civic projects entail little more then what Otto Neugebauer (1969) calls "elementary household arithmetic which no mathematician would call mathematics." Neugebauer concludes: "Ancient science was the product of a very few men; and these few happened not to be Egyptian." Of far greater interest is what was happening in Babylon.

Babylon

Several hundred baked-clay tablets about the size of the palm of the hand, incised with neatly crowded rows of cuneiform inscriptions, provide more information about the mathematical sciences in Babylon circa 1900 to 1600 bce than exists for any other place or time preceding the Elements of Euclid, circa 320 bce. Standard tables of multiplication of reciprocals and a place value notation on base sixty facilitated computation at a level Neugebauer compares with that of Europe in the early Renaissance, more than three thousand years later.

Babylonian geometry, like that of Egypt, was still "applied algebra or arithmetic in which numbers are attached to figures" (see figure 3). The ratio between the side and diagonal of a square (i.e., the square root of two) was computed correctly to about one part in a million (see figure 4). Ratios between the areas of a pentagon, hexagon, and heptagon and those of squares built on one side were closely approximated, as was the value of π. A geometric concept of similarity is applied to circles, and perhaps also to triangles. An angle inscribed in a semicircle is known to be a right angle (see figure 5). The Pythagorean theorem (which holds that the square on the hypotenuse of a right triangle is equal to the sum of the squares on the other two sides) was understood in all its generality a thousand years before Pythagoras. One tablet, known as Plimpton 322, develops a set of fifteen "Pythagorean triplets" (three numbers defining right triangles, such as 3,4,5) in a sequence in which acute angles vary by approximately one degree. This "prototrigonometry," unsuspected until the tablet was translated, is one of the most astonishing mathematical achievements of the ancient world. It demonstrates empirical knowledge of a general formula:

Assuming integers p and q with p > q, and taking p = 2 and q = 1, the 3,4,5 triplet emerges as follows:

It was from the Babylonians, rather than from the Egyptians, that the Greeks inherited the fund of empirical insights that they transformed into an exact science. The example just cited—in which the ratio of the musical octave, 2:1, is transformed into the Pythagorean triplet 3,4,5—is the foundation of Plato's cosmogony, and it may turn out to be one of the most important clues to the numerology of the ancient world.

Greek Transformation of Egyptian and Babylonian Knowledge

Researchers are still trying to unravel the story of how the Greeks, in less than three centuries (600–300 bce), transformed geometry—inherited essentially as an art of making arithmetical relations visible—into a science based on definitions, postulates, and theorems (and appealing therefore to an invisible logos ). Material inherited from Egypt and Babylon through Thales (c. 585 bce), and Pythagoras (c. 550 bce) was riddled with confusion between exactness and approximation. A new "dialectical" spirit arose with Parmenides (c. 475 bce) and his followers. Precise definition, always elusive, was the new goal that Socrates (d. 399 bce) applied to moral and ethical questions by appealing to "harmonic" examples from the Pythagorean geometry of the vibrating string. Plato's affection for dialectics and his emphasis on abstraction bore unexpected fruit in Aristotle's rejection of his teacher's Pythagorean methods: "Dialectics is merely critical where philosophy claims to know" (Metaphysics 1004b). Aristotelian "first philosophy" developed a new syllogistic rigor in the generation after Theaetetus (c. 414–c. 369 bce) and Eudoxus (c. 400–c. 350 bce), who changed the method and enlarged the scope of mathematics. Another generation later, Euclid's Elements completed the mathematical transformation so successfully that the only traces of earlier Greek mathematics that survive must be gleaned from his work. Little else was deemed worth copying.

For current understanding of geometrical symbolism in ancient religions, the Greek transformation has almost no meaning. Although Christianity and Islam are historically young enough to have been affected by the transformation, their holy books show virtually no influence of the new Greek science. (A possible notable exception is the opening line of the Gospel of John: "In the beginning was the Logos.") Plato, the first author in history whose works have survived intact, made an extended commentary on the mathematical bias in his philosophy and offers the richest insight into the intentionality that inserted so many mathematical elements in ancient mythology. The musical geometry at the heart of his mathematics was common to both East and West; its simplest pragmatic formula is found in China.

