Gauss          and          Euler

#### 07/28/2017

MATHEMATICIANS  (Also see http://aleph0.clarku.edu/~djoyce/mathhist/chronology.html )

Pythagoras of Samos (c. 570-c. 495 BC): (Greek)  He was an Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so that very little reliable information is known about him.  Around 530 BC, he moved to Croton, a Greek colony in southern Italy, and there set up a religious sect. His followers pursued the religious rites and practices developed by Pythagoras, and studied his philosophical theories. The society took an active role in the politics of Croton, but this eventually led to their downfall. The Pythagorean meeting-places were burned, and Pythagoras was forced to flee the city. He is said to have ended his days in Metapontum. He is often revered as a great mathematician, mystic and scientist, and he is best known for the Pythagorean theorem which bears his name.

Euclid of Alexandria (300 BC-       ):  (Greek)  He was often referred to as the "Father of Geometry."             His "Elements" is one of the most influential works in mathematics, serving as the main textbook for teaching mathematics, especially geometry, from the time of its publication until the late 19th or early 20th century.           Euclid  would love the following web sites:      and https://en.wikipedia.org/wiki/Missing_square_puzzle and https://www.brainbashers.com/missinganswer.asp  and http://mathusiasts.wordpress.com/2011/04/16/missing-square-puzzle-solution/ but Fibonacci would have preferred http://jaysdesktop.blogspot.com/2009/07/solution-to-missing-square-problem.html

Geometry  Are the eight balls moving in a circle or a straight line?
http://showyou.com/v/y-pNe6fsaCVtI/crazy-circle-illusion?u=multimotion

Archimedes of Syracuse (c.287-c.212 B.C):  (from Sicily) A mathematician and inventor. He determined the exact value of pi, is also known for his strategic role in ancient war and the development of military techniques. "Give me a place to stand and I will move the earth" was his boast when he discovered the laws of levers and pulleys.  His mechanical inventions defeated the Roman fleet of Marcellus.  The word "eureka" comes from the story that when Archimedes figured out a way to determine whether the king (Hiero II of Syracuse), a possible relative, had been duped by measuring the buoyancy of the king's supposedly solid gold crown in water, he became very excited and exclaimed the Greek (Archimedes' native language) for "I have found it": Eureka. Archimedes requested that his tombstone be decorated with a sphere contained in the smallest possible cylinder and inscribed with the ratio of the cylinder's volume to that of the sphere. Archimedes considered the discovery of this ratio the greatest of all his accomplishments.

Heron of Alexandria    Not sure of when he lived. Probably around 62 AD. Spent time in the Library of Alexandria.  Heron also proved his famous formula of the area, A, of a triangle knowing the three sides is:

# Heron's Formula

## Area of a Triangle from Sides

You can calculate the area of a triangle if you know the lengths of all three sides, using a formula that has been known for nearly 2000 years.

It is called "Heron's Formula" after Hero of Alexandria

 Step 1: Calculate "s" (half of the triangles perimeter): Step 2: Then calculate the Area:

It has been suggested that Archimedes knew the formula over two centuries earlier.    Other references:  www.math.umn.edu/~rejto/1151/1151_heron.pdf and mathworld.wolfram.com/HeronsFormula.html

Diophantus of Alexandria (200 and 214 -- 284 and 298): (Greek) Sometimes called "the father of algebra", was an Alexandrian Greek mathematician and the author of a series of books called Arithmetica. These texts deal with solving algebraic equations, many of which are now lost. In studying Arithmetica. Pierre de Fermat concluded that a certain equation considered by Diophantus had no solutions, and noted without elaboration that he had found "a truly marvelous proof of this proposition," now referred to as Fermat's Last Theorem. This led to tremendous advances in number theory, and the study of Diophantine equations ("Diophantine geometry") and of Diophantine approximations remain important areas of mathematical research. Diophantus was the first Greek mathematician who recognized fractions as numbers; thus he allowed positive rational numbers for the coefficients and solutions. In modern use, Diophantine equations are usually algebraic equations with integer coefficients, for which integer solutions are sought. Diophantus also made advances in mathematical notation.

Leonardo Pisano Fibonacci (1170?-1250):  (Italian)  Fibonacci is considered to be one of the most talented mathematicians for the Middle Ages. Few people realize that it was Fibonacci that gave us our decimal number system (Hindu-Arabic numbering system) which replaced the Roman Numeral system. When he was studying mathematics, he used the Hindu-Arabic (0-9) symbols instead of Roman symbols which didn't have 0's and lacked place value. In fact, when using the Roman Numeral system, an abacus was usually required. There is no doubt that Fibonacci saw the superiority of using Hindu-Arabic system over the Roman Numerals. He shows how to use our current numbering system in his book Liber abaci. And he gave us the Fibonacci Series. Fibonacci was known as Leonardo of Pisa. He was born in Pisa, home of the famous leaning tower and his statue is located there.

In his famous "Rabbit Problem" he produces the Fibonacci Series as the answer: 1 1 2 3 5 8 13 21 34 55 etc., where each term is equal to the sum of the two previous terms.  The Fibonacci sequence obeys the recursion relation F(n) = F(n-1) + F(n-2). The ratio of the current term to the previous term approaches the golden ratio or (1 + sq rt of 5)/2, about 1.618...  This ratio is called the "golden ratio".  The German Adolph Zeising claimed the front of the Parthenon is in proportion to the golden ratio. There is no documentary evidence that Phidias, used the golden ratio in any of his work related to the Parthenon.  However around 1909, the American mathematician Mark Barr, named the golden ratio the Greek letter "phi" for Phidias..  When phi is expressed as a continued fraction it looks like this:

Continued fractions provide mathematicians with a way of rating how irrational a number might be. Since the expression for phi contains only 1s, it is the purest continued fraction that there is, and hence is considered the most irrational number.

IRRATIONAL NUMBERS See http://www.ams.org/samplings/feature-column/fcarc-irrational1

An irrational number by definition is one which cannot be written as the ratio of whole numbers. So it would seem that all irrational numbers are equally irrational. All pigs are equal, Orwell said, but some are more equal than others. And in fact there is a precise sense in which some irrational numbers are more irrational than others. This phenomenon has had important consequences in the organization of the natural world. In packing seeds around a core, many plants choose the strategy of placing each one at the most irrational angle possible to the one directly below it.

The Pythagoreans did not believe in irrational numbers or incommensurability.  But in $a^{2}+b^{2}=c^{2}\!\,$  we can look at a square with a and b equal to 1 and c = square root of 2.  One rectangle has a =1, c = 2, and b = square root of 3.  Another rectangle has a =1, b = 2, and c = square root of 5.

These irrational numbers are also found in other geometrical shapes.  The pentagram which contains  .  Sir Thomas Mallory in La Morte d'Arthur placed it on Sir Gawain's shield. In Dan Brown'sThe Da Vinci Code, the dying curator of the Louvre drew a pentagram in his own blood on his abdomen as a clue to identify his murderer.  The hexagram was also mentioned by Dan Brown in the same novel.  It contains 3, which is the length of the shorter diagonal of a hexagram whose sides equal 1.

In studying "Sacred Geometry" (http://sacredgeometryinternational.com/the-meaning-of-sacred-geometry-part-3-the-womb-of-sacred-geometry# )we see the Vessel of Fish. This diagram is usually shown as two identical circles in which the right hand circle passes thru the center of the first circle on the left.  The square roots of 3 and 5 are included. The many objects of Sacred Geometry may be achieved by means of the figure known as the Vesica Piscis.

## The most irrational number

The most irrational number turns out to be a number already well known in geometry. It is the number

Phi = = ( + 1)/2 = 1.618033...

which is the length of the diagonal in a regular pentagon of side length 1. This number, known as the "golden mean," has played a large role in mathematical aesthetics. It is not clear whether its supreme irrationality has anything to do with its artistic applications.

It is the only number that is 1 more than its reciprocal. x = 1/x  -1.

The golden mean satisfies the equation x2 - x - 1 = 0, so its continued fraction expansion is the simplest of all:

A representation in terms of a nested radical is

is the "worst" real number for rational approximation because its continued fraction is:

=

 The Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, ... (add the last two to get the next) The Golden Section numbers are ±0·61803 39887... and ±1·61803 39887...   Phi is the only number that is 1 less than its reciprocal.  Phi -1 = 1/Phi The Golden String is a fractal string of 0s and 1s that grows in a Fibonacci-like way as follows: 1                                                                                                                                                                                                  10                                                                                                                                                                                               101                                                                                                                                                                                                                        10110                                                                                                                                                                                                          10110101                                                                                                                                                                                           1011010110110                                                                                                                                                                                             101101011011010110101   After the first two lines, all the others are made from the two latest lines in a similar way to each Fibonacci numbers being a sum of the two before it. Each string (list of 0s and 1s) is a copy of the one above it followed by the one above that. The resulting infinitely long string is the Golden String or Fibonacci Word or Rabbit Sequence.  It is is closely related to the Fibonacci numbers and the golden section. There is a relationship between fractals and the "golden string".         See http://www.youtube.com/watch?v=ZDGGEQqSXew

Leonardo Da Vinci called the golden ratio  the "divine proportion" and featured it in many of his paintings.

Nicolaus Copernicus (1473-1543):  (Prussia) He was a Renaissance astronomer and the first person to formulate a comprehensive heliocentric cosmology, which displaced the Earth from the center of the universe.  Copernicus' epochal book, De revolutionibus orbium coelestium (On the Revolutions of the Celestial Spheres), published just before his death in 1543, is often regarded as the starting point of modern astronomy and the defining epiphany that began the scientific revolution. His heliocentric model, with the Sun at the center of the universe, demonstrated that the observed motions of celestial objects can be explained without putting Earth at rest in the center of the universe. His work stimulated further scientific investigations, becoming a landmark in the history of science that is often referred to as the Copernican Revolution.

Niccolò Fontana "Tartaglia" (1499/1500-1557): (Italian)  He published many books, including the first Italian translations of Archimedes and Euclid, and an acclaimed compilation of mathematics. Tartaglia was the first to apply mathematics to the investigation of the paths of cannonballs, known as ballistics, in his Nova Scientia, “A New Science;” his work was later validated by Galileo's studies on falling bodies. He also published a treatise on retrieving sunken ships.

When the French invaded Brescia, a French soldier sliced Niccolò's jaw and palate with a saber. This made it impossible for Niccolò to speak normally, prompting the nickname "Tartaglia" ("stammerer"), which he adopted.

His best known work is his treatise General Trattato di numeri, et misure published in Venice 1556–1560. This has been called the best treatise on arithmetic that appeared in the sixteenth century. Not only does Tartaglia have complete discussions of numerical operations and the commercial rules used by Italian arithmeticians in this work, but he also discusses the life of the people, the customs of merchants and the efforts made to improve arithmetic in the 16th century.

Tartaglia is known for his conflicts with Gerolamo Cardano. Cardano cajoled Tartaglia into revealing the solution to the cubic equations, by promising not to publish them. Tartaglia divulged the secrets of the solutions of three different forms of the cubic equation in verse.  Even though Cardano credited his discovery, Tartaglia was extremely upset. He responded by publicly insulting Cardano.  Mathematical historians now credit both with the paternity of the formula to solve cubic equations, referring to it as the "Cardano-Tartaglia Formula".

