41. Fractals, Chaos, And Music archytas of tarentum, a Pythagorean mathematician who lived around 400350 BC, waseven able to work out the relationships between notes in the enharmonic scale http://www.geocities.com/SoHo/Square/7921/fmusic.html

Fractals, Chaos, and Music Abstract Many relations have been discovered in recent years between fractals and music. This paper will cover some of the research that has been done on these relations, including some of the controversies over conflicting discoveries. It will show how some artists are currently using fractals to generate the basic melodies in their compositions. A few computer programs which can be used to generate these melodies will be discussed. This paper will also examine possible practical uses for software which can be used to hear patterns in fractals. The algorithms behind the music-generating algorithms will be discussed, as well as some of the basic relations between mathematics and music. Some of the work of music theorist Joseph Schillinger, whose ideas in the 1920's and 1930's about generating music by recursive and chaotic means were far ahead of their time, will also be covered. Historical perspective The mathematical study of music is certainly nothing new, dating back to ancient Greece. Around the 5th century BC, the Pythagoreans formulated a scientific approach to music, expressing musical intervals as numeric proportions. This was probably done by observing the tones produced by plucked strings of different lengths; for example, the tone produced by a string held at the middle is an octave higher than that of the whole string. They went on to calculate the intervals for several different scales, including the chromatic and diatonic scales. Archytas of Tarentum, a Pythagorean mathematician who lived around 400-350 BC, was even able to work out the relationships between notes in the enharmonic scale, which includes quarter tones. (Britannica)

42. Eudoxus Eudoxus was born in 408 BC in Cnidus. He was a Greek geometer and astronomer. Eudoxusstudied at Plato's Academy and was a student of archytas of tarentum. http://www.angelfire.com/ca5/ancientgreecescience/eudoxus

Eudoxus Scientists Home Page Anaxagoras Archimedes Aristarchus ... Pythagoras Eudoxus was born in 408 B.C. in Cnidus. He was a Greek geometer and astronomer. Eudoxus studied at Plato's Academy and was a student of Archytas of Tarentum. He spent over a year in Egypt and then, on his return, established a school that competed with Plato. There is ample evidence to suggest that Eudoxus had little respect for Plato's analytic ability. Eudoxus proposed a heliocentric system for the solar system; a very important contribution. Other important contributions were to the theory of proportion, where he made a definition of equal ratios similar to cross multiplying, and early work on integration with the theory of exhaustion. Eudoxus also devised a theory of planets carried on glassy spheres that were nested around the Earth in mountings like compass gimbals: rotations on these explained observed motions. The kampyle curve was studied by Eudoxus also in relation to the classical problem of duplication of the cube. Eudoxus found formulas for measuring pyramids cones and cylinders. Books V and XII of Euclid's Elements are attributed to Eudoxus by some experts. In 355 B.C. Eudoxus died in Cnidus.

43. Plutarch According to such principles Pericles regulated his life; so did likewisearchytas of tarentum and Dion the Syracusan and Epaminondas the Theban. http://home.uchicago.edu/~ahkissel/plutarch.html

Plutarch Excerpts from Plutarch's writings on education Adam Kissel (using Charles William Super's tr., Plutarch on Education (C. W. Bardeen: Syracuse, N.Y., 1910)) "The Education of Boys" (ps.-Plutarch) We may remark in general terms regarding virtue what we are accustomed to say concerning the arts and sciences, namely, that three factors are essential to the formation of a well rounded character: phusis logos , and ethos . By instruction I understand the acquisition and imparting of knowledge; by ethos . Natural endowments are inborn; progress is a matter of education; application, of (exercise); while the highest excellence is the result of all combined. In so far as any of these is wanting, is necessarily defective. Natural endowments without education are blind; education, where there are no natural endowments, is inefficacious; and practice apart from both is incomplete and must fail of its end. In confirmation of these views I might say that the three combined and cooperated in the psychic powers of the men of glorious memory such as Pythagoras and Socrates and Plato and all who have won imperishable renown. Fortunate and favored of the gods is every one upon whom the gods have bestowed all these gifts. If any one thinks that lack of natural endowments can not be supplied by suitable instruction and practice in virtue he is very much, yes, altogether mistaken. For disuse destroys the best natural endowments while improves even weak ones. (49-50)

