Pat'sBlog

Wednesday, 31 October 2012

Tuesday, 30 October 2012

Reposting, Roy G Biv

I like to post this one at least once each year. Enjoy
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Every kid learns it in science class.. (or on "They Might be Giants", see below) the colors of the rainbow (visible light spectrum) are given by the mnemonic ROY G BIV for Red, Orange, Yellow, Green, Blue, Indigo, and Violet.
But really, before you knew the mnemonic if someone asked, "What colors do you see?" what would you have said?

I don't know if anyone has ever tested young kids before they are exposed to science classes, but it seems it used to be common to describe four colors, Red, Yellow, Green, and Blue... So why do we see (or why are we told to see) seven?

Turns out, it was old Isaac Newton, and his dedication to the ancient Greek ideas. Here is a clip from a blog by Patricia Fara, a senior lecturer at Cambridge, and author of "Science: A Four Thousand Year History" at the Nature web site.

Consider Isaac Newton. He believed so firmly in the Greek idea of a harmonic universe that he divided the rainbow into seven colours to correspond with the musical scale. Before then, although opinions varied, artists mostly showed rainbows with four colours. It is, of course, impossible to make any objective decision about the correct number, because the spectrum of visible light varies continuously: there is no sharp cut-off between bands of different colours, so how you think about a rainbow affects how you see it. Be honest - can you tell the difference between blue, indigo and violet?

I found this on Wikipedia: "In Classical Antiquity, Aristotle had claimed there was a fundamental scale of seven basic colors. In the Renaissance, several artists tried to establish a new sequence of up to seven primary colors from which all other colors could be mixed. In line with this artistic tradition, Newton divided his color circle, which he constructed to explain additive color mixing, into seven colors.^[1] His color sequence with the unusual color indigo is still kept alive today by the Roy G. Biv mnemonic. Originally he used only five colors, but later he added orange and indigo, in order to match the number of musical notes in the major scale."

So I can't find out who first came up with the mnemonic, but I did find a song about it...lyrics are here. And Aaron Wagner sent me a link to a video of a different ROY G BIV song by "They Might Be Giants" that his four-year old highly endorses. Thanks to Aaron and his child for the tip.

They Might Be Giants - Roy G Biv from They Might Be Giants on Vimeo.

On This Day in Math - October 30

'Mathematics is the science that uses easy words for hard ideas.'

~ Edward Kasner
The 304th day of the year; 304 is the sum of six consecutive primes starting with 41, and also the sum of eight consecutive primes starting with 23. (and for those who keep up with such things, it is also the record number of wickets taken in English cricket season by Tich Freeman in 1928.)

EVENTS
1613 Kepler married his second wife (the ﬁrst died of typhus). She was ﬁfth on his slate of eleven candidates. The story that he used astrology in the choice is doubtful.*VFR Kepler married the 24-year-old Susanna Reuttinger. He wrote that she, "won me over with love, humble loyalty, economy of household, diligence, and the love she gave the stepchildren.According to Kepler's biographers, this was a much happier marriage than his first. *Wik

1710 William Whiston, whom Newton had arranged to succeeded him as Lucasian Professor at Cambridge in 1701, was deprived of the chair and driven from Cambridge for his unorthodox religious views. Whiston was removed from his position at Cambridge, and denied membership in the Royal Society for his “heretical” views. He took the “wrong” side in the battle between Arianism (a unitarian view) and the Trinitarian view, but his brilliance still made the public attend to his proclamations. When he predicted the end of the world by a collision with a comet in October 16th of 1736 the Archbishop of Canterbury had to issue a denial to calm the panic (VFR put it this way, "it is not acceptable to be a unitarian at the College of the Whole and Undivided Trinity".
His translation of the works of Flavius Josephus may have contained a version of the famous Josephus Problem, and in 1702 Whiston's Euclid discusses the classic problem of the Rope Round the Earth, (if one foot of additional length is added, how high will the rope be). I am not sure of the dimensions in Whiston's problem, and would welcome input, I have searched the book and can not find the problem in it, but David Singmaster has said it is there, and he is not an easy source to reject. It is said that Ludwig Wittgenstein was fascinated by the problem and used to pose it to students regularly.

