String, Straightedge, and Shadow the Story of Geometry

by Julia E. Diggins

  To impress this on newcomers, perhaps the first thing they were shown was how to make a mystic ‘pentagram’, the emblem that members of the Order wore on their clothing. By means of a secret device (which we will explain later) a five-sided figure, or pentagon, was traced on cloth. Then its points were connected with diagonals to make a five-pointed star. Finally, around the five points of the star were placed the letters of the Greek word for health, ὑγιεια (hygeia), from which we get the word ‘hygiene’.

  This was the sacred symbol of the Pythagorean Order – the ‘magic pentacle’ that remained a favourite device of sorcerers and conjurors for many centuries. But it was also an experimental discovery: the first known use of letters on a geometric figure.

  Possibly the next experiment shared with newcomers was a basic one with tiles. Ordinary floor tiles had yielded the easiest example of the Pythagorean theorem. So the Secret Order went on with a painstaking study of these close-fitting forms that covered many Greek floors.

  They made loose tiles of various shapes and placed them in patterns on the ground. And they reached a striking conclusion. There were only three regular shapes of tiles that would fit together perfectly to cover a flat area completely: triangles (three sides), squares (four sides), hexagons (six sides).

  If they tried pentagons, they got a beautiful blossomlike design, but there were gaps between the tiles, and tiles of more than six sides would always overlap. No other regular geometric forms of the same size and shape could be so combined.

  They explained this mystery to the newcomers: ‘Since there are four right angles (360°) around a point, you can only use forms whose corner angles together will make that total. There are just three possibilities: six equilateral triangles with 60° angles, four squares with 90° angles, and three hexagons with 120° angles.’

  Regular solids

  From this simple experiment came the fascinating idea of making ‘solid angles’ by fastening tiles together with mortar, or gluing together shapes of wood, or sewing together pieces of leather. And this led to building shapes with the solid angles.

  They called them regular solids because all the edges and faces and angles in each solid were equal. And after much experimenting, as we have said, they found five of these solids. The first two had been known from the most ancient times, but the next two were shapes that people had never seen before. As for the fifth, it was such a startling discovery that they thought they had upset the order of the universe!

  The Cube

  They mortared three square tiles into an angle, and fitted on three more tiles to form a cube with six square faces, which they called a hexahedron.

  The Regular Pyramid

  They put together three equilateral triangles into a solid angle, then added one more, to make the base of their four-faced tetrahedron.

  The Octahedron

  This was made with two solid angles of four equilateral triangles each, so they gave this eight-faced figure the name octahedron.

  The Icosahedron

  Here was a real challenge. When they put together five equilateral triangles, they got a surprise. The open base of this solid angle was a pentagon.

  Now they could trace one perfectly for their emblem, instead of just drawing it freehand (of course, the method was kept secret). But how could they make a regular solid, with five equilateral triangles around each vertex? All their early attempts were failures. Finally, someone got the right inspiration – five equilateral triangles for the top and five triangles for the bottom, and then a centre band of ten more triangles based on the old Babylonian pattern. They had made an icosahedron with twenty triangular faces.

  The Dodecahedron

  The last form was made with the pentagons that were so dear to the Pythagorean Order. They used that flower-like pattern of a single pentagon surrounded by five others – the tiles that would not fit together on the flat floor. But if the surrounding ones were lifted up, then all six pentagons fitted perfectly in a solid cuplike shape. This could be capped by an inverted one just like it to yield the most difficult and beautiful of the five regular solids: the dodecahedron with its twelve pentagonal faces.

  These five shapes created a great stir among the first geometers who studied them. People examined them in fascination and awe, handling them, turning them around in different positions, looking through them as if they were glass. It was somewhat inevitable that the Pythagoreans would assign them mystical meanings.

  The mysteries of the regular solids

  By that time the Pythagorean Order had spread to many towns of Sicily and southern Italy. The Sicilian members were friendly with another strange teacher who lived near Mount Etna – Empedocles, who dressed all in purple, gave away his money and did scientific experiments. Empedocles taught that the world was made of earth, water, air and fire, and the first four regular solids came to be identified with these ‘elements’. We know the identification from a famous passage in one of Plato’s Dialogues, where it is made by a Pythagorean from Locri in the south of Italy. There were many strange stories about the fifth solid, and indeed its existence was kept secret as it seemed to require a fifth ‘element’.

  The esoteric reasoning, as repeated later, was as follows:

  The cube, standing firmly on its base, corresponds to the stable earth. The octahedron, which rotates freely when held by its two opposite corners, corresponds to the mobile air.

  Since the regular pyramid has the smallest volume for its surface, and the almost spherical icosahedron the largest, and these are the qualities of dryness and wetness, the tetrahedron stands for fire and the icosahedron for water.

  As for the last-found regular solid, with its twelve faces: why not let the dodecahedron represent the whole universe, since the zodiac has twelve signs!