Gerolamo Cardano (1501-1576): (French)  He was an Italian Renaissance mathematician, physician, astrologer and gambler. Today, he is best known for his achievements in algebra. He published the solutions to the cubic and quartic equations in his 1545 book Ars Magna. The solution to one particular case of the cubic, x3 + ax = b (in modern notation), was communicated to him by Niccolo Fontana Tartaglia (who later claimed that Cardano had sworn not to reveal it, and engaged Cardano in a decade-long fight), The quartic was solved by Cardano's student Lodovico Ferrari. Both were acknowledged in the foreword of the book, as well as in several places within its body. In his exposition, he acknowledged the existence of what are now called imaginary numbers, although he did not understand their properties (Mathematical field theory was developed centuries later). In Opus novum de proportionibus he introduced the binomial coefficients and the binomial theorem

Cardano was notoriously short of money and kept himself solvent by being an accomplished gambler and chess player. His book about games of chance, Liber de ludo aleae ("Book on Games of Chance") , written in 1526, but not published until 1663, contains the first systematic treatment of probability, as well as a section on effective cheating methods. Cardano invented several mechanical devices including the combination lock, the gimbal consisting of three concentric rings allowing a supported compass or gyroscope to rotate freely, and the Cardan shaft with universal joints, which allows the transmission of rotary motion at various angles and is used in vehicles to this day. He studied hypocycloids, published in de proportionibus 1570. The generating circles of these hypocycloids were later named Cardano circles or cardanic circles and were used for the construction of the first high-speed printing presses.

Franciscus Vieta (1540-1603): (French) His work on new algebra was an important step towards modern algebra, due to its innovative use of letters as parameters in equations. He was a lawyer by trade, and served as a privy councillor to both Henry III and Henry IV.

Galileo Galilei(1564-1642): (Italian)  A physicist, mathematician, astronomer and philosopher who played a major role in the Scientific Revolution. His achievements include improvements to the telescope and consequent astronomical observations, and support for Copernicanism.  Galileo has been called the "father of modern observational astronomy", the "father of modern physics", the "father of science", and "the Father of Modern Science". Stephen Hawking says, "Galileo, perhaps more than any other single person, was responsible for the birth of modern science."  Read about his "square cube" law:    http://dinosaurtheory.com/scaling.html

Johannes Kepler (1571-1630): (German)  A mathematician, astronomer and astrologer, and key figure in the 17th century scientific revolution. He is best known for his eponymous laws of planetary motion, codified by later astronomers, based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican Astronomy. These works also provided one of the foundations for Isaac Newton's theory of universal gravitation.

Rene Descartes (1596-1650): (French)  The inventor of Analytical Geometry.  He was a philosopher, mathematician, physicist and writer. He has been dubbed the "Father of Modern Philosophy".     If you want interesting info on conic sections, see  https://mysite.du.edu/~jcalvert/    (James B Calvert is a retired professor of Engineering at the University of Denver and has an excellent web site).

Pierre de Fermat (1601-1665): (French)  A lawyer and amateur mathematician who contributed to Number Theory and known for "Fermat's Last Theorem".  Fermat was the first person known to have evaluated the integral of general power functions. Using an ingenious trick, he was able to reduce this evaluation to the sum of geometric series. The resulting formula was helpful to both Newton and Leibnitz in developing calculus.

John Wallis (1616-1703): (English) A  mathematician who is given partial credit for the development of infinitesimal calculus. Between 1643 and 1689 he served as chief cryptographer for Parliament and, later, the royal court. He is also credited with introducing the symbol ∞ for infinity.  Wallis made significant contributions to trigonometry, calculus, geometry, and the analysis of infinite series. In his Opera Mathematica I (1695) Wallis introduced the term "continued fraction".  He is generally credited as the originator of the idea of the number line where numbers are represented geometrically in a line with the positive numbers increasing to the right and negative numbers to the left.  In 1655, Wallis published a treatise on conic sections in which they were defined analytically. This was the earliest book in which these curves are considered and defined as curves of the second degree. It helped to remove some of the perceived difficulty and obscurity of Rene Descartes' work on analytic geometryArithmetica Infinitorum, the most important of Wallis's works, was published in 1656. In this treatise the methods of analysis of Descartes and Cavalieri were systematised and extended.  in 1659, Wallis published a tract containing the solution of the problems on the cycloid which had been proposed by Blaise Pascal. In this he incidentally explained how the principles laid down in his Arithmetica Infinitorum could be used for the rectification of algebraic curves; and gave a solution of the problem to rectify (i.e. find the length of) the semi-cubical parabola x3 = ay2, which had been discovered in 1657 by his pupil William Neile. Since all attempts to rectify the ellipse and hyperbola had been (necessarily) ineffectual, it had been supposed that no curves could be rectified, as indeed Descartes had definitely asserted to be the case. The logarithmic spiral had been rectified by Evangelista Torricelli, and was the first curved line (other than the circle) whose length was determined, but the extension by Neil and Wallis to an algebraic curve was novel. The cycloid was the next curve rectified; this was done by Wren in 1658.

Blaise Pascal (1623-1662): (French) He helped create two major new areas. He wrote a significant treatise on  projective geometry  at the age of sixteen.   Pascal's development of probability theory was his most influential contribution to mathematics, a subject on which he corresponded with Fermat.  Pascal continued to influence mathematics throughout his life. In 1653 he described a convenient tabular presentation for binomial coefficients, now called Pascal's triangle

Sir Isaac Newton (1643-1727):  (British) His theory of gravity unified the force  that keeps our feet on the ground, with the force that holds planets in their orbits. His 1687 publication of the Philosophiae Naturalis Principia Mathematica  is considered to be among the most influential books in the history of science.  In this work, Newton described universal gravitation and the three laws of motion. Newton shares the credit with Leibnitz for the development of differential and integral calculus. He also demonstrated the generalized binomial theorem and contributed to the study of power series.

Gottfried Wilhelm Leibnitz (1646-1716): (German) He invented infinitesimal calculus independently of Newton, and his notation has been in general use since then. He also invented the binary system, the foundation of virtually all modern computer architectures.

Bernoulli Family of Swiss Mathematicians: Three were : Jacob Bernoulli (1654-1705), his brother Johann Bernoulli (1667-1748) and Johann's son Daniel Bernoulli (1700-1787).

Jacob wrote the Art of Conjecture.  In this work, he described the known results in probability theory and in enumeration, often providing alternative proofs of known results. This work also includes the application of probability theory to games of chance and his introduction of the theorem known as the law of large numbers. The terms Bernoulli trial and Bernoulli numbers result from this work.  He.chose a figure of a logarithmic spiral  and the motto Eadem mutata resurgo ("Changed and yet the same, I rise again") for his gravestone. He called it the spiral mirabilis, the wonderful spiral. The spiral executed by the stonemasons was, however, an Archimedean spiral.  Just like a fractal, a logarithmic spiral is self similar:  That is, any smaller piece of a larger spiral is identical in shape to the larger piece.

Johann studied the function y = xx and he also investigated series using the method of integration by parts. Integration to Bernoulli was simply viewed as the inverse operation to differentiation and with this approach he had great success in integrating differential equations. He summed series, and discovered addition theorems for trigonometric and hyperbolic functions using the differential equations they satisfy. Johann was known as the "Archimedes of his age" and this is indeed inscribed on his tombstone.

Daniel was a Dutch Swiss mathematician. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics and for his pioneering work in probability and statistics. Bernoulli's work is still studied at length by many schools of science throughout the world. The Bernoulli Principle that was used to explain lift applicable to airplane wings was developed by Daniel Bernoulli.

Thomas Bayes (1702-1761):  An English mathematician, statistician and Presbyterian minister, known for having formulated a specific case of the theorem that bears his name.   Bayes never published what would become his most famous accomplishment.  His notes were edited and published after his death by Richard Price.   Bayes Theorem deals with "conditional probabilities".  http://plato.stanford.edu/entries/bayes-theorem/

There were prominent mathematicians that did not embrace the logic of Bayes Theorem .

ead Sharon Birtsch McGraynes book on "The Theory That Would Not Die" and see her on  Also her book is summarized on eview.html?pagewanted=all&_r=0

Leonhard Euler (1707-1783):  One of his many contributions was called "Euler's Formula".  The formula states that, for any real number  x$e^{ix} = \cos x + i\sin x \!$   where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions, with the argument x given in radians. The formula is still valid if x is a complex number.   Richard Feynman called Euler's formula "our jewel" and "one of the most remarkable, almost astounding, formulas in all of mathematics".

Euler Line

In the 18th century, the Swiss mathematician Leonhard Euler noticed that three of the  centers of a triangle are always collinear (they always lie on a straight line). The three centers that have this surprising property are the triangle's centroid (where the three medians of the triangle's sides meet), circumcenter (where the perpendicular bisectors of the triangle's sides meet) and the orthocenter (where the three altitudes to the vertices of the triangle meet).  The distance from the orthocenter to the centroid is two times the distance from the centroid to the circumcenter.  (Another center, the incenter, where the bisectors of the three angles meet, is not on this line.)

 Acute Triangle
 Obtuse Triangle
 Equilateral Triangle

Euler and the Nine Point Circle

The nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant concyclic points defined from the triangle. These nine points are:

 The midpoint of each side of the triangle The foot of each altitude The midpoint of the line segment from each vertex of the triangle to the orthocenter (where the three altitudes meet; these line segments lie on their respective altitudes).

The nine-point circle is also known as Feuerbach's circle, Euler's circle,  and Terquem's circle.

To construct the nine point circle of a triangle, see http://jwilson.coe.uga.edu/EMT668/EMT668.Folders.F97/Anderson/geometry/geometry1project/construction/construction.html

.1. Draw a triangle ABC and construct the midpoints of the three sides. Label them as L, M, N.

2. Construct the feet of the altitudes of the triangle ABC. Label them as D, E, F. Label the point of intersection of the three altitudes as H. This is also called the orthocenter.

3. Construct the midpoints of the segments AH, BH, CH. Label them as X, Y, Z.

4. Notice the nine points, L,M,N,D,E,F,X,Y, Z, lie in a circle called the Nine-Point Circle..

5. Construct the circumscribed circle for triangle LMN. Label the center of that circle U.

The center U of the circumscribed circle for triangle LMN will also be the center of the Nine-Point Circle.

More on Nine-Point Circle at http://en.wikipedia.org/wiki/Triangle_center where we learn  the following:

Let A, B, C denote the vertex angles of the reference triangle, and let x : y : z be a variable point in trilinear coordinates; then an equation for the Euler line is

$\sin 2A \sin(B - C)x + \sin 2B \sin(C - A)y + \sin 2C \sin(A - B)z = 0.\,$
 Center Trilinear Coordiates On Euler Line? Orttocenter Secant A: Secant B : Secant C Yes Centroid Cosecant A: Cosecant B: Cosecant C Yes Circumcenter Cosine A: Cosine B: Cosine C Yes Nine Point Circle Cosine (B - C): Cosine (C - A): Cosine (A - B) Yes In Center 1:1:1 Only if Isosceles

Euler and polyhedrons

A platonic solid is a regular, convex polyhedron with congruent faces of regular polygons and the same number of faces meeting at each vertex.  There are five regular polyhedrons that  meet those criteria, and each is named after its number of faces.:

 Tetrahedron Hexahedron Octahedron Dodecahedron Icosahedron 4 Triangles 4 Squares 8 Triangles 12 Pentagons 20 Triangles

Euler's formula for polyhedrons is:   V - E + F = 2  That is the number of vertices, minus the number of edges, plus the number of faces, is equal to two.

