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Geometry – History FileThales of Miletus

Born: about 624 BC in Miletus, Asia Minor (now Turkey)Died: about 547 BC in Miletus, Asia Minor (now Turkey)

Thales seems to be the first known Greek philosopher, scientist and mathematician although his occupation was that of an engineer. He is believed to have been the teacher of Anaximander (611 BC - 545 BC) and he was the first natural philosopher in the Milesian School.

http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Thales.html

The Theorems Attributed to Thales

Five Euclidean theorems have been explicitly attributed to Thales, and the testimony is that Thales successfully applied two theorems to the solution of practical problems. Thales did not formulate proofs in the formal sense. What Thales did was to put forward certain propositions which, it seems, he could have 'proven' by induction: he observed the similar results of his calculations: he showed by repeated experiment that his propositions and theorems were correct, and if none of his calculations resulted in contrary outcomes, he probably felt justified in accepting his results as proof. Thalean 'proof' was often really inductive demonstration. The process Thales used was the method of exhaustion. This seems to be the evidence from Proclus who declared that Thales 'attacked some problems in a general way and others more empirically'.

DEFINITION I.17: A diameter of the circle is a straight line drawn through the centre and terminated in both directions by the circumference of the circle; and such a straight line also bisects the circle (Proclus, 124). >

PROPOSITION I.5: In isosceles triangles the angles at the base are equal; and if the equal straight lines are produced further, the angles under the base will be equal (Proclus, 244). It seems that Thales discovered only the first part of this theorem for Proclus reported: We are indebted to old Thales for the discovery of this and many other theorems. For he, it is said, was the first to notice and assert that in every isosceles the angles at the base are equal, though in somewhat archaic fashion he called the equal angles similar (Proclus, 250.18-251.2).

PROPOSITION I.15: 'If two straight lines cut one another, they make the vertical angles equal to one another' (Proclus, 298.12-13). This theorem is positively attributed to Thales. Proof of the theorem dates from the Elements of Euclid (Proclus, 299.2-5).

PROPOSITION I.26: 'If two triangles have the two angles equal to two angles respectively, and one side equal to one side, namely, either the side adjoining the equal angles, or that subtending one of the equal angles, they will also have the remaining sides

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equal to the remaining sides and the remaining angle equal to the remaining angle' (Proclus, 347.13-16). 'Eudemus in his history of geometry attributes the theorem itself to Thales, saying that the method by which he is reported to have determined the distance of ships at sea shows that he must have used it' (Proclus, 352.12-15). Thales applied this theorem to determine the height of a pyramid. The great pyramid was already over two thousand years old when Thales visited Gizeh, but its height was not known. Diogenes recorded that 'Hieronymus informs us that [Thales] measured the height of the pyramids by the shadow they cast, taking the observation at the hour when our shadow is of the same length as ourselves' (D.L. I.27). Pliny (HN, XXXVI.XVII.82) and Plutarch (Conv. sept. sap. 147) also recorded versions of the event. Thales was alerted by the similarity of the two triangles, the 'quality of proportionality'. He introduced the concept of ratio, and recognized its application as a general principle. Thales's accomplishment of measuring the height of the pyramid is a beautiful piece of mathematics. It is considered that the general principle in Euclid I.26 was applied to the ship at sea problem, would have general application to other distant objects or land features which posed difficulties in the calculation of their distances.

PROPOSITION III.31: 'The angle in a semicircle is a right angle'. Diogenes Laertius (I.27) recorded: 'Pamphila states that, having learnt geometry from the Egyptians, [Thales] was the first to inscribe a right-angled triangle in a circle, whereupon he sacrificed an ox'. Aristotle was intrigued by the fact that the angle in a semi-circle is always right. In two works, he asked the question: 'Why is the angle in a semicircle always a right angle?' (An. Post. 94 a27-33; Metaph. 1051 a28). Aristotle described the conditions which are necessary if the conclusion is to hold, but did not add anything that assists with this problem.

It is testified that it was from Egypt that Thales acquired the rudiments of geometry. However, the evidence is that the Egyptian skills were in orientation, measurement, and calculation. Thales's unique ability was with the characteristics of lines, angles and circles. He recognized, noticed and apprehended certain principles which he probably 'proved' through repeated demonstration.

http://www.utm.edu/research/iep/t/thales.htm

Eratosthenes of Cyrene

A versatile scholar, Eratosthenes of Cyrene lived approximately 275-195 BC. He was the first to estimate accurately the diameter of the earth. For several decades, he served as the director of the famous library in Alexandria. He was highly regarded in the ancient world, but unfortunately only fragments of his writing have survived. Eratosthenes died at an advanced age from voluntary starvation, induced by despair at his blindness.

http://www.math.utah.edu/~alfeld/Eratosthenes.html

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A consummate Greek scholar whose status as second best in each field earned him the nickname "Beta." He served as librarian at the great library in Alexandria, and wrote works of mathematics, geography, philosophy, and astronomy. He also wrote a poem called Hermes which described the fundamentals of astronomy in verse! Although most of Eratosthenes' writings are lost, many are preserved through the writings of commentators.

Among Eratosthenes' accomplishments was the accurate measurement the diameter of the Earth by observing that, on the day of the summer solstice, the Sun was directly overhead in Syene while it was 7° from the zenith in Alexandria, which he assumed was due north of Syene (Dunham 1990). Unfortunately, since the original work On the Measurement of the Earth was lost, the details of Eratosthenes' procedure are not known. Eratosthenes also determined the obliquity of the ecliptic, prepared a star map containing 675 stars, suggested that a leap day be added every fourth year, tried to construct an accurately-dated history, and developed the "sieve of Eratosthenes " method of finding prime numbers. At the age of 80, blind and weary, he died of voluntary starvation.

Euclid of Alexandria

Born: about 325 BCDied: about 265 BC in Alexandria, Egypt

Euclid of Alexandria is the most prominent mathematician of antiquity best known for his treatise on mathematics The Elements. The long lasting nature of The Elements must make Euclid the leading mathematics teacher of all time. However little is known of Euclid's life except that he taught at Alexandria in Egypt.

http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Euclid.html

Euclid and his Elements

One of the most influential mathematicians of ancient Greece, Euclid flourished around 300 B.C. Not much is known about the life of Euclid. One story which reveals something about Euclid's character concerns a pupil who had just completed his first lesson in geometry. The pupil asked what he would get from learning geometry. So Euclid told his slave to give the pupil a coin so he would be gaining something from his studies. Included in the many works of Euclid is Data, concerning the solution of problems through geometric analysis, On Divisions (of Figures), the Optics, the Phenomena, a treatise on spherical geometry for astronomers, several lost works on higher geometry, and the Elements, a thirteen volume textbook on geometry. [1]

The Elements, which surely became a classic soon after its publication, eventually became the most influential textbook in the history of civilization. In fact, it has been said that apart from the Bible, the Elements is the most widely read and studied book in the

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world. [2] It has also been said that the Greeks used to post over the doors of their schools the inscription: ``Let no one come to our school who has not learned the Elements of Euclid.'' [3] Probably every great Western mathematician to arise in the last two thousand years has studied Euclid's Elements.