1735 Ben Franklin published “On the Usefulness of Mathematics,” his only published article on mathematics. *VFR

1826 Abel presented a paper to the French Academy of Science that was ignored by Cauchy, who was to serve as referee. The paper was published some twenty years later.*VFR

In 1937, the closest approach to the earth by an asteroid, Hermes, was measured to be 485,000 miles, which, to an astronomer, is a mere hair's width (asteroid now lost).*TIS

1945 The first conference on Digital Computer Technique was held at MIT. The conference was sponsored by the National Research Council, Subcommittee Z on Calculating Machines and Computation. Attended by the Whirlwind team,(The Whirlwind computer was developed at the Massachusetts Institute of Technology. It is the first computer that operated in real time, used video displays for output, and the first that was not simply an electronic replacement of older mechanical systems) it influenced the direction of this computer. *CHM

1978 Laura Nickel and Curt Noll, eighteen year old students at California State at Hayward, show that 2^21,701 − 1 is prime. This was the largest prime known at that time. *VFR (By Feb of the next year, Noll had found another, 2²³²⁰⁹-1. By April, another larger Prime had been found.)

1992 The Vatican announced that a 13-year investigation into the Catholic Church’s condemnation of Galileo in 1633 will come to an end and that Galileo was right: The Copernican Theory, in which the Earth moves around the Sun, is correct and they erred in condemning Galileo. *New York Times for 31 October 1992.

2012 After Hurricane Sandy came ashore in New Jersey on the 29th, the huge weather system was captured with an overlay to emphasize it's Fibonacci-like structure. *HT to Bob Mrotek for sending me this image

BIRTHS
1840 Joseph Jean Baptiste Neuberg (30 Oct 1840 in Luxembourg City, Luxembourg - 22 March 1926 in Liège, Belgium) Neuberg worked on the geometry of the triangle, discovering many interesting new details but no large new theory. Pelseneer writes, "The considerable body of his work is scattered among a large number of articles for journals; in it the influence of A Möbius is clear." *SAU

1844 George Henri Halphen (30 October 1844, Rouen – 23 May 1889, Versailles) was a French mathematician. He did his studies at École Polytechnique (X 1862). He was known for his work in geometry, particularly in enumerative geometry and the singularity theory of algebraic curves, in algebraic geometry. He also worked on invariant theory and projective differential geometry.*Wik

1863 Stanislaw Zaremba (3 Oct 1863 in Romanowka, Poland - 23 Nov 1942 in Kraków, Poland) From very unpromising times up to World War I, with the recreation of the Polish nation at the end of that war, Polish mathematics entered a golden age. Zaremba played a crucial role in this transformation. Much of Zaremba's research work was in partial differential equations and potential theory. He also made major contributions to mathematical physics and to crystallography. He made important contributions to the study of viscoelastic materials around 1905. He showed how to make tensorial definitions of stress rate that were invariant to spin and thus were suitable for use in relations between the stress history and the deformation history of a material. He studied elliptic equations and in particular contributed to the Dirichlet principle.*SAU

1906 Andrei Nikolaevich Tikhonov (30 Oct 1906 in Gzhatska, Smolensk, Russia - November 8, 1993, Moscow) Tikhonov's work led from topology to functional analysis with his famous fixed point theorem for continuous maps from convex compact subsets of locally convex topological spaces in 1935. These results are of importance in both topology and functional analysis and were applied by Tikhonov to solve problems in mathematical physics.
The extremely deep investigations of Tikhonov into a number of general problems in mathematical physics grew out of his interest in geophysics and electrodynamics. Thus, his research on the Earth's crust lead to investigations on well-posed Cauchy problems for parabolic equations and to the construction of a method for solving general functional equations of Volterra type.
Tikhonov's work on mathematical physics continued throughout the 1940s and he was awarded the State Prize for this work in 1953. However, in 1948 he began to study a new type of problem when he considered the behaviour of the solutions of systems of equations with a small parameter in the term with the highest derivative. After a series of fundamental papers introducing the topic, the work was carried on by his students.
Another area in which Tikhonov made fundamental contributions was that of computational mathematics. Under his guidance many algorithms for the solution of various problems of electrodynamics, geophysics, plasma physics, gas dynamics, ... and other branches of the natural sciences were evolved and put into practice. ... One of the most outstanding achievemnets in computational mathematics is the theory of homogeneous difference schemes, which Tikhonov developed in collaboration with Samarskii.
In the 1960s Tikhonov began to produce an important series of papers on ill-posed problems. He defined a class of regularisable ill-posed problems and introduced the concept of a regularising operator which was used in the solution of these problems. Combining his computing skills with solving problems of this type, Tikhonov gave computer implementations of algorithms to compute the operators which he used in the solution of these problems. Tikhonov was awarded the Lenin Prize for his work on ill-posed problems in 1966. In the same year he was elected to full membership of the USSR Academy of Sciences.*SAU

