## Tuesday, 22 May 2018

### On This Day in Math - May 22

 The Le Petit Journal cover, on 1912 April 21, shows eclipse watchers in 1912 along with the solar eclipse of May 22, 1724, the previous total solar eclipse visible from Paris, France

“Biographical history, as taught in our public schools, is still largely a history of boneheads:
ridiculous kings and queens, paranoid political leaders, compulsive voyagers, ignorant generals
- the flotsam and jetsam of historical currents.
The men who radically altered history,
the great scientists and mathematicians,
are seldom mentioned, if at all.”
Martin Gardner

The 142nd day of the year; there are 142 possible planar graphs with six vertices.

142 is the smallest Semi-prime (having exactly 2 prime factors), whose sum of divisors is a cube. 142+71+2+1 = 63

The binary representation of 142 has the same number of zeros and ones.

142 is the number of ways of partitioning 25 into distinct parts... which must be the number of ways of partitioning them into odd parts according to Euler.

EVENTS
1453 A lunar eclipse fulfilled an omen for many prior to one of the red-letter dates in medieval history. On May 22, 1453 a partially eclipsed Moon rose over the city of Constantinople. One can only imagine the fear that it inspired in the embattled city that had already been under siege for a month. It certainly didn’t raise morale that legends had foretold that an eclipse would mark the fall of the Byzantine Empire. In a case of self-fulfilling prophecy, the city fell to the Ottomans seven days later.*listosaur.com

1649 Pascal obtained a monopoly by royal decree for his computing machine. [DSB 10, 332] *VFR

1724 A total solar eclipse occurred on May 22, 1724. A total solar eclipse occurs when the Moon's apparent diameter is larger than the Sun, blocking all direct sunlight, turning day into darkness.
This solar eclipse crossed the United Kingdom near sunset, north-west to south-east track, from southern Wales and Devon in the west, eastwards to Hampshire and Sussex, but passing to the south of London.
It crossed the city Los Angeles, CA in the morning, unfortunately it wasn't settled until after 1771, 47 years later. The next total eclipse over Los Angeles won't occur until April 1, 3290. *Wik

1788 William Herschel reads a paper to the Royal Society describing the observation of two satellites around the "Georgian Planet." *Phil. Trans. R. Soc. Lond. 1788 78, 364-378
On 13 March 1781, when he discovered the new celestial object  he named  it "Georgium Sidus (the Georgian planet)".  It would later become the planet Uranus.
Herschel even claimed in 1797 that he saw rings around the seventh planet and drew a small diagram of the ring and noted that it was "a little inclined to the red". For nearly two centuries the claim was dismissed as a mistake but in 1977 rings around Uranus were detected during an experiment.* bristol@bbc.co.uk

1849, Abraham Lincoln was issued a patent for "buoying boats over shoals" (No. 6,469). He was the first American president to receive a patent. (Note: he was NOT President in 1849) His idea utilized inflated cylinders to float grounded vessels through shallow water. Lincoln had worked as a deck-hand on a Mississippi flat-boat. *TIS

1866 Herman von Helmholtz published his paper “On the facts that underlie the foundations of geometry,” containing an account of elliptic geometry. *VFR

1906, the brothers Orville and Wilbur Wright received a patent for  "new and useful improvements in Flying Machines" (U.S. No. 821,393). This was the first airplane patent in the USA. *TIS

1936  M. C. ESCHER visited the Alhambra on 18‑24 Oct 1922 and was impressed by the patterns, but he didn't really use them in his art until after his second visit on 22-26 May 1936

1973 Robert Metcalfe wrote a memo describing a way to transmit data from the early generation of personal computers to a new device, the laser printer. He called his multipoint data communications system Ethernet, and today it continues to dominate as the standard computer network. A U.S. patent for "a Multipoint data communication system with collision detection" was issued 13 Dec 1977 ( 4,063,220) to Metcalfe, and others who developed the Ethernet. The patent was assigned to the Xerox Corporation. *TIS

1995 astronomers Amanda S. Bosh and Andrew S. Rivkin found two new moons of Saturn in photos taken by the Hubble Space Telescope. *TIS

1999 These beautiful magic squares, consisting of 11-digit palindromic primes, are by Carlos Rivera and Jaime Ayala. They were e-mailed to *Harvey Heinz, Magic-Squares.net

2010 A Pac-Man Mini version, originally created by Google as an animated logo for the game's 30th anniversary on May 22, 2010. Play it here.