  Such notions were typical of that age. And more than two thousand years later, the famous astronomer Kepler was still so awed by the unique properties of the five regular solids that he applied them to the intervals between planetary orbits: he assigned the cube between Saturn and Jupiter, the pyramid between Jupiter and Mars, the dodecahedron between Mars and Earth, the icosahedron between Earth and Venus, and the octahedron between Venus and Mercury.

  Remarkable shapes

  Even today, these regular solids seem almost magical in their beauty and their interrelations.

  In the first place, it is quite startling that there are only five. An infinite number of regular polygons can be inscribed in a circle – their sides becoming so small that they approach the form of the circle itself. But it is not so with regular convex polyhedra inscribed in a sphere. There are only these five possible shapes, and no others.

  And these five shapes are connected with one another in a most remarkable way. All five can be fitted together, one inside the next, like the compartments of some magic box. And they are further linked by a strange inner harmony. They can be inscribed in themselves or each other, in certain endless rhythmic alternations. So it’s no wonder the five regular solids were often referred to as the ‘dice of the gods’.

  14. An Unspeakable Tragedy

  The members of the Pythagorean Order thought they had grasped the key to the cosmos.

  Then everything collapsed. Their whole scheme was destroyed by a fatal discovery, and the Order itself was destroyed by traitors and mob violence. Yet as we retell the sombre tale, we will find that it was not a complete tragedy after all, for the Pythagoreans did enjoy their cosmic key briefly. This key was not found in abstract shapes alone, nor in music, nor in the stars, but in one factor that – they believed – linked all of these: number.

  Pythagoras had said it: ‘Everything is number!’

  So they followed Pythagoras’ teaching that the universe was ruled by whole numbers. That did not mean numbers for ordinary counting or calculating. What interested them was the nature of a number itself – odd, even, divisible, indivisible – and the relations between numbers. This was their arithmetike. And they applied it to their othe
r three fields, and found startling number patterns in each.

  Musical intervals

  In music, for instance, a sensational discovery about the relations of whole numbers and musical intervals was attributed to Pythagoras himself.

  One legend said that on his long voyages he listened to the music of flapping sails and the wind whistling and whining through the ship’s rigging and playing a melody on the ropes, and that he decided then and there to investigate the connection between the tempest of sounds and the vibrating strings.

  Another version said that he was strolling through the town of Crotone, deep in thought, listening to the musical sounds of hammers striking anvils in a blacksmith’s shop, when suddenly, tripping on a taut string that some children had stretched across the street, he got the inspiration for an experiment.

  But the most popular story told that the idea came to him straight from the stringed lyre of his ‘father’ Apollo, who was also the god of music.

  Anyway, Pythagoras experimented with stretched strings of different lengths placed under the same tension. Soon he found the relation between the length of the vibrating string and the pitch of the note. He discovered that the octave, fifth and fourth of a note could be produced by one string under tension, simply by ‘stopping’ the string at different places: at one-half its length for the octave, two-thirds its length for the fifth and three-quarters its length for the fourth.

  Other musical innovations were credited to him, such as a one-string apparatus for the study of harmonics. But his great discovery was the tetrachord, where the most important harmonic intervals were obtained by ratios of the whole numbers: 1, 2, 3, 4. The Order gave this fourfold chord mystical significance and used to say, ‘What is the oracle at Delphi? The tetrachord! For it is the scale of the sirens.’

  Music of the spheres

  In the relation of number and music, the Pythagoreans believed they had found the pattern that guided the ‘wandering’ planets through the heavens. They pictured the sun and the planets as geometrically perfect spheres, moving through the visibly circular sky on perfect circular orbits, separated by harmonic ratios – musical intervals! Theirs was a vision of time and space revealed in lines, tones and mathematical ratios. And they even imagined the brilliant planets emitting harmonious tones, the so-called ‘music of the spheres’.

  Triangular and square numbers

  But it was in the connection of number and geometry, their two completely mathematical subjects, that the Pythagoreans were on surest ground. Numbers, they had discovered, whole numbers, actually had geometric shapes. There were triangular numbers, square numbers, pentagonal numbers, rectangular numbers, and so on.

  This was no wild fantasy like the singing planets. It was a real mathematical discovery and came from the circumstance that they did not do their number work by writing numbers at all. Instead, they placed pebbles on the sand. But the Pythagoreans placed their pebbles in patterns, adding extra rows for each number. Their two most important series were the square numbers and the triangular numbers.

  The most important number of all, to the Pythagoreans, was the fourth triangular number, 10, for it was made up of 1 + 2 + 3 + 4. They called it the Sacred Tetractys, swore by it in their oaths, and attached marvellous properties to it, as ‘the source and root of eternal nature’.

  Irrational numbers

  Everything fitted perfectly: the Tetractys, the tetrachord, the four regular solids representing the four ‘elements’, inscribed in a dodecahedron representing the celestial sphere. But it was all too pat, a jumble of luck, imagination, serious mathematical experiments and old number magic from the East. Just as the Pythagoreans thought they were getting more and more evidence that number was everywhere, the whole system broke down. The entire connection between geometry and number – the foundation of their thinking – was shattered by one disastrous experiment.