1907 Harold Davenport (30 Oct 1907 in Huncoat, Lancashire, England - 9 June 1969 in Cambridge, Cambridgeshire, England) Davenport worked on number theory, in particular the geometry of numbers, Diophantine approximation and the analytic theory of numbers. He wrote a number of important textbooks and monographs including The higher arithmetic (1952)*SAU

1946 William Paul Thurston (October 30, 1946 – August 21, 2012) American mathematician who was awarded the Fields Medal in 1983 for his work in topology. As early as his Ph.D. thesis entitled Foliations of 3-manifolds which are circle bundles (1972) that showed the existence of compact leaves in foliations of 3-manifolds, Thurston had been working in the field of topology. In the following years, Thurston's contributions to the field of foliations were recognized to be of considerable depth, set apart by their originality. This was also true of his subsequent work on Teichmüller space. *TIS

DEATHS
1626 Willebrord van Royen Snell (13 June 1580 in Leiden, Netherlands - 30 Oct 1626 in Leiden, Netherlands) Snell was a Dutch mathematician who is best known for the law of refraction, a basis of modern geometric optics; but this only become known after his death when Huygens published it. His father was Rudolph Snell (1546-1613), the professor of mathematics at Leiden. Snell also improved the classical method of calculating approximate values of π by polygons which he published in Cyclometricus (1621). Using his method 96 sided polygons gives π correct to 7 places while the classical method yields only 2 places. Van Ceulen's 35 places could be found with polygons of 230 sides rather than 262. In fact Van Ceulen's 35 places of π appear in print for the first time in this book by Snell. *SAU

1631 Michael Mästin (30 Sept 1550 in Göppingen, Baden-Würtemberg, Germany
- 30 Oct 1631 in Tübingen, Baden-Würtemberg, Germany) astronomer who was Kepler's teacher and who publicized the Copernican system. Michael Mästin was a German astronomer who was Kepler's teacher and who publicised the Copernican system. Perhaps his greatest achievement (other than being Kepler's teacher) is that he was the first to compute the orbit of a comet, although his method was not sound. He found, however, a sun centered orbit for the comet of 1577 which he claimed supported Copernicus's heliocentric system. He did show that the comet was further away than the moon, which contradicted the accepted teachings of Aristotle. Although clearly believing in the system as proposed by Copernicus, he taught astronomy using his own textbook which was based on Ptolemy's system. However for the more advanced lectures he adopted the heliocentric approach - Kepler credited Mästlin with introducing him to Copernican ideas while he was a student at Tübingen (1589-94).*SAU

1739 Leonty Filippovich Magnitsky (June 9, 1669, Ostashkov – October 30, 1739, Moscow) was a Russian mathematician and educator. From 1701 and until his death, he taught arithmetic, geometry and trigonometry at the Moscow School of Mathematics and Navigation, becoming its director in 1716. In 1703, Magnitsky wrote his famous Arithmetic (Арифметика; 2,400 copies), which was used as the principal textbook on mathematics in Russia until the middle of the 18th century. This book was more an encyclopedia of mathematics than a textbook because most of its content was communicated for the first time in Russian literature. In 1703, Magnitsky also produced a Russian edition of Adriaan Vlacq's log tables called Таблицы логарифмов и синусов, тангенсов и секансов (Tables of Logarithms, Sines, Tangents, and Secants). Legend has it that Leonty Magnitsky was nicknamed Magnitsky by Peter the Great, who considered him a "people's magnet" *Wik