44. Kurzweil Technologies, Inc.: Publications c. 420 BC, archytas of tarentum, a friend of Plato, constructs a wooden pigeonwhose movements are controlled by a jet of steam or compressed air. http://www.kurzweiltech.com/mchron.htm

The Age of Intelligent Machines "Chronology" by Ray Kurzweil The world has changed less since Jesus Christ than it has in the last thirty years. Charles Peguy, 1913 Year Event million years ago Dinosaurs roam the earth. Less than years ago Homo sapiens begin using intelligence to further their goals More than years ago The abacus, which resembles the arithmetic unit of a modern computer, is developed in the Orient. 3000-700 B.C. Water clocks are built in China in 3000 B.C., in Egypt c. 1500 B.C. and in Assyria 700 B.C. 2500 B.C. Egyptians invent the idea of thinking machines: citizens turn for advice to oracles, which are statues with priests hidden inside. b. 469 B.C. Socrates, the mentor of Plato, is the first Western thinker to assert that mental activities occur in the unconscious. 469-322 B.C. Socrates, Plato, and Aristotle establish the essentially rationalistic philosophy of Western culture. 427 B.C. In the Phaedo and later works Plato expresses ideas, several millennia before the advent of the computer, that are relevant to modern dilemmas regarding human thought and its relation to the mechanics of the machine. c. 420 B.C.

45. The Earlier History Of Powered Flight Another tale recounts the invention of a Greek named archytas of tarentum whowas said to have made a wooden bird about four hundred years before Christ. http://www.centennialofflight.gov/essay/Prehistory/earliest_flight/PH1.htm

The Earliest Efforts at Flight In Greek mythology, Icarus flew too close to the sun Other legends about flight abound. Another early one tells of King Bladud, who ruled in Britain in the ninth century B.C.E. Bladud supposedly constructed a pair of wings with which he proposed to fly. But, according to the monk Geoffrey of Monmouth in a history of the British kings, Bladud was dashed to pieces as he landed on top of the Temple of Apollo in the town of Trinovantum. Another tale recounts the invention of a Greek named Archytas of Tarentum who was said to have made a wooden bird about four hundred years before Christ. This bird was powered by steam and supposedly flew about 50 feet (15 meters). "Flight" of Simon the Magician in Roman times Earlier, in 1010, a monk in Malmsesbury, England, attached artificial wings to his body and jumped from the top of his abbey to glide to only two broken legs upon landing. Other tower jumpers suffered death or injuries, while a few achieved some partial success with their glides. In fifteenth-century Italy, the brilliant Italian artist, scientist, and engineer Leonardo da Vinci, who understood some of the basic principles of flight as early as the 1480s, was the first to seriously study aerodynamics. He examined the way birds flew and sketched their wings and muscles in a flying position. His notebooks were filled with drawings and descriptions of birds in flight and with models that had wings like birds. Unfortunately, his ornithopter and helicopter designs could never have left the ground with them. He, and others of his time, failed to realize that human beings lacked the necessary muscle power to imitate bird flight with flapping wings. His works were rediscovered in the 19th century and had little or no influence on the history of flight technology.

46. Plato Plato did not expect the plan to succeed but because both Dion andarchytas of tarentum believed in the plan then Plato agreed. http://www.crystalinks.com/plato.html

Plato (about 427-347 B.C.) Plato is probably one of the greatest philosophers of all times, if not the greatest. Plato was born to an aristocratic family in Athens. His father, Ariston, was believed to have descended from the early kings of Athens. Perictione, his mother, was distantly related to the 6th-century B.C. lawmaker Solon. When Plato was a child, his father died, and his mother married Pyrilampes, who was an associate of the statesman Pericles. Plato's original name was Aristocles, but in his school days he received the nickname Platon (meaning "broad" ) because of his broad shoulders. It was mostly in Pyrilampes' house that Plato was brought up. Aristotle writes that when Plato was a young man he studied under Cratylus who was a student of Heracleitus, famed for his cosmology which is based on fire being the basic material of the universe. It almost certain that Plato became friends with Socrates when he was young, for Plato's mother's brother Charmides was a close friend of Socrates. The Peloponnesian War was fought between Athens and Sparta between 431 BC and 404 BC.

