I. The Mystical Triangle Think back to geometry class, and perhaps you can dredge up a sleepy memory of Pythagoras. He's credited with discovering that the square of the longest side of a right-angled triangle -- remember the hypotenuse? -- is equal to the sum of the squares of the other two sides. That factoid is Pythagoras's most visible legacy. You may know it as the Pythagorean Theorem: a2 + b2 = c2. But hardly anyone who uses the formula knows that this mathematician of ancient Greece, who has helped so many of us navigate right triangles and solve a host of practical problems in the real world, led a bizarre religious cult and died in a fiery blaze when the forces of democracy rallied against him and his mystic band. The fire is very reminiscent of the one in which David Ko-resh and many of his Branch Davidian followers died near Waco, Tex., in 1994. Imagine that Koresh in his spare time had been a world famous philosopher and scientist and also a good friend and personal, spiritual and dietary adviser to, say, Bruce Jenner, the former Olympic decathlon champion. Imagine that the two had been hanging out with other Branch Davidians at Jenner's house when federal agents swarmed the joint. That would be the picture, because, according to historians, that's how Pythagoras died -- at the house of his disciple, Milo, a famous Olympic wrestler. They were attacked by community leaders who thought that Pythagoras' personality cult had gone far enough. Kind of makes geometry and right triangles seem a whole lot sexier, doesn't it? II. WHAT THE ORACLE SAID Mnesarchus, a Greek jeweler, and his wife, Parthenis, a housewife, were affluent worshippers of Apollo and lived on the Greek isle of Samos. Theirs was the quiet existence of a young couple in one of the less bustling regions of ancient Greece. Soon, their peace would change. On a visit to the town of Delphi, according to Iamblichus, the 4th century Syrian philosopher, they consulted one of Greece's most famous religious authorities, a priestess called "the Pythian oracle" and thought able to see the future. She told the young couple they would have a son who would change the world. Perhaps Mnesarchus had visions of a Greek Cal Ripken or Joe Montana because Delphi was the site of a major athletic festival, the Pythian Games, held every four years, like the Olympic Games. Mnesarchus promptly renamed his wife Pythias. Then the two lovers dutifully followed the seer's instructions to visit Phoenicia, in what now is Syria, for the baby's birth. In 582 B.C., at Sidon, Mnesarchus and Pythias had a son. They named him, too, in the Pythian tradition. He was Pythagoras (pith-AG-or-us). The family moved to Samos, which some sources list as Pythagoras's birthplace. At 18, Pythagoras left Samos. He studied in Miletus, a Greek-occupied seaport on the coast of modern-day Turkey, with Thales, the legendary naturalist, and explored his roots in Phoenicia. Then he went to Memphis in Egypt to study for more than 20 years with Egyptian religious mystics. Not until Pythagoras turned 60 did he begin to settle down, and his thoughts on aging were not those of the typical senior citizen. According to Diogenes Laertius, a 3rd century biographer and one of many ancient writers who burnished the legends of Pythagoras, the great man saw life in four stages. "Twenty years a boy, 20 years a youth, 20 years a young man and 20 years an old man," Pythagoras said. Having completed the first three of those epochs, Pythagoras decided to nest. His travels had brought him to Croton, a city in southern Italy inhabited mostly by Greeks. Historians disagree about what he did there, but there is general agreement that Pythagoras combined radiant charisma with a shaman's magnetic charm. He was a teacher of many things, an uptown mystic in a backwater town. Crotonians flocked to bask in his glamour and soak up his wisdom. Pythagoras also had a "golden thigh," according to many ancient writers, who soberly reported this curious characteristic as if the man's leg literally were made of the metal. This led Crotonians to what they considered a natural conclusion: Pythagoras was either the god Apollo or the son of Apollo. There was no one the great man wouldn't teach -- children, the poor, city elders. To all, he disseminated his beliefs on faith, diet and morality. He even spoke to women, whom he treated as equals, atypical behavior for a man in those days that led some historians a few centuries later to discuss parallels between stories about Pythagoras and those about Jesus Christ. For example, Iamblichus claims that, within a few days after Pythagoras arrived at Croton, the teacher saw fishermen pulling in a large haul of fish. Pythagoras told them how many fish they had. When they tallied their catch, they were stunned to learn that he was correct. When Pythagoras ordered the men to return the fish to the sea, they thought it best to comply, Iamblichus says. As if by a further miracle, not one fish died. With such flourishes, Pythagoras spread his doctrine of strict vegetarianism. According to legends, he had long performed these odd, magical feats. For example, traveling as a youth, Pythagoras had been seized by pirates hoping to sell him on the slave market. Oddly, the boy didn't seem to mind. He sat in a corner of the ship in a trance. He didn't eat. He didn't drink. He didn't move for three days. The