BIRTHS

1783 William Sturgeon  (22 May 1783 - 4 December 1850) English electrical engineer who devised the first electromagnet capable of supporting more than its own weight (1825). The 7-oz (200-g) magnet supported 9-lb (4-kg) of iron with a single cell's current. He built an electric motor (1832) and invented the commutator, now part of most modern electric motors. In 1836, he invented the first suspended coil galvanometer, a device for measuring current. Sturgeon also worked on improving the voltaic battery, developing a theory of thermoelectricity, and even atmospheric charge conditions. From 500 kite flights made in calm weather, he found the atmosphere is consistently charged positively with respect to the Earth, and increasingly so at increased height. *TIS

1848 Hermann Schubert (22 May 1848 in Potsdam, Germany – 20 July 1911 in Hamburg, Germany) worked on parts of algebraic geometry that involve a finite number of solutions. This is called Enumerative Geometry. *SAU

1865 Alfred Cardew Dixon (22 May 1865 in Northallerton, Yorkshire, England - 4 May 1936 in Northwood, Middlesex, England) Alfred Dixon graduated from London and Cambridge and then had professorial appointments in Galway

1903 Yves-André Rocard  (Vannes, 22 May 1903 – 16 March 1992 in Paris)  French mathematician and physicist who contributed to the development of the French atomic bomb and to the understanding of such diverse fields of research as semiconductors, seismology, and radio astronomy. During WW II, as Head of the Research Department of the Free French Naval Forces in England, he learnt about radars in England and interference from strong radio emission from the Sun. After the war, Rocard returned to France and proposed that France started a project to conduct radio astronomy. In the last part of his life he studied biomagnetism and dowsing which reduced his standing in the eyes of many of his colleagues. *TIS

1916 Albrecht Fröhlich FRS (22 May 1916 – 8 November 2001) was a mathematician famous for his major results and conjectures on Galois module theory in the Galois structure of rings of integers.
He was born in Munich to a Jewish family. He fled from the Nazis to France, and then to Palestine. He went to Bristol University in 1945, gaining a Ph.D in 1951 with a dissertation entitled On Some Topics in the Theory of Representation of Groups and Individual Class Field Theory under the supervision of Hans Heilbronn. He was a lecturer at the University of Leicester and then at the Keele University, then in 1962 moved as reader to King's College London where he worked until his retirement in 1981 when he moved to Robinson College, Cambridge.
He was elected a Fellow of the Royal Society in 1976. He was awarded the Berwick Prize of the London Mathematical Society in 1976 and its De Morgan Medal in 1992. The Society's Fröhlich Prize is named in his honour.
He is the brother of Herbert Fröhlich. *Wik

1920 Thomas Gold (22 May 1920; 22 Jun 2004 at age 84) Austrian-British-American astronomer known for a steady-state theory of the universe, explaining pulsars, and naming the magnetosphere. In 1948, as a graduate student at Cambridge, he (together with Hermann Bondi and Fred Hoyle) proposed that, a continuous creation of matter in space is gradually forming new galaxies, maintaining the average number of galaxies in any part of the universe, despite its expansion. This is not accepted, as there is more evidence for the Big Bang theory. In 1967, Gold presented his theory on the nature of pulsars (objects in deep space that produce regularly pulsing radio waves). He suggested that they were rotating neutron stars - tiny, extraordinarily massive stars - which emit waves as they spin. *TIS

DEATHS

1626 Caspar Schott SJ, and Gaspar Schott or Kaspar Schott (February 5 1608 in Königshofen, May 22 1666 in Würzburg) was a scientific author and educator.
Schott attended the Würzburg Jesuit High School and entered the Order in 1627. During his studies in Würzburg one of his teachers was Athanasius Kircher. When the Jesuits fled before the approaching Swedish army in 1631,Schott went to Palermo to complete his studies. He stayed in Sicily 20 years as a teacher of mathematics, philosophy, moral theology at the Jesuit school in Palermo. In 1652 was sent to Rome as support in the scientific work of Kircher. He decided to publish Kircher's work. In 1655, he returned as Professor in the Würzburg school, where he taught mathematics and physics. He was Hofmathematker and confessor of the Elector Johann Philipp von Schönborn who had just purchased the vacuum pump invented by Otto von Guericke and used at Magdburg.
He corresponded with leading scientists including Otto von Guericke, Christiaan Huygens, and Robert Boyle . The term "technology" was probably invented by Schott in his "Technica curiosa" which inspired Boyle and Hooke's vacuum experiments.
In the posthumously published work Organum mathematicum he describes his Cistula invented by him, a computing device with which you can multiply and divide. *Wik

1868 Julius Plücker (16 June 1801 – 22 May 1868)  German mathematician and physicist whose work suggested the far-reaching principle of duality, which states the equivalence of certain related types of theorems. He also discovered that cathode rays (electron rays produced in a vacuum) are diverted from their path by a magnetic field, a principle vital to the development of modern electronic devices, such as television. At first alone and later with the German physicist Johann W. Hittorf, Plücker made many important discoveries in spectroscopy. Before Bunsen and Kirchhoff, he announced that spectral lines were characteristic for each chemical substance and this had value to chemical analysis. In 1862 he pointed out that the same element may exhibit different spectra at different temperatures. *TIS

1967 Josip Plemelj (December 11, 1873 – May 22, 1967) was a Slovene mathematician, whose main contributions were to the theory of analytic functions and the application of integral equations to potential theory. He was the first chancellor of the University of Ljubljana.*Wik

1974 Irmgard Flugge-Lotz (6 July 1903 - 22 May 1974)born in Hameln, Germany. Her father encouraged her in mathematics, but she chose engineering because “I wanted a life which would never be boring—a life in which new things would always occur.” She studied applied mathematics at the Technical University of Hanover and in 1929 she became a Doktor-Ingenieur, the equivalent of an American Ph.D. in Engineering. She made contributions to aerodynamics, control theory, and ﬂuid mechanics. In 1960 she became full professor at Stanford. *WM