47. A Brief History Of Automatons is credited by Aristotle with creating moving statues which guarded the entranceto the Labyrinth at Crete, while the philosopher archytas of tarentum, in the http://access.tucson.org/~michael/cb_1.html

48. Ancient History Sourcebook: M. Tullius Cicero: On The Republic, Book 1 I imitate the famous archytas of tarentum, who, when he came to his villa, and foundall its arrangements were contrary to his orders, said to his steward Ah! http://www.fordham.edu/halsall/ancient/cicero-republic1.html

Back to Ancient History Sourcebook Ancient History Sourcebook: M. Tullius Cicero: from On the Republic [Thatcher Introduction]: On the Republic. Book I. Then Laelius said: But you have not told us, Scipio, which of these three forms of government you yourself most approve. Scipio: Laelius: I think so, too, but yet it is impossible to dispatch the other branches of the question, if you leave this primary point undetermined. Scipio: We must, then, I suppose, imitate Aratus, who, when he prepared himself to treat of great things, thought himself in duty bound to begin with Jupiter. Laelius: Why Jupiter? And what is there in this discussion which resembles that poem? Scipio: Why, it serves to teach us that we cannot better commence our investigations than by invoking him whom, with one voice, both learned and unlearned extol as the universal king of all gods and men. Laelius: How so? Scipio: Do you, then, believe in nothing which is not before your eyes? Whether these ideas have been established by the chiefs of states for the benefit of society, that there might be believed to exist one Universal Monarch in heaven, at whose nod (as Homer expresses it) all Olympus trembles, and that he might be accounted both king and father of all creatures; for there is great authority, and there are many witnesses, if you choose to call all many, who attest that all nations have unanimously recognized, by the decrees of their chiefs, that nothing is better than a king, since they think that all the gods are governed by the divine power of one sovereign; or if we suspect that this opinion rests on the error of the ignorant, and should be classed among the fables, let us listen to those universal testimonies of erudite men, who have, as it were, seen with their eyes those things to the knowledge of which we can hardly attain by report.

49. Math Forum: Ask Dr. Math FAQ: "Impossible" Geometric Constructions The Proof of Archytus. Biographies archytas of tarentum MacTutor History of MathematicsArchive; Hippocrates of Chios - MacTutor History of Mathematics Archive. http://mathforum.org/dr.math/faq/faq.impossible.construct.html

Ask Dr. Math: FAQ I mpossible G eometric C onstructions Dr. Math FAQ Classic Problems Formulas Search Dr. Math ... Doubling the cube Three geometric construction problems from antiquity puzzled mathematicians for centuries: the trisection of an angle, squaring the circle, and duplicating the cube. Are these constructions impossible? Whether these problems are possible or impossible depends on the construction "rules" you follow. In the time of Euclid, the rules for constructing these and other geometric figures allowed the use of only an unmarked straightedge and a collapsible compass. No markings for measuring were permitted on the straightedge (ruler), and the compass could not hold a setting, so it had to be thought of as collapsing when it was not in the process of actually drawing a part of a circle. Following these rules, the first two problems were proved impossible by Wantzel in 1837, although their impossibility was already known to Gauss around 1800. The third problem was proved to be impossible by Lindemann in 1882.

50. BBC News | ARTIFICIAL INTELLIGENCE | Timeline: Real Robots computers. 400 BC Philosopher and mathematician archytas of tarentumbuilt a wooden dove that could flap its wings and fly. Early http://news.bbc.co.uk/1/hi/in_depth/sci_tech/2001/artificial_intelligence/153143