pirates became concerned. But when opportune winds blew the ship well ahead of schedule, they concluded that the boy was no mere waif but a god. When the ship reached Egypt, crew members carried him ashore. They erected an altar to him, surrounded him with food and fruits and departed. Pythagoras had that kind of luck. His good fortune continued in Croton whose citizens had built him a school and a temple honoring his patron god, Apollo. His burgeoning coterie of followers became a "state within a state." According to modern scholar Peter Gorman in his book Pythagoras: A Life, the faithful were divided into two ranks -- disciples who lived in a commune, sharing all possessions, and a larger group called "Acousmatics," whose dedication was less consuming. The disciples lived faithfully in step with their guru. Like Pythagoras, they ate lightly, taking neither meat nor fish. They slept sparingly, drank no alcohol, insisted on monogamy. They never ate beans because Pythagoras taught that men's souls were inside the beans. They never traveled the high road, never touched white roosters. They "received from {Pythagoras} laws," Iamblichus says, "as if they were divine precepts, without which they did nothing." III. The Cult of Numbers The Pythagorean community's ranks eventually swelled to more than 2,500 in Croton. Many citizens, particularly the old-guard gentry, were not pleased. Formation of the community, according to Gorman's book, was "gradual, because a sudden conversion of this magnitude would have disturbed the council of leaders." Cults were as feared and distrusted then as now and as much in vogue because this was an age in which old, stable social orders and their trusted religions were breaking down. Society was in ferment. Many people were seeking new values and updated moral guideposts. Pythagoras's cult had a lot in common with Orphic cults sweeping Greece. These were named for Orpheus, the tragic mythological lover. Orphics preached and practiced a potpourri of science, mysticism and monkish self-control. Angry Greek men and women were seeking the same salvation through Orphics and Pythagoras's cult that today's malcontents seek in such modern movements as militias and new religions. Life in that time, according J.B. Bury in his History of Greece, was not like that of the good old days. Gone were the secure, nuclear households of Homer's age three centuries earlier. In a tumultuous era of increasing political democracy, which many saw as promoting mob rule and social debauchery, new cults such as the Pythagoreans and Orphics offered new creeds by which to live, preaching temperance, communal living and oligarchy, government by an enlightened few. Many people in Pythagoras's day were drawn by hushed whispers about life after death in some kind of nether world, a concept that reached Greeks from Egyptian and Eastern cultures. Secrecy and mystical traditions were part of the cults' appeal. It gave them an air of enlightenment and revolution. Each group had its own spin. Orphics emphasized transmigration of souls after death into new bodies. Cults dedicated to the god Dionysius preferred frenzied channeling rituals. Pythagoras's angle was numbers. He was, after all, a mathematician -- some consider him the world's first -- though certainly not in our sense of the word. To him, numbers were divine, the primary elements of all existence. Individual numbers had magical powers. Numbers, he said, are "the cause of gods and demons." Pythagoras started counting with the number 3. One and 2 he considered building blocks for all other numbers, not numbers themselves. Numbers were the building blocks of everything else. "Opportunity was 7," modern scholar Ward Rutherford writes in his account of the Pythagoreans' linkage of numbers with even the most abstract notions. "Justice is 4, masculinity an odd number, femininity an even." Two embodied the female principle, 3 the male and so marriage was 5. Odd as such thinking may seem today, Pythagoreans developed key concepts that influenced development of modern science. One, for example, was that nature, or reality, at its deepest level is mathematical. "All is number," Pythagoras taught, using a formulation surprisingly close to views expressed by Albert Einstein and others: God is a mathematician. This trust in mathematics has been among the most powerful tools of modern science, especially physics. But Pythagoreans went a bit further. They claimed that their numerical mysticism could lead to spiritual purification, ultimately uniting souls of individuals with the divine. Still, the emphasis on math led to practical advances. In Croton, for example, Pythagoras established one of the world's first laboratories, where he tested acoustics by hammering bells of different weights and measuring the pitch of tones they produced. He discovered that musical pitch is related to vibration in string and varies with the string's length. He found musical harmony results from waves in vibrating strings that are precise multiples of one another -- one wave in the vibrating string being exactly twice or three times the length of another with which it harmonizes. He measured the stars and plotted an Earth-centered solar system. In fact, he argued that movements of stars and planets produced a form of musical harmony beyond human comprehension -- the source of the expression "the