1991 Derrick Lehmer  (February 23, 1905 – May 22, 1991) , one of the world's best known prime number theorists, born in Berkeley, California. Before World War II, Lehmer invented a number of electromechanical sieves for finding prime numbers and made many important contributions in prime number theory throughout his life. Prime numbers are of interest in themselves as mathematical curiosities but are also of great importance to cryptography. The Computer Museum History Center has three Lehmer sieves in its permanent collection. Lehmer died in 1991.*CHM Lehmer's peripatetic career as a number theorist, with he and his wife taking numerous types of work in the United States and abroad to support themselves during the Great Depression, fortuitously brought him into the center of research into early electronic computing.His father Derrick Norman Lehmer, known mainly as a pioneer in number theory computing, also made major contributions to combinatorial computing. *Wik

2009 Walter Ledermann (18 March 1911 in Berlin, Germany - 22 May 2009 in London, England) graduated from Berlin but was forced to leave Germany in 1933 to avoid Nazi persecution. He came to St Andrews and studied under Turnbull. He worked at Dundee and St Andrews until after World War II when he moved to Manchester and then to the University of Sussex. He is especially known for his work in homology, group theory and number theory. *SAU

2010  Martin Gardner (October 21, 1914 – May 22, 2010) died.  Gardner more or less single-handedly sustained and nurtured interest in recreational mathematics in the U.S. for a large part of the 20th century. He is best known for his decades-long efforts in popular mathematics and science journalism, particularly through his "Mathematical Games" column in Scientific American. *Wik
It is said that Gardner "Turned children into mathematicians and mathematicians into children.".. For some of us he did each in turn.  More than any classroom teacher I ever had, Martin Gardner shaped my mathematical interests. "For 35 years, he wrote Scientific American's Mathematical Games column, educating and entertaining minds and launching the careers of generations of mathematicians"
Only two days before I learned of his death, I stood in the front yard of my Mother's home in Fort Worth and told Alex, my sister's grandson, aged 12, that if he wanted to nurture his curiosity for math and science he should find anything in the library by Martin Gardner and read it every year for the next ten years of his life, and each year, I promised, he would find something new in the reading.
I can not do justice to the life of a man who was the mathematical Pied-Piper of mathematics for a generation of us; so here is link to the article in Scientific American.

Credits :
*CHM=Computer History Museum
*FFF=Kane, Famous First Facts
*NSEC= NASA Solar Eclipse Calendar
*RMAT= The Renaissance Mathematicus, Thony Christie
*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, 21 May 2018

### On This Day in Math - May 21

Whoever ... proves his point
and demonstrates the prime truth geometrically
should be believed by all the world,
for there we are captured
~Albrecht Durer

The 141st day of the year; 141 is the first non-trivial palindrome appearing in the decimal expansion of Pi, appearing immediately after the decimal point, 3.14159. Tanya Khovanova, Number Gossip

141 is the second n to give a prime Cullen number (of the form n*2n + 1). Cullen numbers were first studied by Fr. James Cullen in 1905. (That prime is 393050634124102232869567034555427371542904833,

141 is the number of lattice paths from (0,0) to (6,6) using steps (2,0), (0,2), (1,1).

EVENTS

1728 The term "mathematical expectation, "l'espérance mathématique," with its modern meaning is found in a letter by Gabriel Cramer to Nicholas Bernoulli.
The first use in English seems to be in A. de Morgan's Essay on Probabilities (1838, p. 97), "The balance is the average required, and is known by the name of the mathematical expectation." (OED). *Jeff Miller, Earliest Known Uses of Some of the Words of Mathematics

1819 the first bicycle in the U.S. was seen in New York City. Such bicycle velocipedes or "swift walkers" had been imported that same year. Shortly thereafter, on 19 Aug 1819, the city's Common Council passed a law to "prevent the use of velocipedes in the public places and on the sidewalks of the city of New York."**TIS (Skateborders take note, you are not the first to be banned from the sidewalks)

1908 Glenn (Hammond) Curtiss was a pioneer in the development of U.S. aviation whose aircraft were widely used during World War I. That the Wrights made the first powered flights has generally been accepted, but the achievements of Curtiss spanned several decades and took the airplane from its wood, fabric and wire beginnings to the forerunners of modern transport aircraft. Curtiss made his first flight on his 30th birthday, 21 May 1908, in White Wing, a design of the Aerial Experiment Association, a group led by Alexander Graham Bell. White Wing was the first plane in America to be controlled by ailerons instead of the wing-warping used by the Wrights. It was also the first plane on wheels in the U.S. *TIS (See 1878 Birth below)

1901 the first U.S. State motor car legislation was an act to regulate the speed of motor vehicle, passed in Connecticut. A limit was established of 12 mph within city limits and 15 mph outside, which were higher than the 8 mph city and 12mph country speeds in the bill as originally presented. Also, the car driver was required to reduce speed upon meeting or passing a horse-drawn vehicle, and if necessary, to stop to avoid frightening the horse.*TIS

This last part about meeting (or passing) a horse, with or without cart, is still essentially the law in England and Ireland.