CATEGORIES TV RADIO COMMUNICATE ... INDEX SEARCH You are in: In Depth: Artificial intelligence Front Page World ... AudioVideo SERVICES Daily E-mail News Ticker Mobiles/PDAs Feedback ... Low Graphics Monday, 10 September, 2001, 10:42 GMT 11:42 UK Timeline: Real robots Robots are not new. They have been around for centuries in various forms. Here's a brief overview of the development of both robots and computers. 400 BC Philosopher and mathematician Archytas of Tarentum built a wooden dove that could flap its wings and fly. Early 16th Century Hans Bullmann creates the first androids - simulated people that can play musical instruments for the delight of paying customers. In his laboratory at Nuremburg, scholar Johann Müller, aka Regiomontanus, is reputed to have created an iron fly and an artificial eagle, both of which could take to the air. In England, John Dee creates a wooden beetle that can fly for an undergraduate production of Aristophanes' Pax. At the Heilbrunn chateau in Germany, a mechanical theatre is created featuring 119 animated figures that perform a play about village life to the accompaniment of a water-powered organ. While training as a Jesuit, Jacques Vaucanson creates flying angels which cause him to be thrown out of the order.

51. Music Philosophy Plato himself was not sanguine of results, but as both Dion and philosopherstatesmanarchytas of tarentum thought the prospect promising he felt bound in http://www.dhpc.org/about/stained/musicand.htm

52. History - Power: The Final Ingredient - Instructor with four vertical vents. Some fifty years later archytas of tarentumused a steam jet to launch a wooden bird. But it was not until http://wings.avkids.com/Book/History/instructor/power-01.html

Power: The Final Ingredient page 1 The earliest recorded use of power to make objects fly was during the classic Greek civilization. About 500 B.C. Heron of Alexandria harnessed steam power when he constructed a rotating boiler with four vertical vents. Some fifty years later Archytas of Tarentum used a steam jet to launch a wooden bird. But it was not until the onset of the Industrial Revolution in England at the beginning of the 18th century that the steam engine became mass-produced and less expensive. Cayley himself had in 1837 designed a steam engine to power the propellers of a streamlined balloon - that is, he had foreshadowed the airship as well as the aeroplane. But he was dissatisfied with the design, which would pass water through a cooled pipe in such a way as to convert it to steam in a single pass. He was generally frustrated by the massive weight of the engines used to propel land and sea vehicles, and stressed the need for lightness of material and fuel. "It is proper to notice the probability that exists of using the expansion of air, by the sudden expansion of inflammable powders or liquids ... an engine of this sort might be produced by a gas-tight apparatus and by firing the inflammable air generated with a due portion of air under a piston." Cayley was anticipating the internal combustion engine, but it would not arrive until after his death. Orville and Wilbur Wright became interested in flight as boys in Ohio when their father gave them a rubber band powered toy glider, a recent invention of Alphonse Penaud. Although the brothers did not go onto higher education they were curious, creative and mechanically inclined. As adolescents they had designed and built their own printing press and published a weekly newspaper. When bicycling became popular in the 1890's they took advantage of the "mania" and ran a successful bicycle repair shop. They made so much money repairing bikes they went on to design and manufacture their own bicycles and, of course, added mechanical innovations they had designed.

53. EXPLORIT Science Center - "Float, Sink Or Swim?" take advantage of air currents and wind to enable them to swim in the air, are thoughtto have been invented by the Greek scientist archytas of tarentum in the http://www.explorit.org/science/bytes/float.html

Science Bytes FLOAT, SINK OR SWIM by Anne Hance T he ancient Chinese propounded the notion during the first century AD, that the sun, moon and stars float freely in space. This notion, while an improvement over earlier ideas, is not acceptable in light of our modern understanding of bodies in space. But they correctly recognized that water particles float in the air and, as part of a water cycle, sink down to earth as rain to replenish lakes, rivers and oceans. S ince a fluid is any substance that takes the shape of any vessel containing it, air and water are both 'fluids'. The physical and chemical properties of air and water have enabled the development of many forms of life ever since life began on this planet. But the purposeful study of these properties had to wait until sentient beings paid deliberate attention to them. O f course, prehistoric peoples recognized that some things would float on water and, without the need for any understanding of science to explain this phenomenon, they built rafts and boats. Experience, rather than a formalized knowledge of 'science', taught them how to take advantage of the physical properties of water and the materials with which they built their craft. L earning to take advantage of the physical properties of air was historically a much slower process. From earliest times, people watched birds and insects use the fluid nature of air to enable them to fly. People could not fly and so they invented things that could. Kites, which take advantage of air currents and wind to enable them to swim in the air, are thought to have been invented by the Greek scientist Archytas of Tarentum in the fourth century BC.