In writing the Elements Euclid collected and extended many of the ideas of other Greek mathematicians before him. The Elements is basically a chain of 465 propositions encompassing most of the geometry, number theory, and geometric algebra of the Greeks up to that time. [4] Book I contains twenty-three definitions, five common notions (axioms), five postulates, and forty-eight propositions of plane geometry.

The definitions of Book I include those of points, lines, planes, angles, circles, triangles, quadrilaterals, and parallel lines.

The five postulates may be translated into the following:

1. Two points determine a straight line. 2. A line segment extended infinitely in both directions produces a straight line. 3. A circle is determined by a center and distance. 4. All right angles are equal to one another. 5. If a straight line falling an two straight lines forms interior angles on the same

side less than 180 degrees, the two straight lines, if produced indefinitely, will meet on that side.

The last of these, commonly known as the ``parallel postulate,'' is by far the most important of the five. Through manipulation, the following statement may be derived: ``The sum of the angles in a triangle is equal to 180 degrees.'' Changing ``equal to'' to ``less than'' or ``greater than'' results in entirely different geometries -- non-Euclidean geometries. In spherical geometry, for example, this would read: ``The sum of the angles in a triangle is greater than 180 degrees.'' In hyperbolic geometry it would read: ``The sum of the angles in a triangle is less than 180 degrees.'' Hyperbolic geometry was invented by the Russian mathematician Nicolai Ivanovitch Lobachevsky. [5]

Postulates, by definition, are not and cannot be proven. However, some mathematicians have claimed that postulate four can be proven; [6] and many have believed that postulate five, partly because of its length and complexity, can be proven. [7] Lobachevsky's geometry grew out of his unsuccessful attempts to prove Euclid's parallel postulate. [8] Zeno of Sidon in the first century B.C. believed that Euclid's list of postulates was incomplete. He claimed that one must postulate that two distinct straight lines cannot have a segment in common. Without this, he claimed, some of the propositions in Book I are fallacious. [9]

Unlike the specialized nature of the postulates, the five common notions, or axioms, were essentially taken to be universal truths in all of mathematics and the sciences. The fifth

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axiom breaks down when exposed to the concept of infinite sets. For example, the set of all integers is not larger than the set of all even integers. [10]

The final section of Book I includes the forty-eight postulates. Included in these are the familiar results on triangles, such as proposition 5 [that the angles at the base of an isosceles triangle are equal], as well as the four congruence theorems for triangles: side-angle-side (prop. 4), side-side-side (prop. 8), angle-side-angle (prop. 26), and side-angle-angle (prop. 26, also). The last two propositions are the Pythagorean theorem and its converse.

http://www.obkb.com/dcljr/euclidhs.html

Pythagoras of Samos

Born: about 569 BC in Samos, IoniaDied: about 475 BC

http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Pythagoras.html

Pythagoras (fl. 530 BCE) must have been one of the world's greatest men, but he wrote nothing, and it is hard to say how much of the doctrine we know as Pythagorean is due to the founder of the society and how much is later development. It is also hard to say how much of what we are told about the life of Pythagoras is trustworthy; for a mass of legend gathered around his name at an early date. Sometimes he is represented as a man of science, and sometimes as a preacher of mystic doctrines, and we might be tempted to regard one or other of those characters as alone historical. The truth is that there is no need to reject either of the traditional views. The union of mathematical genius and mysticism is common enough. Originally from Samos, Pythagoras founded at Kroton (in southern Italy) a society which was at once a religious community and a scientific school. Such a body was bound to excite jealousy and mistrust, and we hear of many struggles. Pythagoras himself had to flee from Kroton to Metapontion, where he died.

It is stated that he was a disciple of Anaximander, his astronomy was the natural development of Anaximander's. Also, the way in which the Pythagorean geometry developed also bears witness to its descent from that of Miletos. The great problem at this date was the duplication of the square, a problem which gave rise to the theorem of the square on the hypotenuse, commonly known still as the Pythagorean proposition (Euclid, I. 47). If we were right in assuming that Thales worked with the old 3:4:5 triangle, the connection is obvious.

Pythagoras (fl. 530 BCE) must have been one of the world's greatest men, but he wrote nothing, and it is hard to say how much of the doctrine we know as Pythagorean is due to the founder of the society and how much is later development. It is also hard to say how much of what we are told about the life of Pythagoras is trustworthy; for a mass of legend gathered around his name at an early date. Sometimes he is represented as a man of

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science, and sometimes as a preacher of mystic doctrines, and we might be tempted to regard one or other of those characters as alone historical. The truth is that there is no need to reject either of the traditional views. The union of mathematical genius and mysticism is common enough. Originally from Samos, Pythagoras founded at Kroton (in southern Italy) a society which was at once a religious community and a scientific school. Such a body was bound to excite jealousy and mistrust, and we hear of many struggles. Pythagoras himself had to flee from Kroton to Metapontion, where he died.

It is stated that he was a disciple of Anaximander, his astronomy was the natural development of Anaximander's. Also, the way in which the Pythagorean geometry developed also bears witness to its descent from that of Miletos. The great problem at this date was the duplication of the square, a problem which gave rise to the theorem of the square on the hypotenuse, commonly known still as the Pythagorean proposition (Euclid, I. 47). If we were right in assuming that Thales worked with the old 3:4:5 triangle, the connection is obvious.

http://www.utm.edu/research/iep/p/pythagor.htm

René Descartes

Born: 31 March 1596 in La Haye (now Descartes),Touraine, FranceDied: 11 Feb 1650 in Stockholm, Sweden

René Descartes was a philosopher whose work, La géométrie, includes his application of algebra to geometry from which we now have Cartesian geometry.

Descartes was educated at the Jesuit college of La Flèche in Anjou. He entered the college at the age of eight years, just a few months after the opening of the college in January 1604. He studied there until 1612, studying classics, logic and traditional Aristotelian philosophy. He also learnt mathematics from the books of Clavius. While in the school his health was poor and he was granted permission to remain in bed until 11 o'clock in the morning, a custom he maintained until the year of his death.

School had made Descartes understand how little he knew, the only subject which was satisfactory in his eyes was mathematics. This idea became the foundation for his way of thinking, and was to form the basis for all his works.

Descartes spent a while in Paris, apparently keeping very much to himself, then he studied at the University of Poitiers. He received a law degree from Poitiers in 1616 then enlisted in the military school at Breda. In 1618 he started studying mathematics and mechanics under the Dutch scientist Isaac Beeckman, and began to seek a unified science of nature. After two years in Holland he travelled through Europe. Then in 1619 he joined the Bavarian army.