More on Euler

In analytical mathematics, Euler's identity (also known as Euler's equation) is the equality:

 eiπ + 1 = 0 e is Euler's number, the base of natural logarithms i is the imaginary unit, which satisfies i2 = −1 π is pi, the ratio of the circumference of a circle to its diameter

Euler's  number e is an important mathematical constant, approximately equal to 2.71828, that is the base of the natural logarithm.  It is the limit of (1 + 1/n)n as n becomes large, an expression that arises in the study of compound interest, and can also be calculated as the sum of the infinite series:

$e = \displaystyle\sum\limits_{n = 0}^{ \infty} \dfrac{1}{n!} = 1 + \frac{1}{1} + \frac{1}{1\cdot 2} + \frac{1}{1\cdot 2\cdot 3} + \cdots$

e is the unique real number such that the value of the derivative (slope of the tangent line) of the function f(x) = ex at the point x = 0 is equal to 1. The function ex so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base e.

The number e is of eminent importance in mathematics, alongside 0, 1, π and i. All five of these numbers play important and recurring roles across mathematics, and are the five constants appearing in one formulation of Euler's identity. Like the constant π, e is irrational: it is not a ratio of integers; and it is transcendental: it is not a root of any non-zero polynomial with rational coefficients. The numerical value of e truncated to 50 decimal places is 2.71828182845904523536028747135266249775724709369995...

The Euler–Mascheroni constant (also called Euler's constant) is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter $\gamma$(Gamma). It is defined as the limiting difference between the harmonic series and the natural logarithm:

$\gamma = \lim_{n \rightarrow \infty } \left( \sum_{k=1}^n \frac{1}{k} - \ln(n) \right)=\lim_{b \rightarrow \infty } \int_1^b\left({1\over\lfloor x\rfloor}-{1\over x}\right)\,dx.$

Here, $\lfloor x\rfloor$ represents the floor function. The numerical value of this constant, to 50 decimal places, is 0.57721566490153286060651209008240243104215933593992 ...  Euler established this formula in 1734. It is outrageous that we cannot decide if $\gamma$ is a rational number or not. Even though over 1,000,000 digits of this number have been calculated, it is not yet known if it is a rational number (the ratio of two integers a/b). But if it is rational, the denominator (b) must have more than 244,663 digits!

The sum of the reciprocals of all integers: 1/1 +1/2 + 1/3+ 1/4 + 1/5 + 1/6+ 1/7  etc equals Ln(N) + Gamma                                          The sum of the reciprocals of the odd integers: 1 + 1/3 + 1/5 + 1/7 etc equals  {Ln(N) + Ln(2) + Gamma}/2                                           The sum of the reciprocals of the even integers:  1/2 +1/4+1/6+1/8 etc equals {Ln(N) -Ln(2) - Gamma}/2

The Basel problem is a famous problem in mathematical analysis with relevance to number theory, first posed by Pietro Mengoli in 1644 and solved by Leonhard Euler in 1735. Since the problem had withstood the attacks of the leading mathematicians of the day, Euler's solution brought him immediate fame when he was twenty-eight. Euler generalised the problem considerably, and his ideas were taken up years later by Bernhard Riemann in his seminal 1859 paper On the Number of Primes Less Than a Given Magnitude, in which he defined his zeta function and proved its basic properties. The problem is named after Basel, hometown of Euler as well as of the Bernoulli family who unsuccessfully attacked the problem.

The Basel problem asks for the precise summation of the reciprocals of the squares of the natural numbers, i.e. the precise sum of the infinite series:

$\sum_{n=1}^\infin \frac{1}{n^2} = \lim_{n \to +\infty}\left(\frac{1}{1^2} + \frac{1}{2^2} + \cdots + \frac{1}{n^2}\right).$

The series is approximately equal to 1.644934. The Basel problem asks for the exact sum of this series (in closed form), as well as a proof that this sum is correct. Euler found the exact sum to be π2/6 and announced this discovery in 1735. His arguments were based on manipulations that were not justified at the time, and it was not until 1741 that he was able to produce a truly rigorous proof.

Joseph-Louis Lagrange (1736-1813): (Italian) Lagrange was one of the creators of the calculus of variations, deriving the Euler Lagrange equations. Lagrange invented the method of solving differential equations known as variation of parameters, applied differential calculus to the theory of probabilities and attained notable work on the solution of equations. He proved that every natural number is a sum of four squares. His treatise Theorie des fonctions analytiques laid some of the foundations of group theory, anticipating Galois. In calculus, Lagrange developed a novel approach to interpolation and Taylor series. He studied the three-body problem for the Earth, Sun, and Moon and the movement of Jupiter's satellites. In 1772 found the special-case solutions to this problem that are now known as Lagrangian points. He transformed Newtonian mechanics into a branch of analysis, Lagrangian mechanics as it is now called. One of Lagrange's more famous books is the Analytical Mechanics, which, he boasted proudly, contains no pictures.

Caspar Wessel (1745-1818): (Danish-Norwegian)  Wessel was a mathematician who was born in Norway. In 1763, having completed secondary school, he went to Denmark for further studies (Norway having no university at  the time). In 1778 he acquired the degree of candidatus juris. From 1794, however, he was employed as at ttp://www.maa.org/sites/default/files/pdf/upload_library/22/Chauvenet/mccartin.pdf surveyor (from 1798 as Royal inspector of Surveying).

It was the mathematical aspect of surveying that led him to exploring the geometrical significance of complex numbers. His fundamental paper, Om directionens analytiske betegning, was published in 1799 by the Royal Danish Academy of Sciences and Letters. Since it was in Danish, it passed almost unnoticed, and the same results were later independently found by Argand and Gauss.

One of the more prominent ideas presented in "On the Analytical Representation of Direction" was that of vectors. Even though this wasn't Wessel's main intention with the publication, he felt that a geometrical concept of numbers, with length and direction, was needed. Wessel's approach on addition was: "Two straight lines are added if we unite them in such a way that the second line begins where the first one ends and then pass a straight line from the first to the last point of the united lines. This line is the sum of the united lines". This is the same idea as used today when summing vectors.

Wessel's priority to the idea of a complex number as a point in the complex plane is today universally recognized. His paper was re-issued in French translation in 1899, and in English in 1999 as "On the analytic representation of direction".

Pierre-Simon, marquis de Laplace (1749-1827):  (French) He is remembered as one of the greatest scientists of all time, sometimes referred to as a French Newton or Newton of France, with a phenomenal natural mathematical faculty superior to any of his contemporaries.  Laplace's writing of Celestial Mechanics, an enormous, five volume tome of celestial mechanics, established him as the Prince of Celestial Mechanicians. When presented with a copy of some of the initial volumes, Napoleon is said to have remarked, "I see no mention of God in this work". Laplace is said to have replied, "Sir, I have no need of that hypothesis." (In an addition to the story, the tale was related to Lagrange, who added "Ah, but it is such a beautiful hypothesis; it explains a great many things!"

Jean Baptiste Joseph Fourier (1768-1830): (French)  A mathematician and physicist  best known for initiating the investigation of Fourier series and their applications to problems of heat transfer and vibrations. The Fourier transform and Fourier's Law are also named in his honour. Fourier is also generally credited with the discovery of the greenhouse effect.

Marie-Sophie Germain (1776-1831):  (French) A French mathematician, physicist, and philosopher. Despite initial opposition from her parents and difficulties presented by society, she gained education from books in her father's library including ones by Leonhard Euler and from correspondence with famous mathematicians such as Lagrange, Legendre, and Gauss. One of the pioneers of elasticity theory, she won the grand prize from the Paris Academy of Sciences for her essay on the subject. Her work on Fermat's Last Theorem provided a foundation for mathematicians exploring the subject for hundreds of years after.  Because of prejudice against her sex, she was unable to make a career out of mathematics, but she worked independently throughout her life.   At the centenary of her life, a street and a girls' school were named after her. The Academy of Sciences established The Sophie Germain Prize in her honor.

Carl Friedrich Gauss (1777-1855): (German) Called the Prince of Mathematicians and the greatest mathematician since antiquity. He is ranked as one of history's most influential mathematicians. He referred to mathematics as the Queen of Sciences.  Gauss proved the Fundamental Theorem of Algebra. Gauss claimed to have discovered the possibility of non Euclidean Geometries but never published it.

Simeon Denis Poisson (1781-1840): (French)  A mathematician, geometer, and physicist.  In probability theory and statistics, the Poisson distribution (or Poisson law of small numbers) is a discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate and independently of the time since the last event. (The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.)

Baron Augustin-Louis Cauchy (1789-1857): ( French) He was an early pioneer of analysis. He started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner. He also gave several important theorems in complex analysis and initiated the study of permutation groups in abstract algebra. A profound mathematician, Cauchy exercised a great influence over his contemporaries and successors. His writings cover the entire range of mathematics and mathematical physics.

Michael Faraday (1791-1867) and James Clerk Maxwell (1831-1879):  They proved that electric and magnetic forces are the same force in different guises.

Nikolai Ivanovich Lobachevsky (Никола́й Ива́нович Лобаче́вский) (1792-1856):  (Russian)  A mathematician and geometer, renowned primarily for his pioneering works on hyperbolic geometry.

Niels Henrik Abel (1802-1829):  (Norwegian)  At the age of 16, Abel gave a proof of the binomial theorem valid for all numbers, extending Euler's result which had only held for rational numbers. At age 19, he showed there is no general algebraic solution for the roots of a quintic equation, or any general polynomial equation of degree greater than four, in terms of explicit algebraic operations. To do this, he invented (independently of Galois) an extremely important branch of mathematics known as group theory, which is invaluable not only in many areas of mathematics, but for much of physics as well. Among his other accomplishments, Abel wrote a monumental work on elliptic functions which, however, was not discovered until after his death. When asked how he developed his mathematical abilities so rapidly, he replied "by studying the masters, not their pupils."

Janos Bolyai (1802-1860): (Hungarian) He was known for his work in non-Euclidean geometryBetween 1820 and 1823 he prepared a treatise on a complete system of non-Euclidean geometry. Bolyai's work was published in 1832 as an appendix to a mathematics textbook by his father.

Gauss, on reading the Appendix, wrote to a friend saying "I regard this young geometer Bolyai as a genius of the first order". In 1848 Bolyai discovered not only that Lobachevsky had published a similar piece of work in 1829, but also a generalization of this theory. As far as is known, Lobachevsky published his work a few years earlier than Bolyai, but it contained only hyperbolic geometry. Bolyai and Lobachevsky did not know each other or each other's works. In addition to his work in the geometry, Bolyai developed a rigorous geometric concept of complex numbers as ordered pairs of real numbers. Although he never published more than the 24 pages of the Appendix, he left more than 20,000 pages of mathematical manuscripts when he died.

Carl Gustav Jacob Jacobi (1804-1851) (German) A mathematician, widely considered to be the most inspiring teacher of his time and one of the greatest mathematicians of all time.  One of Jacobi's greatest accomplishments was his theory of elliptic functionsHe also made fundamental contributions in the study of differential equations.  It was in algebraic development that Jacobi's peculiar power mainly lay, and he made important contributions of this kind to many areas of mathematics, as shown by his long list of papers in Crelle's Journal and elsewhere from 1826 onwards. One of his maxims was: 'Invert, always invert' ('man muss immer umkehren'), expressing his belief that the solution of many hard problems can be clarified by re-expressing them in inverse form.  He was also one of the early founders of the theory of determinants.

Johann Peter Gustav Lejeune Dirichlet (1805-1859): (German) He was credited with the modern formal definition of a functionDirichlet's brain is preserved in the anatomical collection of the University of Gottingen, along with the brain of Gauss.