Harmonic Cosmology in China and Greece

The natural, or counting, numbers (1, 2, 3, …, infinity) are a primordial image of order. Developed systematically into Pythagorean triplets, they lead to a prototrigonometry of the plane. Applied systematically to the geometry of a vibrating string, they link the magical realm of tone with the numbers that measure the world. To use economical modern concepts, the octave ratio 1:2 becomes the cyclic module (Plato's matrix, or "universal mother") in which the even numbers are "modular activities" (doubling and halving merely produce further "octave identities") and the odd numbers are "modular residues" (meaning that they define new pitches within the octave matrix). To build a scale, the simplest procedure is to follow the old Chinese rule of adding or subtracting one-third (from any reference pipe or string length). This is the geometrical analogue of the musical procedure of tuning by ear: A subtraction of one-third correlates with the musical interval of an ascending perfect fifth (3:2); an addition of one-third correlates with the descending perfect fourth (4:3). To avoid fractions in the arithmetic, the reference length must contain one factor of three for every "tone child" to be generated. The Chinese pentatonic (five-tone) scale must therefore be generated from 3;s4 = 81.

Rearranged into scale order, this number sequence has a reciprocal "twin" that defines frequency ratios:

Both Chinese and Greek cosmology are projections from this tonal geometry, reducible to continued operations with the prime number 3. Note that the defining operation started on 34 = 92 = 81. Ancient China was conceived as 1/81 part of the whole world, that is, as 1/9 of one of the nine "great continents." China was also considered to be divided into nine provinces, so that each Chinese province was 1/729 of the whole world; now 36 = 93 = 729 is the base for the same tuning calculation when it is extended through seven tones for the complete diatonic scale, also standard in China in the fifth century bce. At about the same time, Philolaus, a Greek Pythagorean philosopher, conceived the year as made up of "729 days and nights," a number that would seem to come from nowhere but such a musical cosmology. Plato linked the seven tones in this set to the sun, moon, and five planets; later, Ptolemy (second century ce) linked the scale to the zodiac.

The numbers 64 and 81, on which the alternate scale progressions commence, and the number 108, largest in the set of pentatonic frequency values, are immortalized in various ways. In China there are 64 hexagrams in the Yi jing divination text. The numbers 82 = 64 and 92 = 81 have been the favored squares on which to construct the Hindu fire altar since Vedic times (c. 1500 bce). The number 108, upper limit in the set, is the number of beads in the Buddhist rosary. The tuning pattern itself has recently been discovered (but without numbers) on an Old Babylonian cuneiform tablet from circa 1800 bce.

The set of twelve consecutive tones generated by the above procedure constitutes a chromatic scale. In ancient China each twelve tones in turn became the tonic of the standard pentatonic scale for a particular calendric period. Throughout Chinese history the bureau of standards remained wedded to a tonal geometry: The length of a pitch pipe (an end-blown hollow tube) sounding the reference tone determined the standard for both weights and volumes. Each new regime established a new reference pitch; today there is the record of dozens of succesives changes in the bureau of standards as the reference pitch oscillated over the range of about the interval of a sixth.

Unless one knows the musical procedure, the Daoist formula for the creation of the creation of the world sounds mystical: "The Dao [the Way] produced one, the one produced two, the two produced three, and three produced the ten thousand things [everything]." The creation myth related by Plato in the Timaeus similary develops the world's harmonical soul and body from the numbers 1,2, 3. Pythagoreans frankly announced that, to them, "All is number," and Aristotle quotes them as saying, "The world and all that is in it is determined by the number of three." Plutarch describes planetary ditances in the Philoaus system as "a geometrical progression with three as the common ratio." For eight hundred years Greek astronomers toyed with varations on this planetary motion (c. 1600 ce) while still looking for the right tones to associate with each celestial body. East and West, the "geometry" of heaven and earth was musical and profoundly trintarian while astronomy was being gestated.