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From 1620 to 1628 Descartes travelled through Europe, spending time in Bohemia (1620), Hungary (1621), Germany, Holland and France (1622-23). He spent time in 1623 in Paris where he made contact with Mersenne, an important contact which kept him in touch with the scientific world for many years. From Paris he travelled to Italy where he spent some time in Venice, then he returned to France again (1625).

By 1628 Descartes tired of the continual travelling and decided to settle down. He gave much thought to choosing a country suited to his nature and chose Holland. It was a good decision which he did not seem to regret over the next twenty years.

http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Descartes.html

Descartes's chief contributions to mathematics were his analytical geometry and his theory of vortices, and it is on his researches in connection with the former of these subjects that his mathematical reputation rests.

Analytical geometry does not consist merely (as is sometimes loosely said) in the application of algebra to geometry; that had been done by Archimedes and many others, and had become the usual method of procedure in the works of the mathematicians of the sixteenth century. The great advance made by Descartes was that he saw that a point in a plane could be completely determined if its distances, say x and y, from two fixed lines drawn at right angles in the plane were given, with the convention familiar to us as to the interpretation of positive and negative values; and that though an equation f(x,y) = 0 was indeterminate and could be satisfied by an infinite number of values of x and y, yet these values of x and y determined the co-ordinates of a number of points which form a curve, of which the equation f(x,y) = 0 expresses some geometrical property, that is, a property true of the curve at every point on it. Descartes asserted that a point in space could be similarly determined by three co-ordinates, but he confined his attention to plane curves.

It was at once seen that in order to investigate the properties of a curve it was sufficient to select, as a definition, any characteristic geometrical property, and to express it by means of an equation between the (current) co-ordinates of any point on the curve, that is, to translate the definition into the language of analytical geometry. The equation so obtained contains implicitly every property of the curve, and any particular property can be deduced from it by ordinary algebra without troubling about the geometry of the figure. This may have been dimly recognized or foreshadowed by earlier writers, but Descartes went further and pointed out the very important facts that two or more curves can be referred to one and the same system of co-ordinates, and that the points in which two curves intersect can be determined by finding the roots common to their two equations.

http://www.maths.tcd.ie/pub/HistMath/People/Descartes/RouseBall/RB_Descartes.html

Archimedes of Syracuse

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Born: 287 BC in Syracuse, SicilyDied: 212 BC in Syracuse, Sicily

Archimedes was a native of Syracuse, Sicily. It is reported by some authors that he visited Egypt and there invented a device now known as Archimedes' screw. This is a pump, still used in many parts of the world. It is highly likely that, when he was a young man, Archimedes studied with the successors of Euclid in Alexandria. Certainly he was completely familiar with the mathematics developed there, but what makes this conjecture much more certain, he knew personally the mathematicians working there and he sent his results to Alexandria with personal messages. He regarded Conon of Samos, one of the mathematicians at Alexandria, both very highly for his abilities as a mathematician and he also regarded him as a close friend.

http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Archimedes.html

Archimedes is considered one of the three greatest mathematicians of all time along with Newton and Gauss. In his own time, he was known as "the wise one," "the master" and "the great geometer" and his works and inventions brought him fame that lasts to this very day. He was one of the last great Greek mathematicians.

Born in 287 B.C., in Syracuse, a Greek seaport colony in Sicily, Archimedes was the son of Phidias, an astronomer. Except for his studies at Euclid's school in Alexandria, he spent his entire life in his birthplace. Archimedes proved to be a master at mathematics and spent most of his time contemplating new problems to solve, becoming at times so involved in his work that he forgot to eat. Lacking the blackboards and paper of modern times, he used any available surface, from the dust on the ground to ashes from an extinguished fire, to draw his geometric figures. Never giving up an opportunity to ponder his work, after bathing and anointing himself with olive oil, he would trace figures in the oil on his own skin.

Much of Archimedes fame comes from his relationship with Hiero, the king of Syracuse, and Gelon, Hiero's son. The great geometer had a close friendship with and may have been related to the monarch. In any case, he seemed to make a hobby out of solving the king's most complicated problems to the utter amazement of the sovereign. At one time, the king ordered a gold crown and gave the goldsmith the exact amount of metal to make it. When Hiero received it, the crown had the correct weight but the monarch suspected that some silver had been used instead of the gold. Since he could not prove it, he brought the problem to Archimedes. One day while considering the question, "the wise one" entered his bathtub and recognized that the amount of water that overflowed the tub was proportional the amount of his body that was submerged. This observation is now known as Archimedes' Principle and gave him the means to solve the problem. He was so excited that he ran naked through the streets of Syracuse shouting "Eureka! eureka!" (I have found it!). The fraudulent goldsmith was brought to justice. Another time, Archimedes stated "Give me a place to stand on and I will move the earth." King Hiero, who was absolutely astonished by the statement, asked him to prove it. In the harbor was a ship that had proved impossible to launch even by the combined efforts of all the men

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of Syracuse. Archimedes, who had been examining the properties of levers and pulleys, built a machine that allowed him the single-handedly move the ship from a distance away. He also had many other inventions including the Archimedes' watering screw and a miniature planetarium.

Though he had many great inventions, Archimedes considered his purely theoretical work to be his true calling. His accomplishments are numerous. His approximation of between 3-1/2 and 3-10/71 was the most accurate of his time and he devised a new way to approximate square roots. Unhappy with the unwieldy Greek number system, he devised his own that could accommodate larger numbers more easily. He invented the entire field of hydrostatics with the discovery of the Archimedes' Principle. However, his greatest invention was integral calculus. To determine the area of sections bounded by geometric figures such as parabolas and ellipses, Archimedes broke the sections into an infinite number of rectangles and added the areas together. This is known as integration. He also anticipated the invention of differential calculus as he devised ways to approximate the slope of the tangent lines to his figures. In addition, he also made many other discoveries in geometry, mechanics and other fields.

The end of Archimedes life was anything but uneventful. King Hiero had been so impressed with his friend's inventions that he persuaded him to develop weapons to defend the city. These inventions would prove quite useful. In 212 B.C., Marcellus, a Roman general, decided to conquer Syracuse with a full frontal assault on both land and sea. The Roman legions were routed. Huge catapults hurled 500 pound boulders at the soldiers; large cranes with claws on the end lowered down on the enemy ships, lifted them in the air, and then threw them against the rocks; and systems of mirrors focused the sun rays to light enemy ships on fire. The Roman soldiers refused to continue the attack and fled at the mere sight of anything projecting from the walls of the city. Marcellus was forced to lay siege to the city, which fell after eight months. Archimedes was killed by a Roman soldier when the city was taken. The traditional story is that the mathematician was unaware of the taking of the city. While he was drawing figures in the dust, a Roman soldier stepped on them and demanded he come with him. Archimedes responded, "Don't disturb my circles!" The soldier was so enraged that he pulled out his sword and slew the great geometer. When Archimedes was buried, they placed on his tombstone the figure of a sphere inscribed inside a cylinder and the 2:3 ratio of the volumes between them, the solution to the problem he considered his greatest achievement.