1805 Ormbsy MacKnight Mitchel (July 20, 1805 – October 30, 1862) American astronomer and major general in the American Civil War.
A multi-talented man, he was also an attorney, surveyor, and publisher. He is notable for publishing the first magazine in the United States devoted to astronomy. Known in the Union Army as "Old Stars", he is best known for ordering the raid that became famous as the Great Locomotive Chase during the Civil War. He was a classmate of Robert E. Lee and Joseph E. Johnston at West Point where he stayed as assistant professor of mathematics for three years after graduation.
The U.S. communities of Mitchell, Indiana, Mitchelville, South Carolina, and Fort Mitchell, Kentucky were named for him. A persistently bright region near the Mars south pole that was first observed by Mitchel in 1846 is also named in his honor. *TIA

1806 Alexander (Dallas) Bache (July 19, 1806 – February 17, 1867) was Ben Franklin's great grandson. A West Point trained physicist, Bache became the second Superintendent of the Coast Survey (1844-65). He made an ingenious estimate of ocean depth in 1856. He studied records of a tidal wave that had taken 12 hours to cross the Pacific. Knowing that wave speeds depend on depth, he calculated a 2 1/5-mile average depth for the Pacific (within 15% of the right value). Bache created the National Academy of Sciences, securing greater government involvement in science. Through the Franklin Institute he instituted boiler tests to promote safety for steamboats.*TIS

1975 Gustav Hertz (22 July 1887, 30 Oct 1975) German quantum physicist who, with James Franck, received the Nobel Prize for Physics in 1925 for the Franck-Hertz experiment, which confirmed the quantum theory that energy can be absorbed by an atom only in definite amounts and provided an important confirmation of the Bohr atomic model. He was a nephew of Heinrich Hertz. Although he fought on the German side in World War I, being of Jewish descent, he was forced to resign his professorship (1934) when Hitler took power. From 1945 he worked in the Soviet Union, and then in 1955 was a professor of physics in Leipzig, East Germany.*TIS

2007 Juha Heinonen, (23 July 1960 in Toivakka, Finland - 30 Oct 2007 in Ann Arbor, Michigan, USA) Professor of Mathematics passed away on October 30. He arrived in the Department in 1988 as a postdoctoral assistant professor, and became a professor in 2000. He was a leading researcher in geometric function theory, having published two books and numerous articles with many collaborators. Most recently, Juha served as Associate Chair for Graduate Studies in the Department, where he mentored many young mathematicians. *Math at U of M webpage memorial (Heinonen died at the age of 47 'after a brief but courageous battle with kidney cancer'. The Department of Mathematics at the University of Michigan established the Juha Heinonen Memorial Graduate Student Fellowship in his honour. An international conference in his memory Quasiconformal Mappings and Analysis on Metric Spaces was organised at the University of Michigan, Ann Arbor in May 2008.)

Credits
*CHM=Computer History Museum
*FFF=Kane, Famous First Facts
*NSEC= NASA Solar Eclipse Calendar
*SAU=St Andrews Univ. Math History
*TIA= Today in Astronomy
*TIS= Today in Science History
*VFR = V Frederick Rickey, USMA
*Wik = Wikipedia
*WM = Women of Mathematics, Grinstein & Campbell

Monday, 29 October 2012

On This Day in Math - October 29

Allez en avant, et la foi vous viendra
Push on and faith will catch up with you.

~Jean d'Alembert [advice to those who questioned the calculus](probably also great for students struggling with mathematics at any level)

The 303rd day of the year; there are 303 different bipartite graphs with 8 vertices. *What's Special About This Number

EVENTS
1669 Newton, aged twenty-six, appointed Lucasian Professor at Cambridge. This post required Newton to lecture once each week on “some part of Geometry, Astronomy, Geography, Optics, Statics, or some other Mathematical discipline,” and to deposit ten of those lectures in the library each year. The students were required to attend, but like all other requirements they ignored this one too. We know of only three people who attended a lecture at Cambridge by Newton. [Westfall 208–210; Works, 3, xv] *VFR

1675 Leibniz ﬁrst used the integral sign. Also ﬁrst used “d”. He also constructed what he calls the “triangulum characteristicum,” which had been used before him by Pascal and Barrow. [Cajori, History of Mathematical Notations, vol. 2, p. 2; Struik’s Source Book mistakenly has 26 October]

VFR Historical notes for the calculus classroom ,
In these same pages he will write examples of the integrals of x² and x³,and then illustrate that a constant multiple may be taken outside the integral as shown in the image below.