The Greek view of this tonal-planetary geometry points in the direction of a more abstract mathematical system. The Pythagorean "holy tetractys" was a pebble pattern symbolizing continuing geometric progression from a point through a line and a plane to the "solid" dimension (see figure 6). Plato takes advantage of the double meanings of integers (both as whole numbers and as reciprocal fractions) to generate the material for a seven-tone scale at the cube dimension (33=27; see figure 7). Nicomachus (fl. c. 100 ce), writing an introduction to Plato, simplifies the double view of the multiplication table 2 × 3 up to the limit of 36 = 93 = 729 (see figure 8.) Stones, musical tones, planets, numbers, and geometry are all part of one vast Pythagorean synthesis, replete with symbolic cross-reference and a supporting mythology. An exasperated Aristotle mocked it; Euclid made it obsolete.

The problem of "excess and deficiency" with which a musician wrestles in adjusting the geometry of the string by ear had its arithmetical analogue in the ancient problem of making approximations between the areas of a circle and square and the volumes of a sphere and cube. Thus harmonics was a paradigm for an ethics of moderation; "nothing too much," the Greek ideal, had its counterpart in Confucian concepts of morality and behavior. Thousands of years earlier, the Egyptians had conceived the scales of Maat as the "great balance" on which the heart of the deceased was figuratively weighed to test its fitness for immortality. Thus the wisdom literature of the ancient world shows remarkable parallels between cultures.

The historical record is so fragmented, however, that interpretation of geometric symbols remains speculative. Modern studies in the neurophysiology of vision and in related psychological inferences suggest that schematic, geometric relations play decisive roles. The universal acceptance of the octave ratio 1:2 is further evidence of human psychophysical norms that could generate correspondence between cultures that were never in contact. While they can neither fully document the paths of cultural diffusion nor even claim that diffusion is necessary, researchers cannot entirely rid themselves of suspicion that there was a great deal more diffusion than can be proven.

Problems in Ancient Geometric Symbols

It is easy to imagine that the ancient stone circles that abound in Europe and America linked people to events in the sky—that the twenty-eight poles arranged in a circle for the lodge of the Arapaho Indians' Sun Dance, for instance, may correspond to twenty-eight lunar mansions; that the twelve sections of the Crow tribe's lodge for that ceremony may allude to the months of the year; and that other cultures possessed similar symbols of earth, sky, and calendar (Burckhardt, 1976). But many familiar symbols are more puzzling. Why, for instance, did the Pythagoreans take a five-pointed star (see figure 9) as their special symbol? Is it because each line cuts two others in "mean and extreme ratio" (meaning that the whole lines is to the longer segment as the longer is to the shorter) so that the figure symbolizes both "continuing geometric progression" (the world's "best bonds," for Plato) and a victory over the "darkness" of the irrational? Could the Hindu "drum of Śiva" (see figure 10)—with its inverted triangles and the interlocked triangles of the star-hexagon (see figure 11), prevalent in Indian and Semitic cultures—be related to the Pythagorean symbols in figures 6 and 7?

The so-called Pascal triangle (see figure 12), containing the coefficients for the expansion of the binomial (a+b ) 2,3,4,5,6 was known to Pingala (c. 200 bce) as Mount Meru, the Hindu-Buddhist holy mountain. Pingala interpreted this triangle as showing the possible variations of meter built from monosyllables, disyllables, trisyllables, and so on. Could this "Mount Meru" be related to the holy mountains of other Eastern religions? Is the Pythagorean tetractys simply the Greek form of older holy mountains? Is it significant that the Sumerian symbol for mountain is a triangular pile of bricks (see figure 13), aligned in a pattern Pythagoreans found useful for numbers? Is the hourglass shape of the later Buddhist holy mountain simply a geometric variation of the "drum of Śiva"?