1916 Daylight Saving Time was introduced in Britain as a war-time measure to save fuel. The idea began when a London builder, William Willett, presented a scheme of shifting the clock to better use the hours of daylight in summer. He campaigned and published a brochure on the subject in 1907 (in which his proposal was to adjust the clocks in four weekly adjustments of 10-mins). When Parliament did consider a Daylight Saving Bill, to implement a seasonal one-hour change, it failed for lack of support. However, a little more than a year after his death after his death, the idea was finally adopted during WW I for wartime fuel savings. Now most of the countries in the northern hemisphere use a form of daylight saving time. *TIS

1932 Amelia Earhart ﬂew alone across the Atlantic, being the ﬁrst woman to do so. *VFR

1952 IBM Announces Model 701, "Defense Calculator.":
IBM announced its 701 machine and by doing so emphasized its commitment to innovation in electronic computing. The company's first computer designed for scientific computations. The IBM 701 had an electrostatic storage tube memory and kept information on magnetic tape. The company eventually sold 19 of the machines -- more than expected -- to the government and large companies and universities for complex research.*CHM

BIRTHS

429 B.C. Plato born in Athens. He died on the same date in 348 B.C. [Muller] [Should it be 427 B.C.?] *VFR

1471 Albrecht Durer, (21 May 1471 – 6 April 1528) German painter and engraver. Mathematicians are fond of his etching Melancholia for it contains the magic square. Oldstyle numerals are used in the two center squares to emphacize the year that this etching was done by Durer. There is still debate about the shape of the solid in the foreground of the picture. *TIS He also published a book on geometric constructions (1525) using a straight-edge and compass. Although designed to enable artists better represent a natural three-dimensional scene on a canvas, Dürer included careful proofs to establish the validity of the constructions. In this respect, it could be regarded as the oldest surviving text on applied mathematics. He also wrote on the proportions of the human body. *TIS  At a blog by Richard Elwes I found out that Durer was also was one of the first to create a fractal image. In reference to some snowflake fractals at walking randomly he writes,  "It was Dürer who first discovered them, in the second volume of his work Underweysung der Messung (‘Instruction in measurement’) in 1525 (almost 400 years before the discovery of the Koch snowflake)."  He also let me know that "Durer had a hand in the invention of nets, and the rediscovery of Archimedan solids." Thanks Richard. The Renaissance Mathematicus pointed out in his blog that Durer's geometry book was the first true math book printed in the German language.

A close up of the magic square

1792 Gustave-Gaspard Coriolis  (21 May 1792 – 19 September 1843) French engineer and mathematician who first described the Coriolis force, an effect of motion on a rotating body, of paramount importance to meteorology, ballistics, and oceanography. Whereas pressure differences tend to push winds in straight paths, winds follow curved paths across the Earth. In 1835, Coriolis first gave a mathematical description of the effect, giving his name to the Coriolis force. While air begins flowing from high to low pressure, the Earth rotates under it, thus making the wind appear to follow a curved path. In the Northern Hemisphere, the wind turns to the right of its direction of motion. In the Southern Hemisphere, it turns to the left. The Coriolis force is zero at the equator. *TIS

My high-school science teacher told me that the Coriolis effect explains why bathtubs in the Northern Hemisphere drain in a clockwise swirl ... John Cook explains that is a science myth.

1839 Nils Christofer Dunér (21 May 1839; 10 Nov 1914 at age 75) Swedish astronomer who studied the rotational period of the Sun. Although his PhD thesis had been theoretical (the orbit of asteroid Panopea), Dunér mostly worked as an observer. The most outstanding observing astronomer in Swedish 19th century astronomy, he is mostly known for his introduction of new astrophysical techniques. In 1867-75, he made 2679 micrometer measurements of 445 double and multiple stars. After publishing his catalogue of double star measurements in 1876, Dunér turned to spectroscopy, at first specializing in the spectra of red stars. Later, by measuring the Doppler shift of the spectral lines of light from the approaching and receding edges of the sun, he made the significant discovery that the rotational period differs from about 25.5 days near the Sun's equator but up to 38.5 days near the Sun's poles. His career spanned over almost 50 years, from classical astronomy to astrophysics. *TIS

1847 Antonio Favaro, (21 May, 1847 - ? 1922) Professor of Projective Geometry at Padua, editor of  the works of Galileo after a labor of thirty years.