Sir William Rowan Hamilton (1805-1865): (Irish) A  physicist, astronomer, and mathematician, who made important contributions to classical mechanics, optics, and algebra. His studies of mechanical and optical systems led him to discover new mathematical concepts and techniques. His greatest contribution is perhaps the reformulation of Newtonian mechanics, now called Hamiltonian mechanics. This work has proven central to the modern study of classical field theories such as electromagnetism, and to the development of quantum mechanics. In mathematics, he is perhaps best known as the inventor of quaternionsA striking feature of quaternions is that the product of two quaternions is noncommutative, meaning that the product of two quaternions depends on which factor is to the left of the multiplication sign and which factor is to the right. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space or equivalently as the quotient of two vectors. It can also be represented as the sum of a scalar and a vector. http://en.wikipedia.org/wiki/Quaternion  In four-dimensioal space the tesseract, or hype, is the four-dimensional analog of the cube. The tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8 cubical cells. The tesseract is one of the six convex regular 4-polytopes.  A generalization of the cube to dimensions greater than three is called a "hypercube", "n-cube" or "measure polytope". The tesseract is the four-dimensional hypercube'  See http://en.wikipedia.org/wiki/Fourth_dimension

Joseph Liouville (1809-1882): (French) Liouville worked in a number of different fields in mathematics, including number theory, complex analysis, differential geometry and topology, but also mathematical physics and even astronomy. He is remembered particularly for Liouville's theorem, a nowadays rather basic result in complex analysis. In number theory, he was the first to prove the existence of transcendental numbers in 1844 by a construction using continued fractions (Liouville numbers). In mathematical physics, Liouville made two fundamental contributions: the Sturm Liouville theory, which was joint work with Charles Francois Sturm, and is now a standard procedure to solve certain types of integral equations by developing into eigenfunctions, and the fact (also known as Liouville's theorem) that time evolution is measure preserving for a Hamiltonian system. In Hamiltonian dynamics, Liouville also introduced the notion of action-angle variables as a description of completely integrable systems. The modern formulation of this is sometimes called the Liouville-Arnold theorem, and the underlying concept of integrability is referred to as Liouville integrability.

The following number is known as Liouville's constant.  (The exponent is negative j factorial.)

Liouville's constant is a decimal fraction with  1"s and 0"s in each decimal place. In1844 he constructed an infinite class of transcendental numbers using continued fractions, but the above number was the first decimal constant to be proven  by Liouville in 1850 to be  transcendental.  Cantor subsequently proved that "almost all" real numbers are in fact transcendental.

The crater Liouville on the Moon is named after him. So is the Liouville function, an important function in number theory. See http://mathworld.wolfram.com/LiouvilleFunction.html

Ernst Eduard Kummer (1810 – 1893): (German)  Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a gymnasium, the German equivalent of high school, where he inspired the mathematical career of Leopold Kronecker. Kummer made several contributions to mathematics in different areas; he codified some of the relations between different hypergeometric series, known as contiguity relations. The Kummer surface results from taking the quotient of a two-dimensional abelian variety by the cyclic group {1, −1} (an early orbifold: it has 16 singular points, and its geometry was intensively studied in the nineteenth century). Kummer also proved Fermat's last theorem for a considerable class of prime exponents (see regular primeideal class group). His methods were closer, perhaps, to p-adic ones than to ideal theory as understood later, though the term 'ideal' arose here. He studied what were later called Kummer extensions of fields: that is, extensions generated by adjoining an nth root to a field already containing a primitive nth root of unity. This is a significant extension of the theory of quadratic extensions, and the genus theory of quadratic forms (linked to the 2-torsion of the class group). As such, it is still foundational for class field theory.

Evariste Galois (1811-1832):  A symmetry of an object is what you can do to an object to leave it essentially looking like it did before you touched it.  Galois was interested in the collection of all symmetries and seeing what happens if you do one symmetry after another.  He discovered that it is the interactions between the symmetries in a group that encapsulate the essential qualities of the symmetry of an object.  Mathematicians struggled for centuries to find formulas for the solutions of equations of higher degree, but despite the efforts of Euler, Bezout, Malfatti, Lagrange, and others, no general solutions were found. Finally, Ruffini (1799) and Abel (1826) showed that the solution of the general quintic cannot be written as a finite formula involving only the four arithmetic operations and the extraction of roots. Galois developed the theory of Galois groups and described exactly when a polynomial equation is solvable.

Karl Theodor Wilhelm Weierstrass (1815-1897): (German) He is often cited as the "father of modern analysis".

George Boole (1815-1864): (English) A mathematician and philosopher. As the inventor of Boolean logic, the basis of modern digital computer logic, Boole is regarded in hindsight as a founder of the field of computer science. Boole said:  " ... no general method for the solution of questions in the theory of probabilities can be established which does not explicitly recognise ... those universal laws of thought which are the basis of all reasoning".

Arthur Cayley (1821-1895): (British) He helped found the modern British school of pure mathematics. He proved the Cayley-Hamilton theorem: that every square matrix is a root of its own characteristic polynomial. He was the first to define the concept of a group in the modern way: as a set with a binary operation satisfying certain laws. Formerly, when mathematicians spoke of "groups", they had meant permutation groups.

Charles Hermite (1822-1901): (French) He did research on number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra Hermite polynomials, Hermite interpolation, Hermite normal form, Hermitian operators, and cubic Hermite splines are named in his honor. One of his students was Henri Poincare In 1873, Hermite was the first to prove that e, the base of natural logarithms, is a transcendental number. His methods were later used by Ferdinand von Lindemann  to prove in 1882 π is transcendental.  In a letter to Thomas Stieltjes in 1893, Hermite famously remarked: "I turn with terror and horror from this lamentable scourge of continuous functions with no derivatives."       See http://www.pi314.net/eng/lindemann.php

Leopold Kronecker (1823 – 1891): (German) He worked on number theory and algebra. He criticized Cantor's work on set theory, and was quoted by Weber (1893) as having said, "Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk" ("God made the integers, all else is the work of man."). Kronecker was a student and lifelong friend of Ernst Kummer.  For several years Kronecker focused on business, and although he continued to study mathematics as a hobby and corresponded with Kummer, he published no mathematical results. For several years Kronecker focused on business, and although he continued to study mathematics as a hobby and corresponded with Kummer, he published no mathematical results. In 1853 he wrote a memoir on the algebraic solvability of equations extending the work of Évariste Galois on the theory of equations.

Bernhard Riemann (1826-1866):  If the "Riemann Hypothesis" is true,  the exact number of primes less than a given number N, or  Pi(N), can be calculated exactly.   Although thought to be correct, this hypothesis is unproven.    Karl Friedrich Gauss (1777-1855) had an approximation to Pi(N), equal to N/ln(N), where ln is the natural logarithm.   Adrien-Marie Legendre (1752-1833) improved on Gauss's estimate using  Pi(N) = N/{ln(N) - 1.08366}   Gauss then improved upon that estimate using Li(N) , which he called the logarithmic integral. (not shown here)  Leonard Euler (1707-1783) showed that the Riemann Zeta Function: Z(s) =  The sum of 1/n raised to the s power for n = 1 to infinity, is also equal to a product series involving primes.  Z(s) = The product of (1 + 1/p to the s + 1/p to the 2s + 1/p to the 3s  + 1/p to the 4s + 1/p to the 5s +...) over all primes.  It is important  to note: "s" is a "complex number". Riemann then hypothesized that Z(s) = 0 for only complex numbers where the real part = 1/2.  The Riemann Hypothesis has not been proven, but computers have shown the first 6.3 billion zeros all lie on the line s = 1/2 +ki.  If the Riemann Hypothesis is correct, then Riemann has a formula for calculating Pi(N) exactly!    Pi(N) =  R(N) minus an Adjustment.  R(N) is a formula involving the logarithmic integral and the Adjustment is expressed in terms of the zeros of the Zeta Function. The function R(N) was named in honor of Riemann.

The Riemann zeta function ζ(s) is a function of a complex variable s = σ + it. (The notation with s, σ, and t is traditionally used in the study of the ζ-function)

The following infinite series converges for all complex numbers s with real part greater than 1, and defines ζ(s) in this case:

$\zeta(s) = \sum_{n=1}^\infty n^{-s} = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \cdots \;\;\;\;\;\;\; \sigma = \mathfrak{R}(s) > 1. \!$

The Riemann zeta function is defined as the analytic continuation of the function defined for σ > 1 by the sum of the preceding series.  When s =2 , this function equals

Riemann's hypothesis in 1859 is a deep mathematical conjecture which states that the nontrivial Riemann zeta function zeros, i.e., the values of other than , , , ... such that (where is the Riemann zeta function) all lie on the "critical line" (where denotes the real part of ).  It has never been proved or disproved.

 Riemann zeta function ζ(s) in the complex plane. The color of a point s encodes the value of ζ(s): colors close to black denote values close to zero, while hue encodes the value's argument. The white spot at s = 1 is the pole of the zeta function; the black spots on the negative real axis and on the critical line Re(s) = 1/2 are its zeros. Values with arguments close to zero including positive reals on the real half-line are presented in red.

Julius Wilhelm Richard Dedekind (1831-1916): ( German) He did important work in abstract algebra (particularly ring theory), algebraic number theory and the foundations of the real numbers.

James Clerk Maxwell (1831-1879): (Scottish) A physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and even optics into a consistent theory. Maxwell's equations demonstrated that electricity, magnetism and even light are all manifestations of the same phenomenon, namely the electromagnetic field. Subsequently, all other classic laws or equations of these disciplines became simplified cases of Maxwell's equations. Maxwell's achievements concerning electromagnetism have been called the "second great unification in physics", after the first one realized by Isaac Newton.

Maxwell demonstrated that electric and magnetic fields travel through space in the form of waves, and at the constant speed of light. In 1864 Maxwell wrote A Dynamical Theory of the Electromagnetic Field. It was with this that he first proposed that light was in fact undulations in the same medium that is the cause of electric and magnetic phenomena. His work in producing a unified model of electromagnetism is one of the greatest advances in physics.

Samuel Loyd (1841-1911): (American)  He was an American chess player, chess composer, puzzle author, and recreational mathematician

As a chess composer, he authored a number of chess problems, often with interesting themes. At his peak, Loyd was one of the best chess players in the US, and was ranked 15th in the world, according to chessmetrics.com.  Loyd was inducted into the US Chess Hall of Fame.

Following his death, his book Cyclopedia of 5000 Puzzles was published (1914) by his son.

Loyd is widely acknowledged as one of America's great puzzle-writers and popularizers, often mentioned as the greatest—Martin Gardner called him "America's greatest puzzler", and The Strand in 1898 dubbed him "the prince of puzzlers". As a chess problemist, his composing style is distinguished by wit and humour.

However, he is also known for lies and self-promotion, and criticized on these grounds—Martin Gardner's assessment continues "but also obviously a hustler". Canadian puzzler Mel Stover called Loyd "an old reprobate", and Matthew Costello called him "puzzledom's greatest celebrity ... popularizer, genius", but also "huckster ... and fast-talking snake oil salesman".

He collaborated with puzzler Henry Dudeney for a while, but Dudeney broke off the correspondence and accused Loyd of stealing his puzzles and publishing them under his own name. Dudeney despised Loyd so intensely he equated him with the Devil.