http://www.shu.edu/projects/reals/history/archimed.html

Birth of Algebra

  Muhammad the Prophet (570 - 632) founded Islam in the Arabian peninsula; by 661, the armies of the Muslim caliphs had already spread to Persia in the east and Egypt in the west, and would soon overrun all of North Africa and Spain.  Constantinople was taken and lost more than once over the next century, and Muslim armies were finally held back from further European conquests by their loss at the Battle of Tours to Frankish forces

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under Charles Martel in 732.  In the early 800's the caliph al-Ma'mun founded the Bayt al-Hikma (House of Wisdom), an institute of higher learning and scholarship, in Baghdad, where Arabic translations of Greek and Indian works in natural philosophy, mathematics and astronomy were made.  Here and elsewhere throughout the Muslim empire, the mathematics of the ancients was studied and improved, and the Western world is indebted to these Arabic scholars for being largely responsible for the later transmission of this body of knowledge into Europe.     Muhammad ibn-Musa al-Khwarizmi (780? - 850?) was one of the earliest scholars at the Bayt al-Hikma; his most famous work was entitled Al-kitab al-muhtasar fi hisab al-jabr w'al-muqabala (The Condensed Book of Calclation by Restoration and Comparison).  In it he describes rules for solving problems involving an unknown quantity, and it represents the first true work of algebra ever written.  In fact, Latin scholars who learned of this work centuries later identified the methods found in this book by the transliterated words in the title: algebra and almucabala; eventually only the first of these terms was retained.  (Other scholars used the term algorismus, from the Latinized form of the author's name.  Today, the word "algorithm" is used to describe any well-defined procedure for calculation.)

http://cerebro.xu.edu/math/math147/02f/algebra/algebra.html

MC Escher

Maurits Cornelis Escher, who was born in Leeuwarden, Holland in 1898, created unique and fascinating works of art that explore and exhibit a wide range of mathematical ideas.          While he was still in school his family planned for him to follow his father's career of architecture, but poor grades and an aptitude for drawing and design eventually led him to a career in the graphic arts. His work went almost unnoticed until the 1950’s, but by 1956 he had given his first important exhibition, was written up in Time magazine, and acquired a world-wide reputation. Among his greatest admirers were mathematicians, who recognized in his work an extraordinary visualization of mathematical principles. This was the more remarkable in that Escher had no formal mathematics training beyond secondary school.          As his work developed, he drew great inspiration from the mathematical ideas he read about, often working directly from structures in plane and projective geometry, and eventually capturing the essence of non-Euclidean geometries, as we will see below. He was also fascinated with paradox and "impossible" figures, and used an idea of Roger Penrose’s to develop many intriguing works of art. Thus, for the student of mathematics, Escher’s work encompasses two broad areas: the geometry of space, and what we may call the logic of space.

Regular divisions of the plane, called “tessellations,” are arrangements of closed shapes that completely cover the plane

without overlapping and without leaving gaps. Typically, the shapes making up a tessellation are polygons or similar regular shapes, such as the square tiles often used on floors. Escher, however, was fascinated by every kind of tessellation – regular

Alhambra sketch (62k)

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and irregular – and took special delight in what he called “metamorphoses,” in which the shapes changed and interacted with each other, and sometimes even broke free of the plane itself.          His interest began in 1936, when he traveled to Spain and viewed the tile patterns used in the Alhambra.Whether or not this is fair to the mathematicians, it is true that they had shown that of all the regular polygons, only the triangle, square, and hexagon can be used for a tessellation. (Many more irregular polygons tile the plane – in particular there are many tessellations using irregular pentagons.) Escher exploited these basic patterns in his tessellations, applying what geometers would call reflections, glide reflections, translations, and rotations to obtain a greater variety of patterns. He also elaborated these patterns by “distorting” the basic shapes to render them into animals, birds, and other figures. These distortions had to obey the three, four, or six-fold symmetry of the underlying pattern in order to preserve the tessellation. The effect can be both startling and beautiful.

http://www.mathacademy.com/pr/minitext/escher/index.asp

John Harrison

John Harrison (March 1693 - March 24, 1776) was an English clock designer, who developed and built the world's first successful maritime clock, one whose accuracy was great enough to allow the determination of longitude over long distances.

Harrison was born at Foulby in Yorkshire, the eldest son of a carpenter. A carpenter by initial trade, Harrison built and repaired clocks in his spare time. Legend has it that he was given a watch when he was six to amuse him while in bed with smallpox, spending hours listening to it and studying its moving parts. Scholars today, however, consider this unlikely to be true, as clocks and watches of all kinds were rare and expensive at the time, and Harrison came from a family of fairly modest means.

He was a man of many skills and used these to improve on the way clocks were built. For example, he developed the gridiron pendulum, consisting of alternating brass and steel rods assembled so that the different expansion and contraction rates cancelled each other out. Another example of his inventive genius was the grasshopper escapement -- a control device for the step-by-step release of a clock's driving power. Being almost frictionless, it required no oiling.

In 1728 Harrison packed up full scale models of his inventions and drawings for a proposed marine clock to compete for the Longitude Prize and headed for London seeking financial assistance. He met with Edmond Halley, the Astronomer Royal, and presented his ideas. Halley sent him to meet George Graham, the country's foremost horologist (clockmaker). He must have been impressed with Harrison, for Graham personally loaned him money and told him to build a model of his marine clock.

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It took Harrison seven years to build Harrison Number One or H1. He presented it to members of the Royal Society who spoke on its behalf to the Board of Longitude. The board was so skeptical of any such design, after 14 years of failures, that they demanded a sea trial. Harrison boarded a small ship to Lisbon and back, and on their return the captain and navigator both praised the design. The navigator noted that his own calculations estimated they were 90 miles offshore on their return to Britain, but the H1 put them just offshore right when the shoreline appeared.

This was not the transatlantic voyage demanded by the Board of Longitude, but the Board was impressed enough to grant Harrison 500 pounds for continued work. Harrison moved on to develop a more rugged version, H2. H2 was ready in 1741 after three years of building and two of on-land testing, but Britain was at war with Spain at the time in the War of Austrian Succession and the mechanism was deemed too important to risk it falling into Spanish hands. He was granted another 500 pounds by the Board while waiting for the war to end, and he used it to work on H3. Five years later he abandoned it, unhappy with its performance.

He then proposed to build two new designs, H4 and H5. It was H4 that would become the masterpiece Harrison's designs are known as today. All of his early designs were heavy instruments needing to be slung from a beam in a ship and all were designed to keep time in a ship rolling and pitching in the worst storm. H4 was an instrument of beauty, being of the shape of a large pocketwatch.

H4 took 13 years to construct, and Harrison, now 68 years old, sent it on its transatlantic trial in the care of his son, William, in 1761. When the ship reached Jamaica it was only two miles in error. When the ship returned Harrison waited for the 20,000 pound prize, but the Board refused to believe the accuracy was not just luck, and demanded another trial. The Harrisons were outraged and demanded their prize, a matter that eventually worked its way to Parliament, which offered 5000 pounds for the design. They refused and planned another trip to Barbados to settle the matter.