1858 Édouard (-Jean-Baptiste) Goursat - (21 May 1858 – 25 November 1936) French mathematician and theorist whose contribution to the theory of functions, pseudo- and hyperelliptic integrals, and differential equations influenced the French school of mathematics. The Cauchy-Goursat theorem states the integral of a function round a simple closed contour is zero if the function is analytic inside the contour. Cauchy had established the theorem with the added condition that the derivative of the function was continuous. In 1891, he wrote Leçons sur l'intégration des équations aux dérivées partielles du premier ordre. Goursat's best known work is Cours d'analyse mathématique (1900-10) which introduced many new analysis concepts. *Wik

1878 Glenn (Hammond) Curtiss (May 21, 1878 – July 23, 1930) was a pioneer in the development of U.S. aviation.. (see 1908 in Events above)

1923 Armand Borel (21 May 1923 –11 August 2003) was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993. He worked in algebraic topology, in the theory of Lie groups, and was one of the creators of the contemporary theory of linear algebraic groups. *Wik

1958 Curtis Tracy McMullen (21 May 1958- ) is Professor of Mathematics at Harvard University. He was awarded the Fields Medal in 1998 for his work in complex dynamics, hyperbolic geometry and Teichmüller theory.
McMullen graduated as valedictorian in 1980 from Williams College and obtained his Ph.D. in 1985 from Harvard University, supervised by Dennis Sullivan. He held post-doctoral positions at the Massachusetts Institute of Technology, the Mathematical Sciences Research Institute, and the Institute for Advanced Study, after which he was on the faculty at Princeton University (1987–1990) and the University of California, Berkeley (1990–1997), before joining Harvard in 1997. He received the Salem Prize in 1991 and was elected to the National Academy of Sciences in 2007.
McMullen also has given a proof that backgammon ends with probability one*Wik

DEATHS

1670 Niccolò Zucchi (December 6, 1586 – May 21, 1670) Italian astronomer who, in approximately 1616, designed one of the earliest reflecting telescopes, antedating those of James Gregory and Sir Isaac Newton. A professor at the Jesuit College in Rome, Zucchi developed an interest in astronomy from a meeting with Johannes Kepler. With this telescope Zucchi discovered the belts of the planet Jupiter (1630) and examined the spots on Mars (1640). He also demonstrated (in 1652) that phosphors generate rather than store light. His book Optica philosophia experimentalis et ratione a fundamentis constituta (1652-56) inspired Gregory and Newton to build improved telescopes. *TIS

1686 Otto von Guericke (originally spelled Gericke) (November 20, 1602 – May 11, 1686 (Julian calendar); November 30, 1602 – May 21, 1686 (Gregorian calendar)) was a German scientist, inventor, and politician. He is best remembered for his invention of the Magdeburg hemispheres, popularized in the writings of Caspar Schott. His major scientific achievements were the establishment of the physics of vacuums, the discovery of an experimental method for clearly demonstrating electrostatic repulsion, and his advocacy of the reality of "action at a distance" and of "absolute space". *Wik

1825 William Nicholson (13 December 1753 – 21 May 1815) was a renowned English chemist and writer on "natural philosophy" and chemistry, as well as a translator, journalist, publisher, scientist, inventor, patent agent and civil engineer.
In 1797 he began to publish and contribute to the Journal of Natural Philosophy, Chemistry and the Arts, generally known as Nicholson's Journal, the earliest monthly scientific work of its kind in Great Britain— the publication continued until 1814. The journal included the first comprehensive descriptions of aerodynamics with George Cayley's "On Aerial Navigation", which inspired the Wright brothers a hundred years later. In May 1800 he with Anthony Carlisle discovered electrolysis, the decomposition of water into hydrogen and oxygen by voltaic current. The two were then appointed to a chemical investigation committee of the new Royal Institution. But his own interests shortly turned elsewhere.
Besides considerable contributions to the Philosophical Transactions, Nicholson wrote translations of Fourcroy's Chemistry (1787) and Chaptal's Chemistry (1788), First Principles of Chemistry (1788) and a Chemical Dictionary (1795); he also edited the British Encyclopaedia, or Dictionary of Arts and Sciences (6 vols., London, 1809).
Nicholson died in Bloomsbury at the age of 61 on 21 May 1815. *Wik

1826 Georg von Reichenbach (July 21, 1771 – May 21, 1826) German maker of astronomical instruments who introduced the meridian, or transit, circle, (above) a specially designed telescope for measuring both the time when a celestial body is directly over the meridian (the longitude of the instrument) and the angle of the body at meridian passage. By 1796 he was engaged in the construction of a dividing engine, a machine used to mark off equal intervals accurately, usually on precision instruments. *TIS

1848 Pierre Laurent Wantzel (June 5, 1814 in Paris – May 21, 1848 in Paris) was a French mathematician who proved that several ancient geometric problems were impossible to solve using only compass and straightedge. In a paper from 1837,( "Recherches sur les moyens de reconnaître si un Problème de Géométrie peut se résoudre avec la règle et le compas". Journal de Mathématiques Pures et Appliquées) he proved that the problems of
1. doubling the cube
2. trisecting the angle and
3. constructing a regular polygon whose number of sides is not the product of a power of two and any number of distinct Fermat primes (i.e. that does not fulfill the same conditions proven to be sufficient by Carl Friedrich Gauss) the solution to which had been sought for thousands of years, particularly by the ancient Greeks, were all impossible to solve if one uses only compass and straightedge. *Wik

1911 Williamina Paton Stevens Fleming (15 May 1857 - 21 May 1911 at age 53) was a Scottish-American astronomer (née Stevens) who pioneered in the classification of stellar spectra and the first to discover stars called "white dwarfs." She emigrated to Boston at age 21. Prof. Edward Pickering, director of the Harvard Observatory first employed Fleming as a maid, but in 1881 hired her to do clerical work and some mathematical calculations at the Observatory. She further proved capable of doing science. After devising her system of classifying stars by their spectra, she cataloged over 10,000 stars within the next nine years. Her duties were expanded and she was put in charge of dozens of young women hired to do mathematical computations (as now done by computers).*TIS