Translation of the ancient Babylonian mathematical texts now makes clear that the computational sophistication achieved four thousand years ago was so great that the sacred texts of all peoples must be studied with new alertness for evidence of rational—and not merely poetic—inspiration. The old Sumerian-Babylonian gods possessed straightforward numerical "nicknames" (used for scribal shorthand) in sexagesimal (base sixty) notation; the three great gods—Anu-An, whose numerical epithet was 60 (written as a large 1), Ea-Enki, associated with 40, and Enlil, associated with 50—are functional equivalents of Plato's 3,4,5 Pythagorean genetic triad. Could some of the ancient religious mythology turn out to be mathematical allegory?

It seems curious that the ancient Greek altar at Delphi is built on cubic dimensions, as are the chapel of the Egyptian goddess Leto that Herodotos saw in the city of Buto, the Vedic fire altar, the Holy of Holies in Solomon's temple, and the ancient Sumerian ark (first to "rescue a remnant of mankind from the flood"). The name of the most sacred Islamic shrine, the Kaʿbah at Mecca, literally means "cube," and the city of New Jerusalem in the Book of Revelation is also measured in such dimensions. All of these cubic consonances between various religions suggest that poetic religious imagination has had a geometrical, "protoscientific" component for a very long time.

Geometry since Euclid

The thirteen books of Euclid's Elements culminate in a treatment of the five regular, or Platonic solids—tetrahedon, cube, octahedron, dodecahedron, and icosahedron—each with uniform sides and angles and all capable, when replicated, of closely packing three-dimensional, abstract space. Euclid mastered the transformational symmetries of these "rigid" bodies.

In the third century bce Archimedes did important work on the area of the surface of a sphere and of a cylinder and on their respective volumes, and Apollonius carried the study of conic sections (ellipse, parabola, and hyperbola) to its highest development. Later Greek authors made further

advances in geometry, but the great wave of development that had begun scarcely four hundred years earlier was spent.

Later Hindu talent has been mainly arithmetic and algebraic; theorems on the areas and diagonals of quadrilaterals in a circle were contributed by Brahmagupta (c. 628 ce). Important Arab contributions to the solution of cubic equations by the method of intersecting conics were summarized by ʿOmar Khayyam (c. 1100 ce). European development has been rapid since Kepler's time; it was Kepler who introduced the concepts of the infinitely great and the infinitely small, which Euclid had carefully excised from consideration. The invention of analytic geometry by Pierre de Fermat and René Descartes in the seventeenth century led to a new integration of geometry and algebra.

The contributions of Euclid's Elements to the invention and development of the physical sciences during all these centuries is inestimable. His kind of logical, geometrical argument is the basis for Archimedes' formalization of the laws

of the lever; for the Greek development of astronomy as a physical science (by Hipparchus, Ptolemy, and others); for Galileo's work on the dynamics of the inclined plane; for Kepler's laws of planetary motion; for Newton's laws of planetary gravitational dynamics; and for an endless host of related physical sciences. Euclidean geometry, still being improved today, has thus been one of the Western world's most powerful engines of progress. The Elements has been "the most influential textbook of all times" (Boyer, 1968), and it was long assumed to be as certain a guide to geometry as the Bible to absolute truth. God, it was confidently asserted, is a geometer.

During the nineteenth century, which Boyer calls the "golden age of geometry," this almost-perfected world of traditional, Euclidean-inspired mathematical physics exploded with a creative energy, leading to a crisis in the very foundations

of mathematics. For example, one of the several new geometries that appeared, projective geometry, has nothing to do with measurement. Several "non-Euclidean" geometrics, by N. I. Lobachevskii, Wolfgang Bolyai, and G. F. B. Riemann, omit Euclid's famous sixth postulate concerning parallel lines, always somewhat suspect, to create equally logical geometries of even wider generality.