On Harrison's second H4 trial, the Reverend Nevil Maskelyne was asked to accompany the ship and test the Lunar Distances system. Once again H4 proved almost astonishingly accurate, measuring Bridgetown to within 10 miles. Maskelyne's measures were also fairly good, at 30 miles, but required considerable work and calculation in order to use. At a meeting of the Board in 1765 the results were presented, and once again they couldn't believe it wasn't just luck. Once again the matter reached Parliament, which offered 10,000 pounds up front, and the other half once he turned over the design to other watchmakers to duplicate. In the meantime H4 would have to be turned over to the Astronomer Royal for long-term on-land testing.

http://en.wikipedia.org/wiki/John_Harrison

13 Archimedean Solids

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The 13 Archimedean solids are the convex polyhedra that have a similar arrangement of nonintersecting regular convex polygons of two or more different types arranged in the same way about each vertex with all sides the same length (Cromwell 1997, pp. 91-92).

The Archimedean solids are distinguished by having very high symmetry, thus excluding solids belonging to a dihedral group of symmetries (e.g., the two infinite families of regular prisms and antiprisms), as well as the the elongated square gyrobicupola (because that surface's symmetry-breaking twist allows vertices "near the equator" and those "in the polar regions" to be distinguished; Cromwell 1997, p. 92). The Archimedean solids are sometimes also referred to as the semiregular polyhedra.

Seven of the 13 Archimedean solids (the cuboctahedron, icosidodecahedron, truncated cube, truncated dodecahedron, truncated octahedron, truncated icosahedron, and truncated icosahedron) can be obtained by truncation of a Platonic solid. The three truncation series producing these seven Archimedean solids are illustrated above.

Two additional solids (the small rhombicosidodecahedron and small rhombicuboctahedron) can be obtained by expansion of a Platonic solid, and two further solids (the great rhombicosidodecahedron and great rhombicuboctahedron) can be obtained by expansion of one of the previous 9 Archimedean solids (Stott 1910; Ball and

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Coxeter 1987, pp. 139-140). It is sometimes stated (e.g., Wells 1991, p. 8) that these four solids can be obtained by truncation of other solids. The confusion originated with Kepler himself, who used the terms "truncated icosidodecahedron" and "truncated cuboctahedron" for the great rhombicosidodecahedron and great rhombicuboctahedron, respectively. However, truncation alone is not capable of producing these solids, but must be combined with distorting to turn the resulting rectangles into squares (Ball and Coxeter 1987, pp. 137-138; Cromwell 1997, p. 81).

The remaining two solids, the snub cube and snub dodecahedron, can be obtained by moving the faces of a cube and dodecahedron outward while giving each face a twist. The resulting spaces are then filled with ribbons of equilateral triangles (Wells 1991, p. 8).

Pugh (1976, p. 25) points out the Archimedean solids are all capable of being circumscribed by a regular tetrahedron so that four of their faces lie on the faces of that tetrahedron.

The Archimedean solids satisfy

where is the sum of face-angles at a vertex and V is the number of vertices (Steinitz and Rademacher 1934, Ball and Coxeter 1987).

Let the cyclic sequence represent the degrees of the faces surrounding a vertex (i.e., S is a list of the number of sides of all polygons surrounding any vertex). Then the definition of an Archimedean solid requires that the sequence must be the same for each vertex to within rotation and reflection. Walsh (1972) demonstrates that S represents the degrees of the faces surrounding each vertex of a semiregular convex polyhedron or tessellation of the plane iff

1. and every member of S is at least 3,

2. , with equality in the case of a plane tessellation, and

3. for every odd number , S contains a subsequence (b, p, b).

Condition (1) simply says that the figure consists of two or more polygons, each having at least three sides. Condition (2) requires that the sum of interior angles at a vertex must be equal to a full rotation for the figure to lie in the plane, and less than a full rotation for a solid figure to be convex.

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http://mathworld.wolfram.com/ArchimedeanSolid.html

Johann Carl Friedrich Gauss

Born: 30 April 1777 in Brunswick, Duchy of Brunswick (now Germany)Died: 23 Feb 1855 in Göttingen, Hanover (now Germany)

At the age of seven, Carl Friedrich Gauss started elementary school, and his potential was noticed almost immediately. His teacher, Büttner, and his assistant, Martin Bartels, were amazed when Gauss summed the integers from 1 to 100 instantly by spotting that the sum was 50 pairs of numbers each pair summing to 101.

In 1788 Gauss began his education at the Gymnasium with the help of Büttner and Bartels, where he learnt High German and Latin. After receiving a stipend from the Duke of Brunswick- Wolfenbüttel, Gauss entered Brunswick Collegium Carolinum in 1792. At the academy Gauss independently discovered Bode's law, the binomial theorem and the arithmetic- geometric mean, as well as the law of quadratic reciprocity and the prime number theorem.

In 1795 Gauss left Brunswick to study at Göttingen University. Gauss's teacher there was Kaestner, whom Gauss often ridiculed. His only known friend amongst the students was Farkas Bolyai. They met in 1799 and corresponded with each other for many years.

Gauss left Göttingen in 1798 without a diploma, but by this time he had made one of his most important discoveries - the construction of a regular 17-gon by ruler and compasses This was the most major advance in this field since the time of Greek mathematics and was published as Section VII of Gauss's famous work, Disquisitiones Arithmeticae.

Gauss returned to Brunswick where he received a degree in 1799. After the Duke of Brunswick had agreed to continue Gauss's stipend, he requested that Gauss submit a doctoral dissertation to the University of Helmstedt. He already knew Pfaff, who was chosen to be his advisor. Gauss's dissertation was a discussion of the fundamental theorem of algebra.

With his stipend to support him, Gauss did not need to find a job so devoted himself to research. He published the book Disquisitiones Arithmeticae in the summer of 1801. There were seven sections, all but the last section, referred to above, being devoted to number theory.

In June 1801, Zach, an astronomer whom Gauss had come to know two or three years previously, published the orbital positions of Ceres, a new "small planet" which was discovered by G Piazzi, an Italian astronomer on 1 January, 1801. Unfortunately, Piazzi had only been able to observe 9 degrees of its orbit before it disappeared behind the Sun. Zach published several predictions of its position, including one by Gauss which differed greatly from the others. When Ceres was rediscovered by Zach on 7 December 1801 it

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was almost exactly where Gauss had predicted. Although he did not disclose his methods at the time, Gauss had used his least squares approximation method.

Gauss's work never seemed to suffer from his personal tragedy. He published his second book, Theoria motus corporum coelestium in sectionibus conicis Solem ambientium, in 1809, a major two volume treatise on the motion of celestial bodies. In the first volume he discussed differential equations, conic sections and elliptic orbits, while in the second volume, the main part of the work, he showed how to estimate and then to refine the estimation of a planet's orbit. Gauss's contributions to theoretical astronomy stopped after 1817, although he went on making observations until the age of 70.