1953 Ernst Friedrich Ferdinand Zermelo (July 27, 1871 – May 21, 1953) was a German mathematician, whose work has major implications for the foundations of mathematics and hence on philosophy. He is known for his role in developing Zermelo–Fraenkel axiomatic set theory and his proof of the well-ordering theorem.*Wik

1957 Aleksandr Ivanovich Nekrasov (9 Dec 1883 in Moscow, Russia - 21 May 1957 in Moscow, Russia) Nekrasov published important work on the theory of waves, the theory of whirlpools, the theory of jet streams and gas dynamics. He also investigated mathematical questions which were related to these applications, in particular writing important works on non-linear integral equations. In fact his deep understanding of mathematical analysis as developed by mathematicians such as Goursat enabled him to succeed in solving a whole range of concrete problems. *SAU

1958 Wilhelm Süss (7 March 1895 - 21 May 1958) was a German mathematician. He was born in Frankfurt, Germany and died in Freiburg im Breisgau, Germany. He was founder and first director of the Mathematical Research Institute of Oberwolfach.*Wik

1964 James Franck (26 Aug 1882; 21 May 1964) German-born American physicist who shared the Nobel Prize for Physics in 1925 with Gustav Hertz for research on the excitation and ionization of atoms by electron bombardment that verified the quantized nature of energy transfer.*TIS
In 1933, after the Nazis came to power, Franck, being a Jew, decided to leave his post in Germany and continued his research in the United States, first at Johns Hopkins University in Baltimore and then, after a year in Denmark, in Chicago. It was there that he became involved in the Manhattan Project during World War II; he was Director of the Chemistry Division of the Metallurgical Laboratory[5] at the University of Chicago. He was also the chairman of the Committee on Political and Social Problems regarding the atomic bomb; the committee consisted of himself and other scientists at the Met Lab, including Donald J. Hughes, J. J. Nickson, Eugene Rabinowitch, Glenn T. Seaborg, J. C. Stearns and Leó Szilárd. The committee is best known for the compilation of the Franck Report, finished on 11 June 1945, which recommended not to use the atomic bombs on the Japanese cities, based on the problems resulting from such a military application.
When Nazi Germany invaded Denmark in World War II, the Hungarian chemist George de Hevesy dissolved the gold Nobel Prizes of Max von Laue and James Franck in aqua regia to prevent the Nazis from stealing them. He placed the resulting solution on a shelf in his laboratory at the Niels Bohr Institute. After the war, he returned to find the solution undisturbed and precipitated the gold out of the acid. The Nobel Society then recast the Nobel Prizes using the original gold.*Wik

1973 Grigore Constantin Moisil (10 January 1906 in Tulcea, Romania – 21 May 1973 in Ottawa, Canada) was a Romanian mathematician, computer pioneer, and member of the Romanian Academy. His research was mainly in the fields of mathematical logic, (Łukasiewicz-Moisil algebra), Algebraic logic, MV-algebra, algebra and differential equations. He is viewed as the father of computer science in Romania. *Wik

Credits :
*CHM=Computer History Museum
*FFF=Kane, Famous First Facts
*NSEC= NASA Solar Eclipse Calendar
*RMAT= The Renaissance Mathematicus, Thony Christie
*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

## Sunday, 20 May 2018

### White Rabbit Mathematics, Extended

***I first wrote most of this post in 2008, but today an event reminded me of it, and so I thought I would add on to this old, but still interesting post with an additional interesting connection.

One of the things that amazes me, and I think most people who are attracted to math, is the mysterious way that different parts of math come together in unexpected ways. I tried to explain this to someone once using a literary analogy..."It is as if you were reading along in some great drama, or trying to understand the message in some grand poem, and suddenly the White Rabbit from Alice in Wonderland comes running through muttering, "Oh dear! Oh dear! I shall be too late!"
It is not the White Rabbit you see in math, but the effect is the same. Euler must have felt that feeling after he struggled to find the value of the series $\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2}+ ...$.. and finds that it turns out to be $\frac{\pi^2}{6}$. Wait.... Pi is the ratio of the circumference to the diameter of a circle, but there are no circles in the sum of the squares of the reciprocals of the integers; and yet, there it is, the mathematical white rabbit coming seemingly from nowhere. Certainly none of the many mathematicians of great repute who had worked on the problem found (or expected) Pi to appear.