The favored status of Euclidean geometry has evaporated. There are no longer any assumptions that command universal assent, no systems of logic powerful enough to validate themselves even in mathematics. Mathematics has thus lost some of its certainty even while multiplying its powers. Einstein's notion of "space-time," with its curvatures and paradoxes, has relegated Euclidean geometry to the role of a convenient tool for certain modern intuitions within a severely limited local space. Space perception itself has been proved culturally biased; intuition can be taught new tricks. Space has also multiplied its hallowed three dimensions beyond any possibility of imagining; today the number of its dimensions is the number of independent variables in various formulas.

With the shattering of mathematical certainty and the ending of the idolatry of Euclidean rationalism, Western scientists in particular have felt a new attraction to the philosophical skepticism of Vedic poets and to Daoist and Buddhist feeling for how the world behaves. The ancient worldview was created in imagination; today imagination is still proving more powerful than logic. In the ferment of this present age, mathematics and physics are committed, perhaps more clearly than ever—but as an act of faith rather than of reason—to the primordial affection for symmetry, guided less by reason than by aesthetic feeling for elegance and beauty.

See Also

Circle; Numbers; Quaternity; Sacred Space.

Bibliography

For a basic textbook in Euclidean geometry and in the various non-Euclidean modern geometries, see H. S. M. Coxeter's Introduction to Geometry (1961; reprint, New York, 1969), notable for the elegant compression it achieves by rigorous pursuit of symmetry and of the "group" of transformations this includes. The history of geometry is set within the wider context of the whole of mathematics and the contributions of nations and individuals treated with great fairness in Carl B. Boyer's A History of Mathematics (New York, 1968). On the early period, Otto Neugebauer's The Exact Sciences in Antiquity, 2d ed. (New York, 1969), has become a classic. The transformation of Egyptian and Babylonian empirical knowledge into Greek science is studied in B. L. van der Waerden's Science Awakening, translated by Arnold Dresden (New York, 1963). Richard J. Gillings's Mathematics in the Time of the Pharaohs (Cambridge, Mass., 1972) is an effort to balance the somewhat negative views of mathematicians toward Egyptian science.

A splendidly illustrated study of Paleolithic stone and bone counting records is Alexander Marshack's The Roots of Civilization (New York, 1972). Neolithic symmetry in pottery and weaving designs is richly illustrated in Marija Gimbutas's The Goddesses and Gods of Old Europe, 6500–3500 b.c. (Berkeley, Calif., 1982). Modern concepts of symmetry are formalized, with great respect toward ancient craftsmen who explored it intuitively, in Hermann Weyl's Symmetry (Princeton, N.J., 1952).

Applications of traditional geometry to religious art and architecture can be found in Robert Lawlor's Sacred Geometry: Philosophy and Practice (New York, 1982) and Titus Burckhardt's Art of Islam: Language and Meaning, translated by J. Peter Hobson (London, 1976). A stone-by-stone analysis of the geometry employed by the several builders of a great cathedral is masterfully displayed in John James's The Contractors of Chartres, 2 vols. (Beckenham, U.K., 1980).

For a development of the tonal-geometrical symbolism hidden in ancient religious and philosophical texts, see my books The Myth of Invariance (New York, 1976), The Pythagorean Plato (Stony Brook, N.Y., 1978), and Meditations through the Quran (York Beach, Maine, 1981).

New Sources

Gray, Jeremy. Ideas of Space: Euclidean, Non-Euclidean and Relativistic. New York, 1989.

Lachterman, David Rapport. The Ethics of Geometry: A Genealogy of Morality. New York, 1989.

Magnani, Lorenzo. Philosophy and Geometry: Theoretical and Historical Issues. Boston, 2001.

Mlodinow, Leonard. Euclid's Window: The Story of Geometry from Parallel Lines to Hyperspace. New York, 2001.

Romano, David Gilman. Athletics and Mathematics in Corinth: The Origins of the Greek Stadion. Philadelphia, 1993.

Szulakowski, Urszula. The Alchemy of Light: Geometry and Optics in Late Renaissance Alchemical Illustration. Symbola and Emblemata Studies in Renaissance and Baroque Symbolism, vol. 10. Leiden, 2000.

Ernest G. McClain (1987)

Revised Bibliography