music of the spheres." But the Pythagoreans did not share their knowledge. Secrecy was maintained with fervor, including knowledge of various geometric forms considered to have divine properties. These forms included the sphere, the cube and the tetrahedron, a solid with four equilateral triangles as faces. According to Gorman, a disciple named Hiappasos "revealed the properties of the dodecahedron" and was promptly expelled from the community. Who knew, after all, what could happen if the secret plans of this solid form with 12 pentagons as sides were to fall into the wrong hands? For all of his geometric accomplishments, Pythagoras may not have devised the theorem bearing his name. His writings are lost, and almost everything attributed to him stems from writings of his community. Many scholars believe that others in his group produced the theorem. IV. Day of reckoning Like Chicago, Croton soon became a one-star town. Like Michael Jordan, Pythagoras had become omnipotent. He probably was above direct involvement in political frays, but many in his cult were not. As Bury wrote, his adherents turned the order into "an instrument of political power." Occasionally, every leader needs a crisis to rally flagging supporters. Pythagoras found one right around the corner in Sybaris. The neighboring city-state was a perfect foil. The name of its citizens, "Sybarites," is our synonym for libertine. Even in those heady days, Sybarites were known for their love of excess and their military power, which far exceeded that of Croton. When Telis, the Sybarite dictator, demanded return of refugees who had fled to Croton, many Crotonians wanted to comply. They had suffered in recent conflicts with other city-states and were not confident about their power. But Pythagoras was defiant and rallied Croton. His disciple, Milo the wrestler, raised an army and attacked Sybaris. Though outnumbered, Milo's troops crushed Sybaris so completely that it virtually disappeared. Suddenly, Croton was a major power in the region. Yet even in victory, bitter dissension remained. Lingering problems with cults festered across the Greek world. Problems for Pythagoras came to a head after he had been in Croton about 20 years. A nobleman named Cylon asked to join the Pythagoreans, but he had a reputation as a partier, so Pythagoras balked. Spurned, Cylon rallied the population against Pythagoras and ambushed him at Milo's house. In the clash, the house was burned, and most local Pythagoreans were killed. The Pythagorean school, though, did not fade. While the Orphic movement was largely repudiated after the 6th century B.C., the Pythagoreans lasted another 300 years, spreading as far as the Middle East and spawning many thinkers. Pythagoras turned out to be one of the fathers of Western philosophical tradition. His easy blend of science, freewheeling spiritualism and philosophy laid groundwork for a heady wave of ancient Greek philosophers. His biggest fan would be Plato. The demise of Pythagoras has never been clear. Diogenes Laertius reports that he escaped the fire at Milo's, only to come to a bean field, sacred ground. When he refused to cross it, pursuers from Croton caught him and slit his throat. Other reports say he escaped to Metapontum, a Greek-Italian city-state, and died there shortly afterward. Some reports place him last in Delos, consoling a dying teacher. Sightings of Pythagoras, Elvis-like, were reported for years. Whatever his fate, Pythagoras's legacies thrive today, not least in the simple formula that reminds us the real world does, after all, have an intimate, fundamental basis in numbers -- a2 + b2 = c2.

Danny Hakim is a former Washington Post news aide who lives in Denver and is writing a children's book about history. CAPTION: THE PYTHAGOREAN THEOREM Certain shapes in the real world obey mathematical laws with a consistency that Pythagoras and his followers considered mystical. Take the so-called right triangle -- any triangle with one inside angle of 90 degrees. No matter what the lengths of the sides, it is always true that, if you square the length of the longest side (the hypotenuse, which is always opposite the right angle), that number will equal the sum of the squares of the other two sides. The triangle defined by these three square sets of blocks is a right triangle. The hypotenuse is five blocks long. Square it, which we have done quite literally, and you have 25 blocks. The square of one of the other sides contains nine blocks, and that of the third is 16 blocks. Nine plus 16 equals 25. If you call the hypotenuse c and the other sides a and b, the formula is a2 + b2 = c2. CAPTION: One of the most famous utterances on geometry pops out of the supposedly newly acquired brain of the Scarecrow toward the end of "The Wizard of Oz." Ray Bolger, who plays the character, says, "The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side." Sounds pretty good and even resembles the Pythagorean Theorem, but it's geometric gibberish, as phony as the Wizard. The Lion may have found courage, the Tin Man may have found his heart but if the Scarecrow has a brain, it has deluded itself. Think about his statement, and you'll see that it cannot be true, even of an isosceles triangle, which is one that has two sides of equal length. CAPTION: Pythagoras teaching (left foreground with book), in a detail from Raphael's "School of Athens."