On the left is Liebniz integral sign with a vincula in place of todays parentheses to show that he is integrating the quantity (a/b) l Then the open bottomed box is Liebniz symbol for equality,then he shows the constant (a/b) multiplied by the integral of l .

1878 Patent issued for Odhner calculating machine. *VFR Willigot T. Odhner was granted a patent for a calculating machine that performed multiplications by repeated additions. The patent, a modified and compact version of Gottfried von Leibniz stepped wheel, was acquired and embodied in Brunsviga calculators that sold into 1950s.*CHM

1929 "Black Tuesday", the great USA stock market crash. About 16 million shares were traded, and the Dow lost an additional 30 points, or 12%.. "Anyone who bought stocks in mid-1929 and held onto them saw most of his or her adult life pass by before getting back to even." Richard M. Salsman *Wik

1985 On October 29th, 1985, the 329th birthday of Edmond Halley, the British threw a big party in honor of the return of Halley's Comet. The Halley's Comet Royal Gala was held at Wembley Conference Centre, London. It was a combination Variety Show and "Who's Who" in British Society, hosted by Princess Anne of the British Royal Family. *Joseph M. Laufer, Halley's Comet Society, USA

In 1991, space probe Galileo become the first human object to fly past an asteroid, Gaspra, making its closest approach at a distance of 1,604 km, passing at a speed of 8 km/sec (5 mi/sec). The encounter provided much data, including 150 images, which showed Gaspra has numerous craters indicating it has suffered numerous collisions since its formation. Gaspra is about 20-km long and orbits the Sun in the main asteroid belt between Mars and Jupiter. Gaspra, asteroid 951, was discovered by Ukrainian astronomer Grigoriy N. Neujamin (1916) who named it after a Black Sea retreat. In the photograph (left), subtle color variations have been exaggerated by NASA to highlight changes in reflectivity, surface structure and composition. *TIS

1998, Nearly four decades after he became the first American to orbit Earth, John Glenn is relaunched into space. *@HISTORYmag

BIRTHS 1897 Edwin James George Pitman was born in Melbourne on 29 October 1897 and died at Kingston near Hobart on 21 July 1993.
In 1920 he completed the degree course and graduated B.A. (1921), B.Sc. (1922) and M.A. (1923). In the meantime he was appointed Acting Professor of Mathematics at Canterbury College, University of New Zealand (1922-23). He returned to Australia when appointed Tutor in Mathematics and Physics at Trinity and Ormond Colleges and Part-time Lecturer in Physics at the University of Melbourne (1924-25). In 1926 Pitman was appointed Professor of Mathematics at the University of Tasmania, a position he held until his retirement in 1962.
Pitman described himself as 'a mathematician who strayed into Statistics'; nevertheless, his contributions to statistical and probability theory were substantial.
Pitman was active in the formation of the Australian Mathematical Society in 1956. He also took an active part in the Summer Research Institutes organized by the Mathematical Society, and used them as a sounding board for his research on statistical inference.
He was a renowned member of the Statistical Society of Australia, attending its biennial conferences. In 1978 the Statistical society established the Pitman Medal.
Pitman presented the first systematic account of non-parametric inference and lectured extensively on the subject, both in Australia and in the United States. The kernel of the subject, as described by him, is 'Suppose that the sum of two samples A, B is the sample C. Then A, B are discordant if A is an unlikely sample from C.' Again, he writes, 'The approach to the subject, starting from the sample and working towards the population instead of the reverse, may be a bit of a novelty'; and later, 'the essential point of the method is that we do not have to worry about the populations which we do not know, but only about the sample values which we do know'.
The notes of the 'Lectures on Non-parametric Inference' given in the United States, though never published, have been widely circulated and have had a major impact on the development of the subject. Among the new concepts introduced in these Lectures are asymptotic power, efficacy, and asymptotic relative efficiency.
A major contribution to probability theory is his elegant treatment of the behavior of the characteristic function in the neighborhood of the origin, in three papers. This governs such properties as the existence of moments. There are also interesting properties of the Cauchy distribution, and of subexponential distributions.
On his death, on 21 July 1993, Edwin was buried at the Hobart Regional Cemetery in Kingston. He lives on in the memory of many of us who are grateful for his life and legacy.
*Evan J. Williams, Australian Academy of Science