Georg Ferdinand Ludwig Philipp Cantor (1845-1918): (German) He  is best known as the inventor of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between sets, defined infinite and well-ordered sets, and proved that the real numbers are "more numerous" than the natural numbers. In fact, Cantor's theorem implies the existence of an "infinity of infinities". He defined the cardinal and ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact of which he was well aware.  Cantor's theory of transfinite numbers was originally regarded as so counter-intuitive, even shocking, that it encountered resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincare and later from Hermann Weyl and L. E. J. Brouwer

A transcendental number is a (possibly complex) number that is not algebraic—that is, it is not a root of a non-zero polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e. Though only a few classes of transcendental numbers are known (in part because it can be extremely difficult to show that a given number is transcendental), transcendental numbers are not rare. Indeed, almost all real and complex numbers are transcendental, since the algebraic numbers are countable while the sets of real and complex numbers are both uncountable. All real transcendental numbers are irrational, since all rational numbers are algebraic. The converse is not true: not all irrational numbers are transcendental; e.g., the square root of 2 is irrational but not a transcendental number, since it is a solution of the polynomial equation x2 − 2 = 0.

Seth Carlo Chandler, Jr. (1846-1913): (American astronomer and actuary).

He was born in Boston, Massachusetts to Seth Carlo and Mary (née Cheever) Chandler.During his last year in high school he performed mathematical computations for Benjamin Peirce, of the Harvard College Observatory.

After graduating, he became the assistant of Benjamin A. Gould. Gould was director of the Longitude Department of the U.S. Coast Survey program, a geodetic survey program. When Gould left to become director of the national observatory in Argentina, Chandler also left and became an actuary for Continental Life in NY City.  However, he continued to work in astronomy as an amateur affiliated with Harvard College Observatory.

In 1876 Chandler moved his family to Boston where he continued his actuarial work as a consultant to Union Mutual Life Insurance Company.  In 1880 he renewed his association with the Harvard College Observatory.  In 1881 he moved to Cambridge.  He mounted  a telescope to the cupola atop the roof and carried on the duties of Associate Editor of the Astronomical Journal while B. A. Gould was editor.  He later became editor and used his own funds to publish this journal in difficult financial times.

Chandler is best known for his discovery (1884–85) of the Chandler Wobble, a movement in Earth’s axis of rotation that causes latitude to vary with a period of about 433 days. A wandering of the rotation axis had been predicted by Swiss mathematician Leonhard Euler in 1765. Chandler’s detection of this effect was facilitated by his invention of the almucantar, a device for measuring the positions of stars relative to a circle centered at the zenith rather than to the meridian. The North Pole of Earth’s rotation axis wanders in an irregular, quasi-circular path with a radius of about 8–10 metres (26–33 feet).

From 1896 to 1909 he edited The Astronomical Journal.

Chandler also made contributions to other areas of astronomy, including variable stars. He independently co-discovered the nova T Coronae Borealis, improved the estimate of the constant of aberration, and computed the orbital parameters of asteroids and comets.

Chandler was awarded the Gold Medal of the Royal Astronomical Society in 1896 and the James Craig Watson Medal in 1894.  The crater Chandler on the Moon is named after him.

See more on the Chandler Wobble: http://curiosity.discovery.com/question/what-is-the-chandler-wobble

Thomas Alva Edison (1847-1931): (American)  An  inventor, scientist, and businessman who developed many devices that greatly influenced life around the world, including the phonograph, the motion picture camera, and a long-lasting, practical electric light bulb. Dubbed "The Wizard of Menlo Park" . He was born in Milan, Ohio.  His quotations include: "There's a way to do it better - find it!"   "Genius is one percent inspiration and ninety-nine percent perspiration."   "I have not failed. I've just found 10,000 ways that won't work."  "I never did a day's work in my life. It was all fun."   Also: "We will make electricity so cheap that only the rich will burn candles." Edison became the owner of his Milan, Ohio, birthplace in 1906. On his last visit, in 1923, he was shocked to find his old home still lit by lamps and candles. Al Bolisha, a Canadian humorist, said, "Do you realize if it weren't for Edison, we'd be watching TV by candelight?"

Prime Number Theorem states that if you select a large number N, the probability of  it being prime is about 1/Ln(N) was solved independently in 1896 by Jacques-Solomon Hadamard (1865-1963) and Charles de la Vallee Poisson (1866-1962) by showing that the Riemann Zeta Function has no zeros of the form (1 + ki)

Carl Louis Ferdinand von Lindemann (1852-1939): (German) He was a  noted for his proof, published in 1882, that π (pi) is a transcendental number, i.e., it is not a zero of any polynomial with rational coefficients.

Jules Henri Poincare (1854-1912): (French) A mathematician,  theoretical physicist, and a philosopher of science.  Poincare is often described  in mathematics as The Last Universalist, since he excelled in all fields of the discipline as it existed during his lifetime.

George Eastman (1854-1932):  (American) An  inventor and philanthropist. He founded the Eastman Kodak Company and invented roll film    In his final two years, Eastman was in intense pain, caused by a degenerative disorder affecting his spine. He had trouble standing and his walking became a slow shuffle. Today it might be diagnosed as lumbar spinal stenosis, a narrowing of the spinal canal caused by calcification in the vertebrae. Eastman grew depressed, as he had seen his mother spend the last two years of her life in a wheelchair from the same condition. On March 14, 1932, Eastman died by suicide with a single gunshot to the heart, leaving a note which read, "My work is done.  Why wait?"

 The logo from 1987 to 2006.  The letter "K" had been a favorite of Eastman's, he is quoted as saying, "it seems a strong, incisive sort of letter." He and his mother devised the name Kodak with an anagram set. He said that there were three principal concepts he used in creating the name: it should be short, one cannot mispronounce it, and it could not resemble anything or be associated with anything but Kodak.

Andrey (Andrei) Andreyevich Markov (Андрей Андреевич Марков) (1856-1922): (Russian)  He is best known for his work on theory of stochastic processes. His research later became known as Markov chains.

Henry Ernest Dudeney (1857-1930):  (English) An author and mathematician who specialized in logic puzzles and mathematical games. He is known as one of the foremost creators of puzzles.

Max Karl Ernst Ludwig Planck (1858-1947): (German)  A physicist who is regarded as the founder of the quantum theory, for which he received the Nobel Prize in Physics in 1918.

David Hilbert (1862-1943): (German) He was recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. He discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He also formulated the theory of Hilbert spaces, one of the foundations of functional analysis.

Bertrand Russell (1872-1970):  "Physics is mathematical not because we know so much about the physical world, but because we know so little; it is only its mathematical properties we can discover."

Pierre Joseph Louis Fatou (1878 – 1929):  (French): He was a French mathematician and astronomer. He is known for major contributions to several branches of analysis. The Fatou lemma and the Fatou set are named after him.

Albert Einstein (1879-1955):  The speed of light is the same, irrespective of how the source of light or the observer is moving.  Furthermore, space and time cannot be treated as separate entities, rather they are inseparably tethered together by symmetry.  One of the known results of special relativity is that the length of moving bodies, as measured by observers at rest, contracts along their direction of motion.  The contraction is larger,  the higher the speed. Gravity warps and bends spacetime.  One of the key predictions of general relativity was the bending of light rays under the influence of gravity. Guided by principles of symmetry Einstein showed that acceleration and gravity are two sides of the same coin.(If a train is moving very fast to the north and a man in a boxcar drops his keys, they fall to the south.)(If a man in a stationary box car drops his keys, the keys would fall to the south, if gravity was tilted to the south.)

Amalie Emmy Noether (1882-1935): (German) A German mathematician known for her groundbreaking contributions to abstract algebra and theoretical physics. Described by Pavel Alexandrov, Albert Einstein, Jean Dieudonné, Hermann Weyl, Norbert Wiener and others as the most important woman in the history of mathematics, she revolutionized the theories of rings, fields, and algebras. In physics, Noether's theorem explains the fundamental connection between symmetry and conservation laws.

Max Born (1882-1970): (German)  physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a number of notable physicists in the 1920s and 30s. Born won the 1954 Nobel Prize in Physics, shared with Walther Bothe.

Frank Albert Benford, Jr. (1883-1948): (American)  Benford's law, also called the first-digit law, states that in lists of numbers from many (but not all) real-life sources of data, the leading digit is distributed in a specific, non-uniform way. According to this law, the first digit is 1 almost one third of the time, and larger digits occur as the leading digit with lower and lower frequency, to the point where 9 as a first digit occurs less than one time in twenty. This distribution of first digits arises logically whenever a set of values is distributed logarithmically. Measurements of real world values are often distributed logarithmically (or equivalently, the logarithm of the measurements is distributed uniformly). This counter-intuitive result has been found to apply to a wide variety of data sets, including electricity bills, street addresses, stock prices, population numbers, death rates, lengths of rivers, physical and mathematical constants, and processes described by power laws (which are very common in nature). The result holds regardless of the base in which the numbers are expressed, although the exact proportions change. It is named after physicist Frank Benford, who stated it in 1938, although it had been previously stated by Simon Newcomb in 1881.

John Maynard Keynes  (1883-1946):  British Economist and Mathematician.  On the Law of Large Numbers or "long run calculations", he said: "This long run is a misleading guide to current affairs.  In  the long run we are all dead.  Economists set themselves too easy, too useless a task if in tempestuous seasons they can only tell us that when the storm is long past the ocean is flat again."  Other quotes:  "I do not know which makes a man more conservative - to know nothing but the present, or nothing but the past."  " It would be foolish, in forming our expectations, to attach great weight to matters which are very uncertain."  " It is generally agreed that casinos should, in the public interest, be inaccessible and expensive. And perhaps the same is true of Stock Exchanges."  "The outstanding faults of the economic society in which we live are its failure to provide for full employment and its arbitrary and inequitable distribution of wealth and incomes."

Niels Henrik David Bohr (1885-1962):  (Danish) A physicist who made fundamental contributions to understanding atomic structure and quantum mechanics, for which he received the Nobel Prize in Physics in 1922. Bohr mentored and collaborated with many of the top physicists of the century at his institute in Copenhagen. He was part of a team of physicists working on the Manhattan Project. Bohr married Margrethe Norlund in 1912, and one of their sons, Aage Bohr, grew up to be an important physicist who in 1975 also received the Nobel prize. Bohr has been described as one of the most influential scientists of the 20th century.

Srīnivāsa Aiyangār Rāmānujan (1887-1920):  (Indian) He was a self taught genius, with almost no formal training in pure mathematics, made substantial contributions to mathematical analysis, number theory, infinite series and continued fractions Ramanujan's talent was said, by the prominent English mathematician G.H. Hardy, to be in the same league as legendary mathematicians such as Euler, Gauss, Newton and Archimedes.

Gaston Maurice Julia (1893 – 1978):  (French) He was a French mathematician who devised the formula for the Julia set. His works were popularized by French mathematician Benoit Mandelbrot; the Julia and Mandelbrot fractals are closely related.

Enrico Fermi (1901-1954): (Italian-American) A physicist particularly known for his work on the development of the first nuclear reactor, Chicago Pile-1, and for his contributions to the development of quantum theory, nuclear and particle physics, and statistical mechanics. He was awarded the 1938 Nobel Prize in Physics for his work on induced radioactivity.

Fermi is widely regarded as one of the leading scientists of the 20th century, highly accomplished in both theory and experiment.  Along with J. Robert Oppenheimer, he is frequently referred to as "the father of the atomic bomb". He also held several patents related to the use of nuclear power.

Andre John von Neumann (1903-1957): (Hungarian-born) An  American mathematician who made major contributions to a vast range of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, continuous geometry, economics and game theory, computer science, numerical analysis, hydrodynamics (of explosions), and statistics, as well as many other mathematical fields. He is generally regarded as one of the greatest mathematicians in modern history. The mathematician Jean Dieudonne called von Neumann "the last of the great mathematicians",while Peter Lax described him as possessing the most "fearsome technical prowess" and "scintillating intellect" of the century.Weil (1906-1998):  (from France) "God exists since mathematics is consistent, and the Devil exists since we cannot prove it."