The normal distribution is another example; De Moivre takes the binomial probability distribution for flipping a coin and generalizes it toward an infinite number of flips, and POW, the normal or bell-shaped curve that is ubiquitous in intro stats. And what happens? Right there in the middle, the height of the normal curve at Z=0 is .39894... No, NO, NO, NOT JUST .39894.. but the .39894... that is exactly equal to $\frac{1}{\sqrt{2 \pi}}$

Ok, so what brought this sudden rebirth of excitement about mathematical interrelationships? Well recently I came across a blog that referred to another blog that (as these things sometimes do) led me to a paper on just such a mathematical "white rabbit". The paper was about partitions of numbers as powers of two (1, 2, 4, 8, 16, etc..)
It began with a simple question, what is the number of ways to write a number n as a sum of powers of two if each value can be expressed no more than two times. For example, we could express 4 as 4, or as 2+2, or as 2 + 1 + 1 since each value is a power of two, and none appears more than twice. You couldn't use 1+1+1+1 since it appears more than twice. For n= 4 it turns out that the number of partitions, as shown above, is three. If we assume that there is one way to express zero, and one way to express one, and figure out the others we get a string like this

1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4, 7,..
Ok, you don't see a white rabbit yet... but then someone ask you a different question. Is it possible to write out ALL the rational numbers in simplified form without repeating any of them. The answer is "Yes, of course, see the list above."
"What?", you ask, "How?", but there it is... The sequence of rational numbers is formed by taking each of the numbers to be the numerator, and using the number behind it to be the denominator. 1/1; 1/2; 2/1; 1/3; 3/2; ... and you never get a repeat, never get an unsimplified form, and you eventually get them ALL, the entire Infinite Set.....
No way you would expect that partitions of powers of two should give you the rational numbers in their entirety... there is (it would seem) nothing to relate the two questions... and yet... there it is. I think that is what makes math the most exciting area of study in the world.
Prove it you say? Nope, In truth I ain't man enough, but you can find the entire paper
Recounting the rationals, by Neil Calkin and Herb Wilf. Read their proof and Enjoy.

*** So today I was catching up on some old audio podcasts from "My Favorite Theorem," and Jordan Ellenberg   was explaining his choice of a special part of Fermat's Little Theorem, that for any prime p, $2^p \equiv 2 Mod p$.   (or in very primitive terms, if you divide 2p by p, you always get a remainder of 2.  I wondered why he found that so interesting, but then he hit me with, "you can discover at least that it’s true on your own, for instance by messing with Pascal’s Triangle, for example." And of course, in a moment I realized yes, Fermat's Little Theorem, at least this limited case, is elementary true by looking at the rows of Pascal's Triangle. The sum of all the elements of any row add up to a power of two, and the pth row has a sum of 2p. But look at some prime row.....

the 3rd has 1,3,3,1 ;

the fifth has 1,5,10,10,5, 1 ;

and the 7th has 1, 7, 21, 35, 35, 21, 7, 1....

In each row, all the entries are divisible by p, except the two ones. Scan the rest and you notice the same thing. And just importantly, you don't have to go very far to see an exception for the non-primes.

Math has those White Rabbits everywhere.

### On This Day in Math - May 20

Mathematicians are like Frenchmen:
whatever you say to them
they translate into their own language
and forthwith it is something entirely different.
-- Johann Wolfgang von Goethe (Maxims and Reflexions, 1829)

The 140th day of the year; 140 is the sum of the squares of the first seven positive integers. 12 + 22 + 32 + 42 + 52 + 62 + 72 = 140. *Prime Curios

140 is a repdigit in bases 13 (aa), 19(7,7), 27(5,5), 34(4,4), 69(2,2), and 139(1,1).

There are 140 x 1021 (140 followed by 21 zeroes) different configurations of the Rubik's Cube. *Cliff Pickover@pickover

140 is the character limit on Twitter (or was)

A. J. Meyl proved in 1878 that only three tetrahedral numbers are also perfect squares,The largest of these is T(48) =1402 = 19600:

T1 = 1² = 1
T2 = 2² = 4
T48 = 140² = 19600.

EVENTS

1570 Cartographer Abraham Ortelius issues Theatrum Orbis Terrarum, the first modern atlas. *RMAT

 * National Maritime Museum
1608 In a letter to Christopher Clavius, Croation mathematician Marino Ghetaldi says that with his latest parabolic mirror:-
... the sun melts not only lead, but silver.
Ghetaldi had traveled extensively throughout Europe visiting and studying with many of the great science minds, including Viete and Galileo. From Galileo he learned optics and produced a 66cm diameter parabolic mirror which is at the National Maritime Museum in London.
*SAU

1663 Robert Hooke was one of 98 persons who were declared members at a meeting of the Royal Society. He was admitted to society on 3 Jun 1663, and was peculiarly exempted of all payments. Before the Royal Society had been establish in 1660, Hooke was already distinguished for the invention of various astronomical instruments, and the air-pump he contrived for Charles Boyle (whom he had assisted for several years with chemical experiments at the Philosophical Society, Oxford). He invented a balance or pendulum spring (1656-58), one of the greatest improvements in the construction of timepieces. By 1662, he had been appointed curator of experiments to the Royal Society, and on 11 Jan 1664, awarded a salary of £30 per annum for life for that position.*TIS

1665 Newton's earliest use of dots, "pricked letters," to indicate velocities or fluxions is found on a leaf dated May 20, 1665; no facsimile reproduction of it has ever been made.' The earliest printed account of Newton's fluxional notation appeared from his pen in the Latin edition of Wallis' Algebra [Cajori, History of Mathematical Notations, vol. 2, p. 197] *VFR