54. The Age Of Spiritual Machines: Timeline c. 420 bc, archytas of tarentum, who was friends with Plato, constructs a woodenpigeon whose movements are controlled by a jet of steam or compressed air. http://www.kurzweilai.net/articles/art0274.html?printable=1

55. The Complete History Of The Discovery Of Cinematography - The why the Ethiopian has his colour. Herodotus 484425 BC. 4TH CENTURYBC archytas of tarentum, It is documented that this philosopher http://www.precinemahistory.net/900.htm

56. Chemistry & Biochemistry Faculty: Kolks archytas of tarentum. I am very lucky to have a job which allows me tospend as much time as I want reading and thinking about chemistry. http://www.manhattan.edu/science/chembioc/faculty/kolks/kolks.html

School of Science People Curriculum Faculty ... John Wasacz GARY KOLKS, Associate Professor phone: (718) 862-7322 fax: email: gkolks@manhattan.edu office: Hayden Hall 604 EDUCATION: B.S., St. Francis College M.S., M.Phil., Ph.D. Columbia University If a man could ascend to heaven and get a clear view of the natural order of the universe and the beauty of the heavenly bodies, that wonderful spectacle would give him small pleasure, though nothing could be conceived more delightful if he had but someone to tell what he had seen. Archytas of Tarentum I am very lucky to have a job which allows me to spend as much time as I want reading and thinking about chemistry. My goal, which I pursue in my general chemistry and inorganic chemistry courses, is to help my students understand chemistry so that they can get as much pleasure out of it as I do. Chemistry is the most wonderful, but also the most difficult, science to learn. In the laboratory chemists watch pieces of red metal dissolve in a colorless liquid to produce a blue liquid and a poisonous red gas. But we try to understand what we see by using a very abstract language to discuss things – atoms, molecules, ions, and electrons – which we can't see. I think the only way to make the gigantic intellectual leap between chemistry lab-world and the chemists' mind-world is by seeing, and then talking about, experiments which illustrate chemical ideas. I am very interested in designing classroom experiments, not just demonstrations, which help my students understand chemistry. As coordinator of the general chemistry program I am also responsible for developing experiments for our general chemistry laboratory. I've put together a pamphlet

57. Peter Fosl's Philosophical Chronology defeated (413) Plague in Athens (430 429 BCE) Plato (c 428/7 - 347 BCE)(foundsAcademy 387 BCE) archytas of tarentum (Plato's contemporary) Theaetetus (c 414 http://www.transy.edu/homepages/philosophy/chronology.html

TRANSYLVANIA UNIVERSITY PHILOSOPHY PROGRAM A PHILOSOPHICAL CHRONOLOGY By Peter S. Fosl Use your "find in page" command (under "Edit" in your browser) to search for particular entries or use the table below to take you to the indicated date. The Big Bang 400 BCE Homo Sapiens 300 BCE ... 500 BCE s s Big Bang postulated (15-16 billion years ago) Formation of the Earth (c 4,500,000,000 years ago) Precambrian Age (4,000,000,000 - 540,000,000 y.a., origin of life [Archeaozoic era] thought to be 4 billion y.a.) Earliest known life in fossil record (c 3,500,000,000 y.a.) Paleozoic Age (540,000,000 - 200,000,000 y.a.) (insects, chondrichthyes, amphibians, reptiles, plants except angiospermae) Mesozoic Age (200,000,000 - 60, 000,000 y.a.) (bony fish, birds, mammals, angiospermae) Dinosaurs become extinct (c 65,000,000 y.a.) Cenozoic Age begins (60,000,000 y.a.) Australopithecus (2,600,000 y.a.) Pleistocene Era (2,000,000 - 10,000 y.a., development of hominids) Appearance of homo sapiens (c 200,000 BCE) Earliest known artwork (c 29,000 BCE) (Willendorf Venus; painted blocks of La Ferrassie)

58. Index Of Ancient Greek Mathematicians And Astronomers archytas of tarentum (420350 BC). Greek mathematician, astronomer and engineer.Last of the Pythagorians. Plato and Eudoxus was his pupils. http://www.ics.forth.gr/~vsiris/ancient_greeks/classical_period.html