From the early 1800s Gauss had an interest in the question of the possible existence of a non-Euclidean geometry. He discussed this topic at length with Farkas Bolyai and in his correspondence with Gerling and Schumacher. In a book review in 1816 he discussed proofs which deduced the axiom of parallels from the other Euclidean axioms, suggesting that he believed in the existence of non-Euclidean geometry, although he was rather vague. Gauss confided in Schumacher, telling him that he believed his reputation would suffer if he admitted in public that he believed in the existence of such a geometry.

http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Gauss.html

Georg Friedrich Bernhard Riemann

Born: 17 Sept 1826 in Breselenz, Hanover (now Germany)Died: 20 July 1866 in Selasca, Italy

Bernhard Riemann's father, Friedrich Bernhard Riemann, was a Lutheran minister. Friedrich Riemann married Charlotte Ebell when he was in his middle age. Bernhard was the second of their six children, two boys and four girls. Friedrich Riemann acted as teacher to his children and he taught Bernhard until he was ten years old. At this time a teacher from a local school named Schulz assisted in Bernhard's education.

In the spring of 1846 Riemann enrolled at the University of Göttingen. His father had encouraged him to study theology and so he entered the theology faculty. However he attended some mathematics lectures and asked his father if he could transfer to the faculty of philosophy so that he could study mathematics. Riemann was always very close to his family and he would never have changed courses without his father's permission. This was granted, however, and Riemann then took courses in mathematics from Moritz Stern and Gauss.

Riemann moved from Göttingen to Berlin University in the spring of 1847 to study under Steiner, Jacobi, Dirichlet and Eisenstein. This was an important time for Riemann. He learnt much from Eisenstein and discussed using complex variables in elliptic function theory. The main person to influence Riemann at this time, however, was Dirichlet.

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Riemann's work always was based on intuitive reasoning which fell a little below the rigour required to make the conclusions watertight. However, the brilliant ideas which his works contain are so much clearer because his work is not overly filled with lengthy computations. It was during his time at the University of Berlin that Riemann worked out his general theory of complex variables that formed the basis of some of his most important work.

In 1849 he returned to Göttingen and his Ph.D. thesis, supervised by Gauss, was submitted in 1851. However it was not only Gauss who strongly influenced Riemann at this time. Weber had returned to a chair of physics at Göttingen from Leipzig during the time that Riemann was in Berlin, and Riemann was his assistant for 18 months. Also Listing had been appointed as a professor of physics in Göttingen in 1849. Through Weber and Listing, Riemann gained a strong background in theoretical physics and, from Listing, important ideas in topology which were to influence his ground breaking research.

Riemann's thesis studied the theory of complex variables and, in particular, what we now call Riemann surfaces. It therefore introduced topological methods into complex function theory. The work builds on Cauchy's foundations of the theory of complex variables built up over many years and also on Puiseux's ideas of branch points. However, Riemann's thesis is a strikingly original piece of work which examined geometric properties of analytic functions, conformal mappings and the connectivity of surfaces.

In proving some of the results in his thesis Riemann used a variational principle which he was later to call the Dirichlet Principle since he had learnt it from Dirichlet's lectures in Berlin. The Dirichlet Principle did not originate with Dirichlet, however, as Gauss, Green and Thomson had all made use if it. Riemann's thesis, one of the most remarkable pieces of original work to appear in a doctoral thesis, was examined on 16 December 1851.

On Gauss's recommendation Riemann was appointed to a post in Göttingen and he worked for his Habilitation, the degree which would allow him to become a lecturer. He spent thirty months working on his Habilitation dissertation which was on the representability of functions by trigonometric series. He gave the conditions of a function to have an integral, what we now call the condition of Riemann integrability.

To complete his Habilitation Riemann had to give a lecture. He prepared three lectures, two on electricity and one on geometry. Gauss had to choose one of the three for Riemann to deliver and, against Riemann's expectations, Gauss chose the lecture on geometry. Riemann's lecture Über die Hypothesen welche der Geometrie zu Grunde liegen (On the hypotheses that lie at the foundations of geometry), delivered on 10 June 1854, became a classic of mathematics.

There were two parts to Riemann's lecture. In the first part he posed the problem of how to define an n-dimensional space and ended up giving a definition of what today we call a Riemannian space. In fact the main point of this part of Riemann's lecture was the

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definition of the curvature tensor. The second part of Riemann's lecture posed deep questions about the relationship of geometry to the world we live in. He asked what the dimension of real space was and what geometry described real space. The lecture was too far ahead of its time to be appreciated by most scientists of that time.

A newly elected member of the Berlin Academy of Sciences had to report on their most recent research and Riemann sent a report on On the number of primes less than a given magnitude another of his great masterpieces which were to change the direction of mathematical research in a most significant way. In it Riemann examined the zeta function

(s) = (1/ns) = (1 - p-s)-1

which had already been considered by Euler. Here the sum is over all natural numbers n while the product is over all prime numbers. Riemann considered a very different question to the one Euler had considered, for he looked at the zeta function as a complex function rather than a real one. Except for a few trivial exceptions, the roots of (s) all lie between 0 and 1. In the paper he stated that the zeta function had infinitely many nontrivial roots and that it seemed probable that they all have real part 1/2. This is the famous Riemann hypothesis which remains today one of the most important of the unsolved problems of mathematics.

Riemann studied the convergence of the series representation of the zeta function and found a functional equation for the zeta function. The main purpose of the paper was to give estimates for the number of primes less than a given number.

http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Riemann.html

Nikolai Ivanovich Lobachevsky

Born: 1 Dec 1792 in Nizhny Novgorod (was Gorky from 1932-1990), RussiaDied: 24 Feb 1856 in Kazan, Russia

In 1807 Lobachevsky graduated from the Gymnasium and entered Kazan University as a free student. Kazan State University had been founded in 1804, the result of one of the many reforms of the emperor Alexander I, and it opened in the following year, only two years before Lobachevsky began his undergraduate career. His original intention was to study medicine but he changed to study a broad scientific course involving mathematics and physics.

One of the excellent professors who had been invited from Germany was Martin Bartels (1769 - 1833) who had been appointed as Professor of Mathematics. Bartels was a school teacher and friend of Gauss, and the two corresponded. We shall return later to discuss ideas of some historians, for example M Kline, that Gauss may have given Lobachevsky hints regarding directions that he might take in his mathematical work through the letters

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exchanged between Bartels and Gauss. A skilled teacher, Bartels soon interested Lobachevsky in mathematics. We do know that Bartels lectured on the history of mathematics and that he gave a course based on the text by Montucla. Since Euclid's Elements and his theory of parallel lines are discussed in detail in Montucla's book, it seems likely that Lobachevsky's interest in the Fifth Postulate was stimulated by these lectures.