Julius Robert Oppenheimer (1904-1967): (American) A theoretical physicist and professor of physics at the University of California, Berkeley. He is often called the "father of the atomic bomb" for his role as the scientific director of the Manhattan Project, the World War II project that developed the first nuclear weapons. The first atomic bomb was detonated in July 1945 in the Trinity test in New Mexico; Oppenheimer remarked later that it brought to mind words from the Bhagavad Gita: "Now, I am become Death, the destroyer of worlds."

After the war he became a chief adviser to the newly created United States Atomic Energy Commission and used that position to lobby for international control of nuclear power to avert nuclear proliferation and an arms race with the Soviet Union. After provoking the ire of many politicians with his outspoken opinions during the Second Red Scare, he had his security clearance revoked in a much-publicized hearing in 1954. Though stripped of his direct political influence he continued to lecture, write and work in physics. A decade later President John F. Kennedy awarded (and Lyndon B. Johnson presented) him with the Enrico Fermi Award as a gesture of political rehabilitation.

Oppenheimer's notable achievements in physics include the Born Oppenheimer approximation for molecular wavefunctions, work on the theory of electrons and positrons, the Oppenheimer Phillips process in nuclear fusion, and the first prediction of quantum tunneling. With his students he also made important contributions to the modern theory of neutron stars and black holes, as well as to quantum mechanics, quantum field theory, and the interactions of cosmic rays. As a teacher and promoter of science, he is remembered as a founding father of the American school of theoretical physics that gained world prominence in the 1930s. After World War II, he became director of the Institute for Advanced Study in Princeton.

Kurt Goedel (1906-1978):  (Austrian  and later American)  A logician, mathematician, and philosopher. Considered with Aristotle and Gottlob Frege to be one of the most significant logicians in history) Goedel's Incompleteness Theorem: Any consistent axiom system is necessarily incomplete in that there will be true statements that can't be deduced from the axioms.

Gödel published his two incompleteness theorems in 1931 when he was 25 years old, one year after finishing his doctorate at the University of Vienna. The first incompleteness theorem states that for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (for example Peano arithmetic), there are true propositions about the naturals that cannot be proved from the axioms. To prove this theorem, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers.

He also showed that neither the axiom of choice nor the continuum hypothesis can be disproved from the accepted axioms of set theory, assuming these axioms are consistent. The former result opened the door for mathematicians to assume the axiom of choice in their proofs. He also made important contributions to proof theory by clarifying the connections between classical logic, intuitionistic logic, and modal logic.

Olga Taussky-Todd (1906 – 1995):  (Austrian and later Czech-American)  She was born into a Jewish family. She worked first in algebraic number theory, with a doctorate at the University of Vienna supervised by Philipp Furtwängler. During that time in Vienna she also attended the meetings of the Vienna Circle.

According to Gian-Carlo Rota, as a young mathematician she was hired by a group of German mathematicians to find and correct the many mathematical errors in the works of David Hilbert, so that they could be collected into a volume to be presented to him on his birthday. There was only one paper, on the continuum hypothesis, that she was unable to repair.

Later, she started to use matrices to analyze vibrations of airplanes during World War II, at the National Physical Laboratory in the United Kingdom. She became the torchbearer for matrix theory. In 1935, she moved to England and became a Fellow at Girton College, Cambridge University, as well as at Bryn Mawr College. In 1938 she married the British mathematician John Todd (1911-2007), a colleague at the University of London.  In 1945 the Todds emigrated to the United States and worked for the National Bureau of Standards. In 1957 they joined the faculty of California Institute of Technology (Caltech) in Pasadena, California.

Andre Weil (1906-1998): (French) He was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry

James R Newman (1907-1966):  (American) Von Neumann's first significant contribution to economics was the minimax theorem in1928.  He eventually improved and extended the minimax theorem to include games involving imperfect information and games with more than two players. This work culminated in the 1944 classic Theory of Games and Economic Behavior. Von Neumann was one of the pioneers of computer science making significant contributions to the development of logical design. "The Theory of Groups is a branch of mathematics in which one does something to something  and  then compares the result with the result obtained from doing the same thing to something else, or something else to the same thing."

Edward Teller (1908-2003):  (Hungarian-born American) A theoretical physicist, known colloquially as "the father of the hydrogen bomb," even though he did not care for the title. In 1942, Teller was invited to be part of Robert Oppenheimer's summer planning seminar at the University of California, Berkeley for the origins of the Manhattan Project, the Allied effort to develop the first nuclear weapons. A few weeks earlier, Teller had been meeting with his friend and colleague Enrico Fermi about the prospects of atomic warfare, and Fermi had nonchalantly suggested that perhaps a weapon based on nuclear fission could be used to set off an even larger nuclear fusion reaction. Even though he initially explained to Fermi why he thought the idea would not work, Teller was fascinated by the possibility and was quickly bored with the idea of "just" an atomic bomb (even though this was not yet anywhere near completion). At the Berkeley session, Teller diverted discussion from the fission weapon to the possibility of a fusion weapon, what he called the "Super" (an early version of what was later known as a hydrogen bomb).

Stanislaw Marcin Ulam (1909-1984): (Polish-Jewish) He participated in America's Manhattan Project, originated the Teller–Ulam design of thermonuclear weapons, invented the Monte Carlo method of computation, and suggested nuclear pulse propulsion. In pure and applied mathematics, he produced many results, proved many theorems, and proposed several conjectures.

Alan Turing 1912-1954):  A British mathematician, logician, cryptanalyst and computer scientist. He was highly influential in the development of computer science, giving a formalisation of the concepts of "algorithm" and "computation" with the Turing machine, which can be considered a model of a general purpose computer. Turing is widely considered to be the father of theoretical computer science and artificial intelligence.

During World War II, Turing worked for the Government Code and Cypher School (GC&CS) at Bletchley Park, Britain's code breaking centre. For a time he led Hut 8, the section responsible for German naval cryptanalysis. He devised a number of techniques for breaking German ciphers, including the bombe method, an electromechanical machine that could find settings for the Enigma machine.  His algorithms used Bayes Theorem.

Turing had something of a reputation for eccentricity at Bletchley Park. He was known to his colleagues as 'Prof' and his treatise on Enigma was known as 'The Prof's Book'. Jack Good, a cryptanalyst who worked with him, is quoted by Ronald Lewin as having said of Turing:

In the first week of June each year he would get a bad attack of hay fever, and he would cycle to the office wearing a service gas mask to keep the pollen off. His bicycle had a fault: the chain would come off at regular intervals. Instead of having it mended he would count the number of times the pedals went round and would get off the bicycle in time to adjust the chain by hand. Another of his eccentricities is that he chained his mug to the radiator pipes to prevent it being stolen.

While working at Bletchley, Turing, a talented long-distance runner, occasionally ran the 40 miles (64 km) to London when he was needed for high-level meetings, and he was capable of world-class marathon standards.

Turing was prosecuted for homosexual acts in 1952, when such acts were still criminalised in the UK. He accepted being chemically castrated as an alternative to prison. He died in 1954, 42 days before his 42nd birthday from cyanide poison ing.  An inquest determined his death  a suicide, his mother and some others believe it was accidental.  See http://www.cbsnews.com/news/wwii-codebreaker-alan-turing-gets-royal-pardon-for-gay-conviction/

There were prominent mathematicians that did not embrace the logic of Bayes Theorem.  Alan Turing used it to decode the German Enigma Cipher.  http://lesswrong.com/lw/774/a_history_of_bayes_theorem/

Turing worked from 1952 until his death in 1954 on mathematical biology, specifically morphogenisis.  He published one paper on the subject called The Chemical Basis of Morphogenesis in 1952, putting forth the Turing hypothesis of pattern formation (the theory was experimentally confirmed 60 years after his death).  His central interest in the field was understanding Fibonacci phyllotaxis, the existence of Fibonacci numbers in plant structures. He used reaction-diffusion equations which are central to the field of pattern formation.  Later papers went unpublished until 1992 when Collected Works of A. M. Turing was published.  His contribution is considered a seminal piece of work in this field. Removal of Hox genes causes an increased number of digits (up to 14) in mice, demonstrating a Turing-type mechanism in the development of the hand.

Note: In October 2016 Great Britain plans to posthumously pardon thousands of men who were convicted of what decades ago was a crime: having or seeking gay sex.

The measure has been nicknamed Turing’s Law, after Alan Turing, the mathematician central to the development of the computer. He committed suicide in 1954, after being convicted on charges of “gross indecency” with another man

Paul Erdős (1913-1996):  (from Budapest, Hungary)  Erdős published more papers than any other mathematician in history, working with hundreds of collaborators.  His colleague Alfred Renyi said, "a mathematician is a machine for turning coffee into theorems", and Erdős drank copious quantities.

Because of his prolific output, friends created the Erdős number as a humorous tribute; Erdős alone was assigned the Erdős number of 0 (for being himself), while his immediate collaborators could claim an Erdős number of 1, their collaborators have Erdős number at most 2, and so on. Approximately 200,000 mathematicians have an assigned Erdős number, and some have estimated that 90 percent of the world's active mathematicians have an Erdős number smaller than 8.

It is said that Hank Aaron  has an Erdős number of 1 because they both autographed the same baseball when Emory University awarded them honorary degrees on the same day. Erdős numbers have also been assigned to an infant, a horse, and several actors.

Martin Gardner (1914-2010):  (American)  A mathematics and science writer specializing in recreational mathematics, but with many interests (especially the writings of Lewis Carroll. He wrote the Mathematical Games column in Scientific American from 1956 to 1981, the Notes of a Fringe-Watcher column in Skeptical Inquirer from 1983 to 2002, and published over 70 books. See: http://en.wikipedia.org/wiki/Martin_Gardner

Ivan Niven (1915-1999):  (Canadian) Ivan published over sixty papers, some with well-known co-authors such as Samuel Eilenberg, Paul Erdos (6 times), Nathan J. Fine, R. D. James, and H. S. Zuckerman (7 times). His areas of expertise were number theory, especially the areas of diophantine approximation and questions of irrationality and transcendance of numbers, and combinatorics. Ken Ross found two of the articles of special interest: his famous 1947 paper containing a simple proof that pi is irrational and his 1969 Monthly article on formal power series, for which he received the Lester R. Ford Award. Ivan viewed his most significant paper to be, "Uniform distribution of sequences of integers".

Charles Frederick Mosteller (1916 - 2006)  American)  He was known as Frederick Mosteller, one of the most eminent statisticians of the 20th century. He was the founding chairman of Harvard's statistics department, from 1957 to 1971, and served as the president of several professional bodies including the Psychometric Society, the American Statistical Association, the Institute of Mathematical Statistics, the American Association for the Advancement of Science, and the International Statistical Institute.  Mosteller and David Wallace studied the historical problem of who wrote each of the disputed Federalist papers, James Madison or Alexander Hamilton. The Federalist Papers study was conducted in order to demonstrate the power of Bayesian inference and required a great deal of computational power for that time. It was featured in Time Magazine in the September 21st, 1962 edition.  See article at http://www.slideshare.net/FritzFerran/frederick-mosteller-fiftychallengingproblemsibookseeorg  Also see Bio at http://www.nasonline.org/publications/biographical-memoirs/memoir-pdfs/mosteller-frederick.pdf

Richard Feynman (1918-1988):  Feynman said:  "Mathematics is looking for patterns".  "Mathematics is only patterns".  "Nature uses only the longest threads to weave her patterns, so that each small piece of her fabric reveals the organization of the entire tapestry."  Also: "Physics is like sex.  Sure, it may give some practical results, but that's not why we do it".