Classical period (5th - 4th century B.C.) Within this period Athens flourishes under Pericles, the Parthenon is built on the Acropolis, the tragedies of Sophocles, Aeschylus and Euripides are created, the phisolophical schools of Socrates and Plato (known as Academy) are established, and the Lyceum of Athens is founded by Aristotle. In science, the importance of the experimental method is accepted. Philosophers-Scientists Socrates (Athens, 470-399 B.C.). Died from poison after the state found him guilty for corrupting the youth. Theodorus of Cyrene (4th century B.C.). . Pythagorean. Plato's teacher in mathematics. Shows that the square roots of 3, 5, 6, 7, 8, 10, 11, 12, 13, 15, and 17 are irrational. Archytas of Tarentum (420-350 B.C.). Greek mathematician, astronomer and engineer. Last of the Pythagorians. Plato and Eudoxus was his pupils. Built a series of toys, among them a mechanical pigeon propelled by a steam jet. Developed the theory for the pulley. Plato (Athens, 430-350 B.C.) . Greek philosopher. He was the founder of the Academy (named from the hero Academos owner of the grove where the Academy was built). Believed that mathematics played an important role in education. Disregarded practicality, a belief he passed to his students such as Eucledes. He started a three part trilogy :

59. Index Of Ancient Greek Scientists archytas of tarentum (420350 BC). Greek mathematician. Built a seriesof toys, among them a mechanical pigeon propelled by a steam jet. http://www.ics.forth.gr/~vsiris/ancient_greeks/whole_list.html

not complete Agatharchos. Greek mathematician. Discovered the laws of perspectives. Anaxagoras of Clazomenae (480-430 B.C.). Greek philosopher. Believed that a large number of seeds make up the properties of materials, that heavenly bodies are made up of the same materials as Earth and that the sun is a large, hot, glowing rock. Discovered that the moon reflected light and formulated the correct theory for the eclipses. Erroneously believed that the Earth was flat. Links: Anaxagoras of Clazomenae, MIT Anaximander (610-545 B.C.). Greek astronomer and philosopher, pupil of Thales. Introduced the apeiron (infinity). Formulated a theory of origin and evolution of life, according to which life originated in the sea from the moist element which evaporated from the sun ( On Nature ). Was the first to model the Earth according to scientific principles. According to him, the Earth was a cylinder with a north-south curvature, suspended freely in space, and the stars where attached to a sphere that rotated around Earth. Links: Anaximander, Internet Encyclopedia of Philosophy

60. Presocratics.html Diogenes Laertius Lives of the Eminent Philosophers archytas of tarentum(4th Century BCE). IV. Heraclitus of Ephesus (floruit circa 500 BCE). http://www.wbenjamin.org/nc/presocratics.html

Back to homepage "From Myth to Mind" The Presocratics: Names, Dates and Linked Readings I. The Seven Sages: 1. Thales: (floruit circa 585 B.C.E.) "Thales and the Origins of Theoretical Reasoning" (Dmitri Pachenko) 2. Bias: (circa 570 B.C.E.) Pertinent Passages on Bias in Herodotus, Apollodorus and Pausanias (c/o the Perseus Project at Tufts University.) Diodorus Siculus, THE HISTORICAL LIBRARY (c/o the Perseus Project at Tufts University.) 3. Pittacus: (circa 600 B.C.E.) Plato's Protagoras : sections 343a -347. Diodorus Siculus, THE HISTORICAL LIBRARY (c/o the Perseus Project at Tufts University.) 4. Solon: (archon 594 B.C.E.) Herodotus, "Solon and Croesus" from THE HISTORIES. Diodorus Siculus, THE HISTORICAL LIBRARY (c/o the Perseus Project at Tufts University.) 5. Cleobulus: (circa 600 B.C.E.) Diogenes Laertius Lives of the Eminent Philosophers 6. Myson: (circa 600 B.C.E.) Diogenes Laertius Lives of the Eminent Philosophers Diodorus Siculus, THE HISTORICAL LIBRARY (c/o the Perseus Project at Tufts University.) 7. Chilon:

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