Since Euclid's axiomatic formulation of geometry mathematicians had been trying to prove his fifth postulate as a theorem deduced from the other four axioms. The fifth postulate states that given a line and a point not on the line, a unique line can be drawn through the point parallel to the given line. Lobachevsky did not try to prove this postulate as a theorem. Instead he studied geometry in which the fifth postulate does not necessarily hold. Lobachevsky categorised euclidean as a special case of this more general geometry.

His major work, Geometriya completed in 1823, was not published in its original form until 1909. On 11 February 1826, in the session of the Department of Physico-Mathematical Sciences at Kazan University, Lobachevsky requested that his work about a new geometry was heard and his paper A concise outline of the foundations of geometry was sent to referees. The text of this paper has not survived but the ideas were incorporated, perhaps in a modified form, in Lobachevsky's first publication on hyperbolic geometry. He published this work on non-euclidean geometry, the first account of the subject to appear in print, in 1829. It was published in the Kazan Messenger but rejected by Ostrogradski when it was submitted for publication by the St Petersburg Academy of Sciences.

In 1834 Lobachevsky found a method for the approximation of the roots of algebraic equations. This method of numerical solution of algebraic equations, developed independently by Gräffe to answer a prize question of the Berlin Academy, is today a particularly suitable method for using computers to solve such problems. This method is today called the Dandelin-Gräffe method since Dandelin also independently investigated it, but only in Russia does the method appear to be named after Lobachevsky who is the third independent discoverer.

The story of how Lobachevsky's hyperbolic geometry came to be accepted is a complex one and this biography is not the place in which to go into details, but we shall note the main events. In 1866, ten years after Lobachevsky's death, Hoüel published a French translation of Lobachevsky's Geometrische Untersuchungen together with some of Gauss's correspondence on non-euclidean geometry. Beltrami, in 1868, gave a concrete realisation of Lobachevsky's geometry. Weierstrass led a seminar on Lobachevsky's geometry in 1870 which was attended by Klein and, two years later, after Klein and Lie had discussed these new generalisations of geometry in Paris, Klein produced his general view of geometry as the properties invariant under the action of some group of transformations in the Erlanger Programm. There were two further major contributions to Lobachevsky's geometry by Poincaré in 1882 and 1887. Perhaps these finally mark the acceptance of Lobachevsky's ideas which would eventually be seen as vital steps in

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freeing the thinking of mathematicians so that relativity theory had a natural mathematical foundation.

http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Lobachevsky.html

Albert Einstein

Born: 14 March 1879 in Ulm, Württemberg, GermanyDied: 18 April 1955 in Princeton, New Jersey, USA

In 1894 Einstein's family moved to Milan but Einstein remained in Munich. In 1895 Einstein failed an examination that would have allowed him to study for a diploma as an electrical engineer at the Eidgenössische Technische Hochschule in Zurich. Einstein renounced German citizenship in 1896 and was to be stateless for a number of years. He did not even apply for Swiss citizenship until 1899, citizenship being granted in 1901.

Einstein earned a doctorate from the University of Zurich in 1905 for a thesis On a new determination of molecular dimensions. He dedicated the thesis to Grossmann.

In the first of three papers, all written in 1905, Einstein examined the phenomenon discovered by Max Planck, according to which electromagnetic energy seemed to be emitted from radiating objects in discrete quantities. The energy of these quanta was directly proportional to the frequency of the radiation. This seemed to contradict classical electromagnetic theory, based on Maxwell's equations and the laws of thermodynamics which assumed that electromagnetic energy consisted of waves which could contain any small amount of energy. Einstein used Planck's quantum hypothesis to describe the electromagnetic radiation of light.

Einstein's second 1905 paper proposed what is today called the special theory of relativity. He based his new theory on a reinterpretation of the classical principle of relativity, namely that the laws of physics had to have the same form in any frame of reference. As a second fundamental hypothesis, Einstein assumed that the speed of light remained constant in all frames of reference, as required by Maxwell's theory.

Later in 1905 Einstein showed how mass and energy were equivalent. Einstein was not the first to propose all the components of special theory of relativity. His contribution is unifying important parts of classical mechanics and Maxwell's electrodynamics.

The third of Einstein's papers of 1905 concerned statistical mechanics, a field of that had been studied by Ludwig Boltzmann and Josiah Gibbs.

After 1905 Einstein continued working in the areas described above. He made important contributions to quantum theory, but he sought to extend the special theory of relativity to phenomena involving acceleration. The key appeared in 1907 with the principle of equivalence, in which gravitational acceleration was held to be indistinguishable from

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acceleration caused by mechanical forces. Gravitational mass was therefore identical with inertial mass.

In 1908 Einstein became a lecturer at the University of Bern after submitting his Habilitation thesis Consequences for the constitution of radiation following from the energy distribution law of black bodies. The following year he become professor of physics at the University of Zurich, having resigned his lectureship at Bern and his job in the patent office in Bern.

By 1909 Einstein was recognised as a leading scientific thinker and in that year he resigned from the patent office. He was appointed a full professor at the Karl-Ferdinand University in Prague in 1911. In fact 1911 was a very significant year for Einstein since he was able to make preliminary predictions about how a ray of light from a distant star, passing near the Sun, would appear to be bent slightly, in the direction of the Sun. This would be highly significant as it would lead to the first experimental evidence in favour of Einstein's theory.

About 1912, Einstein began a new phase of his gravitational research, with the help of his mathematician friend Marcel Grossmann, by expressing his work in terms of the tensor calculus of Tullio Levi-Civita and Gregorio Ricci-Curbastro. Einstein called his new work the general theory of relativity. He moved from Prague to Zurich in 1912 to take up a chair at the Eidgenössische Technische Hochschule in Zurich.

Einstein returned to Germany in 1914 but did not reapply for German citizenship. What he accepted was an impressive offer. It was a research position in the Prussian Academy of Sciences together with a chair (but no teaching duties) at the University of Berlin. He was also offered the directorship of the Kaiser Wilhelm Institute of Physics in Berlin which was about to be established.

Einstein received the Nobel Prize in 1921 but not for relativity rather for his 1905 work on the photoelectric effect. In fact he was not present in December 1922 to receive the prize being on a voyage to Japan.

By 1930 he was making international visits again, back to the United States. A third visit to the United States in 1932 was followed by the offer of a post at Princeton. The idea was that Einstein would spend seven months a year in Berlin, five months at Princeton. Einstein accepted and left Germany in December 1932 for the United States. The following month the Nazis came to power in Germany and Einstein was never to return there.

What was intended only as a visit became a permanent arrangement by 1935 when he applied and was granted permanent residency in the United States. At Princeton his work attempted to unify the laws of physics. In 1940 Einstein became a citizen of the United States, but chose to retain his Swiss citizenship. He made many contributions to peace during his life. In 1944 he made a contribution to the war effort by hand writing his 1905

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paper on special relativity and putting it up for auction. It raised six million dollars, the manuscript today being in the Library of Congress.

http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Einstein.html

Gaston Maurice Julia

Born: 3 Feb 1893 in Sidi Bel Abbès, AlgeriaDied: 19 March 1978 in Paris, France

When only 25 Gaston Julia published his 199 page masterpiece Mémoire sur l'iteration des fonctions rationelles which made him famous in the mathematics centres of his days.