Murray Gell-Mann commented to the New York Times that the Feynman Algorithm to solve a problem is:
1. Write down the problem
2. Think very hard

Benoit Mandelbrot (1924- 2010):  (from France)  The Father of Fractal Geometry.  He was a Polish-born, French and American mathematician, noted for developing a "theory of roughness" in nature and the field of fractal geometry to help prove it, which included coining the word "fractal". He later discovered the Mandelbrot set of intricate, never-ending fractal shapes, named in his honor. While he was a child, his family fled to France in 1936 to escape the growing Nazi persecution of Jews. From 1945 to 1947 attended the École Polytechnique, where he studied under Gaston Julia and Paul Lévy. From 1947 to 1949 he studied at California Institute of Technology, where he earned a master's degree in aeronautics. Returning to France, he obtained his PhD degree in Mathematical Sciences at the University of Paris in 1952.

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 characterize 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 fulfill this ambition with such remarkable success in so many areas.

As a visiting professor at Harvard University, Mandelbrot began to study fractals called Julia sets that were invariant under certain transformations of the complex plane. Building on previous work by Gaston Julia and Pierre Fatou, Mandelbrot used a computer to plot images of the Julia sets. While investigating the topology of these Julia sets, he studied the Mandelbrot set fractal that is now named after him.

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.  See his bio at http://www-history.mcs.st-and.ac.uk/Biographies/Mandelbrot.html

Learn from Robert Devaney at: http://video.google.com/videoplay?docid=1619313842463920970#docid=-6460544449138143366   Fractal art is shown at: http://www.lifesmith.com/art2008.html and at http://www.lifesmith.com/art2006.html and http://www.lifesmith.com/art2007.html

I presented some pictures of fractals and some background at the Annual Meeting of the Society of Actuaries when I was president in 1992. I may have used Robert Devaney to develop the script.  See Chaos video, but start about 40% of the way through. (OOPs is copywrite protected so will only play on my computer)

Alexander Grothendieck (1928--2014): (German) He is one of the most influential mathematicians of the 20th century. He is known principally for his revolutionary advances in algebraic geometry, and also for major contributions to number theory, category theory and homological algebra, and his early achievements in functional analysis. He was awarded the Fields Medal in 1966. He became the leading figure in the creation of modern algebraic geometry.  His research extended the scope of the field and added elements of commutative algebra, homological algebra, sheaf theory and category theory to its foundations, while his so-called "relative" perspective led to revolutionary advances in many areas of pure mathematics.

Murray Gell-Mann (1929-        ): (American) A physicist and polymath who received the 1969 Nobel Prize in physics for his work on the theory of elementary particles. He is a Distinguished Fellow and co-founder of the Santa Fe Institute and the Presidential Professor of Physics and Medicine at the University of Southern California.

He formulated the quark model of hadronic resonances, and identified the SU(3) flavor symmetry of the light quarks, extending isospin to include strangeness, which he also discovered. He developed the V-A theory of the weak interaction in collaboration with Richard Feynman. He created current algebra in the 1960s as a way of extracting predictions from quark models when the fundamental theory was still murky, which led to model-independent sum rules confirmed by experiment.

Roger Penrose (1931 -         ):  (British)  A mathematician and physicist . www.en.wikipedia.org/wiki/Roger_Penrose

Edward Oakley "Ed" Thorp ( 1932): (American) A mathematics professor, author, hedge fund manager, and blackjack player best known as the "father of the wearable computer" after inventing the world's first wearable computer in 1961.[1] He was a pioneer in modern applications of probability theory, including the harnessing of very small correlations for reliable financial gain.

He is the author of Beat the Dealer, the first book to mathematically prove, in 1962, that the house advantage in blackjack could be overcome by card counting.[2] He also developed and applied effective hedge fund techniques in the financial markets, and collaborated with Claude Shannon in creating the first wearable computer.

Thorp received his Ph.D. in mathematics from the University of California, Los Angeles in 1958, and worked at the Massachusetts Institute of Technology (MIT) from 1959 to 1961. He was a professor of mathematics from 1961 to 1965 at New Mexico State University, and then joined the University of California, Irvine where he was a professor of mathematics from 1965 to 1977 and a professor of mathematics and finance from 1977 to 1982.

Robert Phelan Langlands (1936 -        ): (Canadian) He is best known as the founder of the Langlands Program, a vast web of conjectures and results connecting representation theory and automorphic forms to the study of Galois groups in number theoryHe is an emeritus professor and occupies Albert Einstein's office at the Institute for Advanced Study in Princeton.
See projects.thestar.com/math-the-canadian-who-reinvented-mathematics/ and www.math.ubc.ca/Dept/Newsletters/Robert_Langlands_interview_2010.pdf  and www.math.duke.edu/langlands/ and Frenkel's discription: https://math.berkeley.edu/~frenkel/review.pdf  Also, Frenkel writes: The Langlands Program was launched in the late 60s with the goal of relating Galois representations and automorphic forms. In recent years a geometric version has been developed which leads to a mysterious duality between certain categories of sheaves on moduli spaces of (flat) bundles on algebraic curves. Three years ago, in a groundbreaking advance, Kapustin and Witten have linked the geometric Langlands correspondence to the S-duality of 4D supersymmetric gauge theories. This and subsequent works have already led to striking new insights into the geometric Langlands Program, which in particular involve the Homological Mirror Symmetry of the Hitchin moduli spaces of Higgs bundles on algebraic curves associated to two Langlands dual Lie groups.

Langlands Program

A grand unified theory of mathematics which includes the search for a generalization of Artin reciprocity (known as Langlands reciprocity) to non-Abelian Galois extensions of number fields. In a January 1967 letter to André Weil, Langlands proposed that the mathematics of algebra (Galois representations) and analysis (automorphic forms) are intimately related, and that congruences over finite fields are related to infinite-dimensional representation theory. In particular, Langlands conjectured that the transformations behind general reciprocity laws could be represented by means of matrices (Mackenzie 2000).

In 1998, three mathematicians proved Langlands' conjectures for local fields, and in a November 1999 lecture at the Institute for Advanced Study at Princeton University, L. Lafforgue presented a proof of the conjectures for function fields. This leaves only the case of number fields as unresolved (Mackenzie 2000).
www.ams.org/journals/bull/1984-10-02/S0273-0979-1984-15237-6/S0273-0979-1984-15237-6.pdf

James Harris Simons (1939 -       ): (American)  Jim is an American mathematician, hedge fund manager, and philanthropist. He is a code breaker and studies pattern recognition. Simons is the co-inventor, with Shiing-Shen Chern, of the Chern–Simons form - Chern and Simons (1974), and contributed to the development of string theory by providing a theoretical framework to combine geometry and topology with quantum field theory. Simons was a professor of mathematics at Stony Brook University and was also the former chair of the Mathematics Department at Stony Brook.

Dr. Simons received his doctorate at 23; advanced code breaking for the National Security Agency at 26; led a university math department at 30; won geometry’s top prize at 37; founded Renaissance Technologies, one of the world’s most successful hedge funds, at 44; and began setting up charitable foundations at 56.

In 1982, Simons founded Renaissance Technologies, a private hedge fund investment company based in New York with over $25 billion under management. Simons retired at the end of 2009 as CEO of one of the world's most successful hedge fund companies. Simons' net worth is estimated to be$16.5 billion.

Dr. Simons now runs a tidy universe of science endeavors, financing not only math teachers but hundreds of the world’s best investigators, even as Washington has reduced its support for scientific research. His favorite topics include gene puzzles, the origins of life, the roots of autism, math and computer frontiers, basic physics and the structure of the early cosmos.

Stephen Hawkings (1942-      ):  (British) A physicist from Cambridge wrote "A Brief History of Time".  In it he tells the story of a lady commenting on a statement made in a lecture on astronomy.  She said: "Rubbish, The world is really a flat plate supported on the back of a giant tortoise"  When asked what the tortoise was sitting on, her answer would have made Goedel smile: "You're very clever, young man, very clever.  But its turtles all the way down."

Hawking is an English theoretical physicist, cosmologist, auhor and director of research at the center for Theoretical Cosmology within the University of Cambridge.  Hawking was the first to set forth a cosmology explained by a union of the general theory of relativity and quantum mechantics.  He is a vigorous supporter of the many-worlds interpretation of quantum mechanics.

. With Roger Penrose he showed that Einstein's General Theory of Relativity implied space and time would have a beginning in the Big Bang and an end in black holes.

Ian Stewart (1945-       ):  A British mathematician.  Stewart came to the attention of the mathematics teacher. The teacher had Stewart sit mock A-level examinations without any preparation along with the upper-sixth students; Stewart placed first in the examination. This teacher arranged for Stewart to be admitted to Cambridge on a scholarship to Churchill College, where he obtained a BA in mathematics. Stewart then went to the University of Warwick for his doctorate, on completion of which in 1969 he was offered an academic position at Warwick, where he presently professes mathematics. He is well known for his popular expositions of mathematics and his contributions to catastrophe theory.

While at Warwick he edited the mathematical magazine Manifold. He also wrote a column called "Mathematical Recreations" for Scientific American magazine for several years. Stewart has published more than 140 scientific papers, including a series of influential papers co-authored with Jim Collins on coupled oscillators and the symmetry of animal gaits.  He lists his recreations as science fiction, painting, guitar, keeping fish, geology, Egyptology and snorkeling.

Persi Warren Diaconis (1945-       ): He is the statistician who demonstrated that it takes the average card player no fewer than seven shuffles to create a random order in a deck of cards.

Marilyn vos Savant (1946-       ):  An American magazine columnist, author, lecturer, and playwright. She has written "Ask Marilyn", a Sunday column in Parade magazine in which she solves puzzles and answers questions from readers on a variety of subjects.

Her September 9, 1990 column began with a question now called The Monty Hall problem (Suppose you are on a game show and you are given the choice of three doors.  Behind one door is a car, the others, goats. You pick a door, say #1, and the host, who knows what's behind the doors, opens another door, say #3, which has a goat. He says to you: 'Do you want to pick door #2?' Is it to your advantage to switch doors?

Marilyn vos Savant answered arguing that the selection should be switched to door #2 because it has a 2/3 chance of success, while door #1 has just 1/3. This response provoked letters of thousands of readers, nearly all arguing doors #1 and #2 each have an equal chance of success. A follow-up column reaffirming her position served only to intensify the debate and soon became a feature article on the front page of The New York Times. Among the ranks of dissenting arguments were hundreds of academics and mathematicians.

In a subsequent column, vos Savant offered numerous explanations as to why her solution is correct.  She also called upon elementary teachers to simulate the problem in their class. Numerous elementary school math classes devoted themselves to this experiment, playing the game hundreds of times and reporting their results. Nearly 100% of those classes found that your odds of winning were doubled if you switch doors.

Finally, thanks to the diligence of elementary school children, the controversy subsided.

Dr. Keith Devlin (1947-     ): This professor from Stamford defines Mathematics as  the Science of Patterns.

Robert L. Devaney (circa 1948-        ):   A native of Methuen, Massachusetts, is currently Professor of Mathematics at Boston University. He received his undergraduate degree from the College of the Holy Cross in 1969 and his PhD from the University of California at Berkeley in 1973 under the direction of Stephen Smale. He taught at Northwestern University and Tufts University before coming to Boston University in 1980.  His main area of research is dynamical systems, primarily complex analytic dynamics, but also including more general ideas about chaotic dynamical systems. Lately, he has become intrigued with the incredibly rich topological aspects of dynamics, including such things as indecomposable continua, Sierpinski curves, and Cantor bouquets.