  Presiding was Hippasus of Metapontum, whose name was to loom dark in the future of the Order. The idea was simply to find the numbers that matched the sides of the two right-angled triangles with which Pythagoras had first demonstrated his theorem – the Egyptian triangle and the one from the tiled floor.

  Of course, the Egyptian rope-triangle worked perfectly: its 3–4–5 sides made a beautiful Pythagorean series. They indicated the intervals with pebbles. Now what about the right-angled triangle from the Greek tile design, where the two sides were equal?

  Suppose each side had a length of 1 unit – that would require 1 pebble. Then for the hypotenuse – how many pebbles should they put there? Well, the sum of the squares on the sides would equal the square on the hypotenuse. Therefore,

  12 = 1 (square on one side)

  and 12 = 1 (square on other side)

  and 1 + 1 = 2

  so 2 is the square on the hypotenuse. And the hypotenuse is the square root of 2.

  But what was the square root of 2?

  It couldn’t be a whole number, since there is no whole number between 1 and 2.

  Then was it a ratio of whole numbers between 1 and 2? Hopefully, they tried every possible ratio, multiplying it by itself, to see if the answer would be 2. There was no such ratio.

  After long and fruitless work, the Pythagoreans had to give up. They simply could not find any number for the square root of 2. We write the answer as 1.4141…, a continuing decimal fraction, but they couldn’t do that since they had no concept of zero and of decimals. They could draw the hypotenuse easily, but they could not express its length as a number. It was ‘unutterable’.

  Horrified, the Pythagoreans called √2 an irrational number. After that, they found other irrationals and swore to keep them secret, for the discovery of these ‘irrationals’ wrecked their entire beautifully constructed system of a universe guided by whole numbers. The breakdown in their mystical morale was followed by the breakup of the Order itself.

  Hippasus’ betrayal

  In this final demolition, Hippasus played a decisive role, though his own fate is shrouded in mystery. The Order was already in trouble. Bitter resentment had grown up against its secrecy and exclusiveness, and riots of villagers had driven it out of Crotone. Pythagoras himself had died on a neighbouring island. And now mobs of ‘democrats’ began to attack the aristocratic Pythagorean societies everywhere.

  Against this background, Hippasus took a step that was regarded by the conservative members as sheer betrayal. He broke the oath of secrecy and revealed their most closely guarded discoveries – the dodecahedron and the irrationals. When they promptly expelled him, he set himself up as a public teacher of geometry.

  The traitor’s punishment was swift and terrible. He very soon drowned in a mysterious ‘accident’ at sea, and rumours circulated. Some said that a storm had struck his ship as a direct vengeance from the gods: others, that he had been pushed overboard by agents of the Order. But Hippasus’ death was to no avail. The harm was already done to the Order of Pythagoreans. The remaining secret groups soon collapsed, torn by outer violence and inner dissensions. And more and more ‘mathematicians’ followed Hippasus’ example and came out to earn a living as teachers. Pythagoras’ idea had been demolished: no longer was there a closed Order of followers, bound together by a mystical belief in a cosmos ruled by number. Yet his ideals lived on in this broader field. He had pursued knowledge for its own sake, loving wisdom for itself. He knew learning could be shared without diminishing, that it lasts through life and immortalises the learned after death. And the destruction of the Order gave his legacy to the world.

  Geometry was now out in the open – and it was the new Pythagorean geometry. True, mathematics was still mixed with some magic: number mysticism, cosmic ideas about the regular solids. But there was, in addition, the famous theorem and its applications, the careful study of shapes, the theory of numbers and the discover of irrationals.

  From the Academy to the Museum

  Geometry, Art and Science

  15. The Golden Age and the Golden Mean

  The second
half of the fifth century BC was the Golden Age of Greece. This was the period of its most beautiful art and architecture, as well as some of its wisest thinkers. Both owed much to the popular new study of geometry.

  By the start of the next century, geometry itself was entering its own classic age with a series of great developments, including the Golden Mean. The times were glorious in many ways. The Persian invaders had been driven out of Hellas forever and Pericles was rebuilding Athens into the most beautiful city in the world. At his invitation, Greek mathematicians from elsewhere flocked into the new capital. From Ionia came Anaxagoras, nicknamed ‘the Mind’. From southern Italy and Sicily came learned Pythagoreans and the noted Zeno of Elea.

  High on the hill of the Acropolis rose new marble temples and bronze and painted statues. Crowds thronged the vast new open-air theatre nearby, to hear immortal tragedies and comedies by the greatest Greek playwrights. These splendid public works were completed under the direction of the sculptor Phidias and several architects, all of whom knew and used the principles of geometry and optics.

  ‘Success in art,’ they insisted, ‘is achieved by meticulous accuracy in a multitude of mathematical proportions.’ And their buildings had a dazzling perfection never seen before – the beauty of calculated geometric harmony.