As a soldier in the First World War, Julia had been severely wounded in an attack on the French front designed to celebrate the Kaiser's birthday. Many on both sides were wounded including Julia who lost his nose and had to wear a leather strap across his face for the rest of his life. Between several painful operations he carried on his mathematical researches in hospital.

Later he became a distinguished professor at the École Polytechnique in Paris.

In 1918 Julia published a beautiful paper Mémoire sur l'itération des fonctions rationnelles, Journal de Math. Pure et Appl. 8 (1918), 47-245, concerning the iteration of a rational function f. Julia gave a precise description of the set J(f) of those z in C for which the nth iterate fn(z) stays bounded as n tends to infinity. It received the Grand Prix de l'Académie des Sciences.

http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Julia.html

He was a French mathematician who devised the formula for the Julia set. His works were popularised by French mathematician Benoit Mandelbrot, and the Julia and Mandelbrot fractals are closely related.

http://en.wikipedia.org/wiki/Gaston_Julia

Benoit Mandelbrot

Born: 20 Nov 1924 in Warsaw, Poland

Benoit Mandelbrot was largely responsible for the present interest in fractal geometry. He showed how fractals can occur in many different places in both mathematics and elsewhere in nature.

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Mandelbrot was born in Poland in 1924 into a family with a very academic tradition. His father, however, made his living buying and selling clothes while his mother was a doctor. As a young boy, Mandelbrot was introduced to mathematics by his two uncles.

1958 he left for the United States permanently and began his long standing and most fruitful collaboration with IBM as an IBM Fellow at their world renowned laboratories in Yorktown Heights in New York State.

IBM presented Mandelbrot with an environment which allowed him to explore a wide variety of different ideas. He has spoken of how this freedom at IBM to choose the directions that he wanted to take in his research presented him with an opportunity which no university post could have given him. After retiring from IBM, he found similar opportunities at Yale University, where he is presently Sterling Professor of Mathematical Sciences.

In 1945 Mandelbrot's uncle had introduced him to Julia's important 1918 paper claiming that it was a masterpiece and a potential source of interesting problems, but Mandelbrot did not like it. Indeed he reacted rather badly against suggestions posed by his uncle since he felt that his whole attitude to mathematics was so different from that of his uncle. Instead Mandelbrot chose his own very different course which, however, brought him back to Julia's paper in the 1970s after a path through many different sciences which some characterise as highly individualistic or nomadic. In fact the decision by Mandelbrot to make contributions to many different branches of science was a very deliberate one taken at a young age. It is remarkable how he was able to fulfil this ambition with such remarkable success in so many areas.

With the aid of computer graphics, Mandelbrot who then worked at IBM's Watson Research Center, was able to show how Julia's work is a source of some of the most beautiful fractals known today. To do this he had to develop not only new mathematical ideas, but also he had to develop some of the first computer programs to print graphics.

The Mandelbrot set is a connected set of points in the complex plane. Pick a point z0 in the complex plane.

Calculate:z1= z0

2 + z0

z2 = z12 + z0

z3 = z22 + z0

. . .

If the sequence z0 , z1 , z2 , z3 , ... remains within a distance of 2 of the origin forever, then the point z0 is said to be in the Mandelbrot set. If the sequence diverges from the origin, then the point is not in the set.

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As well as IBM Fellow at the Watson Research Center Mandelbrot was Professor of the Practice of Mathematics at Harvard University. He also held appointments as Professor of Engineering at Yale, of Professor of Mathematics at the École Polytechnique, of Professor of Economics at Harvard, and of Professor of Physiology at the Einstein College of Medicine. Mandelbrot's excursions into so many different branches of science was, as we mention above, no accident but a very deliberate decision on his part. It was, however, the fact that fractals were so widely found which in many cases provided the route into other areas.

http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Mandelbrot.html

Lewis Fry Richardson

Born: 11 Oct 1881 in Newcastle upon Tyne, Northumberland, EnglandDied: 30 Sept 1953 in Kilmun, Argyll, Scotland

Lewis Fry Richardson was born into a Quaker family. He attended Newcastle Preparatory School where his favourite subject was the study of Euclid.

It was Richardson who was the first to apply mathematics, in particular the method of finite differences, to predicting the weather in Weather Prediction by Numerical Process (1922). He first developed his method of finite differences in order to solve differential equations which arose in his work for the National Peat Industries concerning the flow of water in peat. Having developed these methods by which he was able to obtain highly accurate solutions, it was a natural step to apply the same methods to solve the problems of the dynamics of the atmosphere which he encountered in his work for the Meteorological Office.

Making observations from weather stations would provide data which defined the initial conditions, then the equations could be solved with these initial conditions and a prediction of the weather could be made. It was a remarkable piece of work but in a sense it was ahead of its time since the time taken for the necessary hand calculations in a pre-computer age took so long that, even with many people working to solve the equations, the solution would be found far too late to be useful to predict the weather. He calculated himself that it would need 60,000 people involved in the calculations in order to have the prediction of tomorrow's weather before the weather actually arrived. Despite this, Richardson's work laid the foundations for present day weather forecasting.

In addition to his 1922 book, Richardson published about 30 papers on the mathematics of the weather and in these he made contributions to the calculus and to the theory of diffusion, in particular eddy-diffusion in the atmosphere. The 'Richardson number', a fundamental quantity involving gradients of temperature and wind velocity is named after him. His achievements were recognised by election to the Royal Society in 1926.

http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Richardson.html

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Richardson also attempted to apply his mathematical skills in the service of his pacifist principles, in particular in understanding the roots of international conflict. As he had done with weather, he analyzed war using differential equations. Considering the armament of two nations, Richardson posited an idealized system of equations whereby the rate of a nation's armament build-up is directly proportional to the amount of arms its rival has and also to the grievances felt toward the rival, and negatively proportional to the amount of arms it already has itself. Solution of this system of equations allows insightful conclusions to be drawn regarding the nature, and the stability or instability, of various hypothetical conditions which might obtain between nations.

He also originated the theory that the propensity for war between two nations was a function of the length of their common border. And in Arms and Insecurity (1949), and Statistics of Deadly Quarrels (1950), he sought to statistically analyze the causes of war. Factors he assessed included economics, language, and religion.

In collecting data, he realised that there was considerable variation in the various gazetted lengths of international borders. For example, that between Spain and Portugal was variously quoted as 987 or 1214 km while that between The Netherlands and Belgium as 380 or 449 km. This led Richardson to the, rather obvious, realisation that the measured length of a line depends on the length of the 'ruler' used to measure it. However, further consideration of this idea led him to the concept of a line, so 'wriggly' that it fills a plane, a key step in the development of the mathematics of fractals.

http://en.wikipedia.org/wiki/Lewis_Fry_Richardson