1925 Klaus Friedrich Roth (29 Oct 1925, )German-born British mathematician who was awarded the Fields Medal in 1958. His major work has been in number theory, particularly the analytic theory of numbers. He solved in the famous Thue-Siegel problem (1955) concerning the approximation to algebraic numbers by rational numbers (for which he won the medal). Roth also proved in 1952 that a sequence with no three numbers in arithmetic progression has zero density (a conjecture of Erdös and Turán of 1935).*TIS

DEATHS
1783 Jean le Rond D'Alembert (16 Nov 1717, 29 Oct 1783) was abandoned by his parents on the steps of Saint Jean le Rond, which was the baptistery of Notre-Dame, qv in Section 7-A-1. Foster parents were found and he was christened with the name of the saint. [Eves, vol. II, pp. 32 33. Okey, p. 297.] When he became famous, his mother attempted to reclaim him, but he rejected her. *VFR Known for his work in various fields of applied mathematics, in particular dynamics. In 1743 he published his Traité de dynamique (Treatise on Dynamics). The d'Alembert principle extends Newton's third law of motion, that Newton's law holds not only for fixed bodies but also for free moving bodies. D'Alembert also wrote on fluid dynamics, the theory of winds, the properties of vibrating strings and conducted experiments on the properties of sound . His most significant purely mathematical innovation was his invention and development of the theory of partial differential equations. He published eight volumes of mathematical studies (1761-80). He was editor of the mathematical and scientific articles for Denis Diderot's Encyclopédie.*TIS

1917 Giovanni Battista Guccia (21 Oct 1855 in Palermo, Italy - 29 Oct 1914 in Palermo, Italy) Guccia's work was on geometry, in particular Cremona transformations, classification of curves and projective properties of curves. His results published in volume one of the Rendiconti del Circolo Matematico di Palermo were extended by Corrado Segre in 1888 and Castelnuovo in 1897. *SAU

1921 Konstantin Alekseevich Andreev (26 March 1848 in Moscow, Russia - 29 Oct 1921 Near Sevastopol, Crimea) Andreev is best known for his work on geometry, although he also made contributions to analysis. In the area of geometry he did major pieces of work on projective geometry. Let us note one particular piece of work for which he has not received the credit he deserves. Gram determinants were introduced by J P Gram in 1879 but Andreev invented them independently in the context of problems of expansion of functions into orthogonal series and the best quadratic approximation to functions. *SAU

1931 Gabriel Xavier Paul Koenigs (17 January 1858 Toulouse, France – 29 October 1931 Paris, France) was a French mathematician who worked on analysis and geometry. He was elected as Secretary General of the Executive Committee of the International Mathematical Union after the first world war, and used his position to exclude countries with whom France had been at war from the mathematical congresses.
He was awarded the Poncelet Prize for 1913.*Wik

1933 Paul Painlevé worked on differential equations. He served twice as prime-minister of France. *SAU

1951 Robert Aitken (31 Dec 1864, 29 Oct 1951) American astronomer who specialized in the study of double stars, of which he discovered more than 3,000. He worked at the Lick Observatory from 1895 to 1935, becoming director from 1930. Aitken made systematic surveys of binary stars, measuring their positions visually. His massive New General Catalogue of Double Stars within 120 degrees of the North Pole allowed orbit determinations which increased astronomers' knowledge of stellar masses. He also measured positions of comets and planetary satellites and computed orbits. He wrote an important book on binary stars, and he lectured and wrote widely for the public. *TIS

1993 Lipman Bers (May 22, 1914 – October 29, 1993) was an American mathematician born in Riga who created the theory of pseudoanalytic functions and worked on Riemann surfaces and Kleinian groups.*Wik

1993 Robert Palmer Dilworth (December 2, 1914 – October 29, 1993) was an American mathematician. His primary research area was lattice theory; his biography at the MacTutor History of Mathematics archive states "it would not be an exaggeration to say that he was one of the main factors in the subject moving from being merely a tool of other disciplines to an important subject in its own right". He is best known for Dilworth's theorem (Dilworth 1950) relating chains and antichains in partial orders; he was also the first to study antimatroids (Dilworth 1940). Dilworth advised 17 Ph.D. students and as of 2010 has 373 academic descendants listed at the Mathematics Genealogy Project, many through his student Juris Hartmanis, a noted complexity theorist.*Wik