Devaney developed the 8 minute script contained  in the middle of the presentation on Chaos that Don had presented for the 1992 Annual Meeting of the Society of Actuaries. This link only works on Don's computer.

Edward Witten (1951-      ):  One of the researchers at Princeton working on "string theory" which may help with the clash between the central ideas of general relativity and quantum mechanics when it comes to extremely small scales.   He is regarded by many of his peers as one of the greatest living physicists, perhaps even a successor to Albert Einstein In 1990 he was awarded a Fields Medal by the International Union of Mathematics, which is the highest honor in mathematics and often regarded as the Nobel Prize equivalent for mathematics. He is the only physicist to have received this honor.

Sir Andrew John Wiles (1953-      ):  (British)  A professor at Princeton University in Number Theory. He published a flawed proof of Fermat's Last Theorem in 1993.  He corrected the error in 1994.

Simon Kirwan Donaldson (1957-      ):  (British)  An English mathematician famous for his work on the topology of smooth (differentiable) four-dimensional manifolds. He is now Royal Society research professor in Pure Mathematics and President of the Institute for Mathematical Science at Imperial College London.  He used the solutions to the Yang-Mills equations to discover a finger-print which allowed him to distinguish whether two shapes were actually the same.  These finger-prints are called invariants.

Steven Strogatz ( 1959 -        ): (American) He is an applied mathematics professor at Cornell.  He has written an interesting book: The Joy of X.

Marcus Peter Francis du Sautoy (1965 -      ): (born in London) A Professor of Mathematics at the University of Oxford. His academic work concerns mainly group theory and number theory. He is known for his books popularizing mathematics. In 2001 he won the Berwick Prize of the London Mathematical Society, which is awarded every two years to reward the best mathematical research by a mathematician under forty. In March, 2006, his article Prime Numbers Get Hitched was published on Seed Magazine's website.   http://seedmagazine.com/content/article/prime_numbers_get_hitched/  In it he explained how the number 42, mentioned in The Hitchhiker's Guide to the Galaxy as the answer to everything, is related to the Riemann zeta function. See http://www.culturenorthernireland.org/article/2836/belfast-festival-marcus-du-sautoy Also:  http://people.maths.ox.ac.uk/dusautoy/newleft.htm and http://people.maths.ox.ac.uk/dusautoy/newright.htm

Grigori Yakovlevich Perelman (1966-       ):  (from Russia) The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute  in 2000. Currently, six of the problems remain unsolved. A correct solution to any of the problems results in a US\$1,000,000 prize (sometimes called a Millennium Prize) being awarded by the institute. One of the problems, the Poincare' conjecture, was solved by Perelman in  2002  He was also awarded  the Fields Medal in 2006.  He has not accepted either prize.

Edward Vladimirovich Frenkel (1967 circa -     ): (Russian  A mathematician working in representation theory, algebraic geometry, and mathematical physics. He is a professor of mathematics at University of California, Berkeley and author of the bestselling book Love and Math.  As a high school student he studied higher mathematics privately with Evgeny Evgenievich Petrov, although his initial interest was in quantum physics rather than mathematics. He was not admitted to Moscow State University because of discrimination against Jews and enrolled instead in the applied mathematics program at the Gubkin University of Oil and Gas. While a student there, he attended the seminar of Israel Gelfand and worked with Boris Lwowitsch Feigin (de) and Dmitry Fuchs. After receiving his college degree in 1989, he was first invited to Harvard University as a visiting professor, and a year later he enrolled as a graduate student at Harvard He received his Ph.D. at Harvard University in 1991, after one year of study, under the direction of Joseph Bernstein. He was a Junior Fellow at the Harvard Society of Fellows from 1991 to 1994, and served as an associate professor at Harvard from 1994 to 1997. He has been a professor of mathematics at University of California, Berkeley since 1997.  Frenkel's recent work has focused on the Langlands program and its connections to representation theoryintegrable systems, geometry, and physics. Together with Dennis Gaitsgory and Kari Vilonen, he has proved the geometric Langlands conjecture for GL(n). His joint work with Robert Langlands and Ngô Bảo Châu suggested a new approach to the functoriality of automorphic representations and trace formulas. He has also been investigating (in particular, in a joint work with Edward Witten) connections between the geometric Langlands correspondence and dualities in quantum field theory

Maryam Mirzakhani (May 3,1977- July 15,1917.)  (Persian)  was an Iranian mathematician and a professor of mathematics at Stanford University. Her research topics include Teichmüller theory, hyperbolic geometry, ergodic theory, and symplectic geometry.On 13 August 2014, Mirzakhani became both the first woman and the first Iranian honored with the Fields Medal, the most prestigious award in mathematics. The award committee cited her work in "the dynamics and geometry of Riemann surfaces and their moduli spaces".

Nationality Iranian]
Fields Mathematics
Institutions
 Princeton University Stanford University
 Sharif University of Technology (BS) Harvard University (PhD)
Thesis Simple geodesics on hyperbolic surfaces and the volume of the moduli space of curves (2004)
Notable awards
 Blumenthal Award (2009) Satter Prize (2013) Clay Research Award (2014) Fields Medal (2014)
Spouse Jan Vondrák
Children Anahita

## Early life and education

Mirzakhani was born on 3 May 1977 in Tehran, Iran. She attended Farzanegan School there, part of the National Organization for Development of Exceptional Talents.

In 1994, Mirzakhani won a gold medal in the International Mathematical Olympiad, the first female Iranian student to do so. In the 1995 International Mathematical Olympiad, she became the first Iranian student to achieve a perfect score and to win two gold medals.

She obtained her BSc in mathematics (1999) from Sharif University of Technology in Tehran. She went to the United States for graduate work, earning a PhD from Harvard University in 2004, where she worked under the supervision of the Fields Medalist Curtis McMullen. She was also a 2004 research fellow of the Clay Mathematics Institute and a professor at Princeton University.

## Research work

Mirzakhani has made several contributions to the theory of moduli spaces of Riemann surfaces. In her early work, Mirzakhani discovered a formula expressing the volume of a moduli space with a given genus as a polynomial in the number of boundary components. This led her to obtain a new proof for the formula discovered by Edward Witten and Maxim Kontsevich on the intersection numbers of tautological classes on moduli space,[7] as well as an asymptotic formula for the growth of the number of simple closed geodesics on a compact hyperbolic surface, generalizing the theorem of the three geodesics for spherical surfaces. Her subsequent work has focused on Teichmüller dynamics of moduli space. In particular, she was able to prove the long-standing conjecture that William Thurston's earthquake flow on Teichmüller space is ergodic.

Most recently as of 2014, with Alex Eskin and with input from Amir Mohammadi, Mirzakhani proved that complex geodesics and their closures in moduli space are surprisingly regular, rather than irregular or fractal.The closures of complex geodesics are algebraic objects defined in terms of polynomials and therefore they have certain rigidity properties, which is analogous to a celebrated result that Marina Ratner arrived at during the 1990s. The International Mathematical Union said in its press release that, "It is astounding to find that the rigidity in homogeneous spaces has an echo in the inhomogeneous world of moduli space."

Mirzakhani was awarded the Fields Medal in 2014 for "her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces". The award was made in Seoul at the International Congress of Mathematicians on 13 August.

At the time of the award, Jordan Ellenberg explained her research to a popular audience:

... [Her] work expertly blends dynamics with geometry. Among other things, she studies billiards. But now, in a move very characteristic of modern mathematics, it gets kind of meta: She considers not just one billiard table, but the universe of all possible billiard tables. And the kind of dynamics she studies doesn't directly concern the motion of the billiards on the table, but instead a transformation of the billiard table itself, which is changing its shape in a rule-governed way; if you like, the table itself moves like a strange planet around the universe of all possible tables ... This isn't the kind of thing you do to win at pool, but it's the kind of thing you do to win a Fields Medal. And it's what you need to do in order to expose the dynamics at the heart of geometry; for there's no question that they're there.

## Personal life

Mirzakhani was diagnosed with breast cancer in 2013. After four years, it spread to her bone marrow. She died on 15 July 2017. She was survived by her husband Jan Vondrák, a Czech theoretical computer scientist and applied mathematician who is an associate professor at Stanford University,[28] and a daughter named Anahita.[

Awards and honors

 IPM Fellowship, Tehran, Iran, 1995–99 Merit fellowship Harvard University, 2003 Harvard Junior Fellowship Harvard University, 2003 Clay Mathematics Institute Research Fellow 2004 AMS Blumenthal Award 2009 Invited to talk at the International Congress of Mathematicians in 2010, on the topic of "Topology and Dynamical Systems & ODE" The 2013 AMS Ruth Lyttle Satter Prize in Mathematics. "Presented every two years by the American Mathematical Society, the Satter Prize recognizes an outstanding contribution to mathematics research by a woman in the preceding six years. The prize was awarded on Thursday, 10 January 2013, at the Joint Mathematics Meetings in San Diego." Named one of Nature's ten "people who mattered" of 2014. Clay Research Award 2014 Plenary speaker at the International Congress of Mathematicians (ICM 2014) Fields Medal 2014 Elected foreign associate to the French Academy of Science in 2015 Elected to the American Philosophical Society in 2015. National Academy of Sciences 2016 Elected to the American Academy of Arts and Sciences in 2017.

RANKING OF GREAT MATHEMATICIANS  See www.fabpedigree.com/james/mathmen.htm

 Isaac Newton Archimedes Carl Gauss Leonhard Euler Bernhard Riemann Henri Poincaré J.-L. Lagrange David Hilbert Euclid G.W. Leibniz Alex. Grothendieck Pierre de Fermat

1. Henri Poincaré
2. Joseph-Louis Lagrange
3. Euclid of Alexandria
4. David Hilbert
5. Gottfried W. Leibniz

1. Kurt Gödel
2. Diophantus of Alexandria
3. Bháscara Áchárya
4. Eudoxus of Cnidus
5. Blaise Pascal

 Albert Einstein Galileo Galilei James C. Maxwell Aristotle Girolamo Cardano

1. Alan M. Turing
2. Michael F. Atiyah
3. Siméon-Denis Poisson
4. Pappus of Alexandria
5. Atle Selberg

 John Willard Milnor John Horton Conway Thabit ibn Qurra Hippocrates of Chios Nicolai Lobachevsky

 Jean Gaston Darboux Andrei A. Markov Nasir al-Din al-Tusi Sofia Kovalevskaya Paul Erdös

 Oliver Heaviside James Gregory Leopold Kronecker John Napier of Merchiston Norbert Wiener

 Emil Artin Georg F. Frobenius Lennart A.E. Carleson Tullio Levi-Civita Gérard Desargues

 Alexis C. Clairaut J. Müller `Regiomontanus' Alfred Clebsch Oswald Veblen Abu Rayhan al-Biruni

 Colin Maclaurin Ptolemy of Alexandria S.G. Vito Volterra Qin Jiushao & Zhu Shiejie Brook Taylor

 R. Maurice Fréchet Michel F. Chasles Henri P. Cartan Thoralf A. Skolem Samuel Eilenberg

 Henry J.S. Smith Augustus F. Möbius Luigi Cremona Rafael Bombelli Edmund G.H. Landau

 Leonardo da Vinci Paul A.M. Dirac Jamshid Al-Kashi William K. Clifford Nicole Oresme

This site was last updated 07/28/17