1716 In a letter written to Leibniz, May 20, 1716, John Bernoulli discussed the equation:d2y/dx2 = 2y/x2
where the general solution when written in the form
y = x2/a + b2/3x
involves three cases: When b approaches zero the curves are parabolas; when a approaches infinity, they are hyperbolas; otherwise, they are of the third order. *John E. Sasser, HISTORY OF ORDINARY DIFFERENTIAL EQUATIONS THE FIRST HUNDRED YEARS

1875 The International Bureau of Weights and Measures established by the International Metric Convention, Sevres, France. The bureau is the repository for the “International Prototype Meter” and the “International Prototype Kilogram.” *SAU= St. Andrews Univ

And for your (in case you thought that 3D movie technology was new, file)
In 1901, Claude Grivolas, one of Pathe's main shareholders in Paris, France, invented a projector that produced three-dimensional pictures.*TIS

1927 At 7:40 a.m., Charles Lindbergh took off from Roosevelt Field in Long Island, N.Y., aboard the "Spirit of St. Louis" monoplane on his historic first solo flight across the Atlantic Ocean. He arrived in France thirty-three and one-half hours later. *TIS

1930 The Institute for Advanced Study incorporated. Two and a half years later Albert Einstein and Oswald Veblen were appointed the ﬁrst professors. [Goldstein, The Computer from Pascal to von Neumann, p. 77]*VFR

In 1956, the first hydrogen fusion bomb (H-bomb) to be dropped from an airplane exploded over Namu Atoll at the northwest edge of the Bikini Atoll. The fireball was four miles in diameter. It was designated as "Cherokee," as part of "Operation Redwing."*TIS

1961 France issued a stamp honoring Charles Coulomb (1736–1806) [Scott #B 352].

1968 A team of six high school students from Upstate New York went to London to participate in the Fourth British Mathematical Olympiad. This was the ﬁrst time a team from the U.S. participated in an international mathematical competition. [The College Mathematics Journal, 16 (1985), p. 331] *VFR

1975 Norway issued a stamp for the centenary of the International Meter Convention in Paris. It pictures Ole Jacob Broch (1818– 1889), the ﬁrst director of the International Bureau of Weights and Measures. [Scott #655] *VFR At least ten other countries issued stamps to commemorate the same event, including Bulgaria, Romania, France, the Soviet Union..... but not the USA. (see 1975 below for another)

1975 Sweden issued a stamp picturing a metric tape measure to honor the centenary of the International Meter Convention in Paris. [Scott #1121] *VFR

1990 the Hubble Space Telescope sent its first photograph from space, an image of a double star 1,260 light years away. *TIS

2012 Solar Eclipse, A total of 154 U.S. national parks will provide views of the eclipse, from partial to full annularity. Many western parks will offer solar observing as a ranger-led program or host a solar party with the help of local amateur astronomy clubs.

Poster advertising viewing the eclipse from Glen Canyon National Recreation Area. Poster © Tyler Nordgren.

2063 99 year old Johnny Depp rolls down red-carpet in wheel-chair for opening of Pirates of the Caribbean sequel #42, “Sun City Pirates”.

BIRTHS

1782 William Emerson (14 May 1701 – 20 May 1782), English mathematician, was born at Hurworth, near Darlington, where his father, Dudley Emerson, also a mathematician, taught a school. William himself had a small estate in Weardale called Castle Gate situated not far from Eastgate where he would repair to work throughout the Summer on projects as disparate as stonemasonry and watchmaking. Unsuccessful as a teacher, he devoted himself entirely to studious retirement. Possessed of remarkable energy and forthrightness of speech, Emerson published many works which are singularly free from errata.
He was early influence in the life of Jeremiah Fenwicke Dixon, of Mason-Dixon fame.

In The Principles of Mechanics (1754) he shows a wind-powered vehicle in which the vertically mounted propeller gives direct power to the front wheels via a system of cogs. In mechanics he never advanced a proposition which he had not previously tested in practice, nor published an invention without first proving its effects by a model. He was skilled in the science of music, the theory of sounds, and the ancient and modern scales; but he never attained any excellence as a performer. He died on 20 May 1782 at his native village, where his gravestone bears epitaphs in Latin and Hebrew.

Emerson dressed in old clothes and his manners were uncouth. He wore his shirt back to front and his legs wrapped in sacking so as not to scorch them as he sat over the fire. He declined an offer to become FRS because it would cost too much after all the expense of farthing candles he had been put to in the course of his life of study. Emerson rode regularly into Darlington on a horse like Don Quixote's, led by a hired small boy. In old age, plagued by the stone, he would alternately pray and curse, wishing his soul 'could shake off the rags of mortality without such a clitter-me-clatter.' *Wik

1825 George Phillips Bond ((May 20, 1825 {sometimes given 1826} – February 17, 1865) Astronomer who made the first photograph of a double star, discovered a number of comets, and with his father discovered Hyperion, the eight moon of Saturn.*TIS

1861 Henry Seely White (May 20, 1861 - May 20, 1943) worked on invariant theory, the geometry of curves and surfaces, algebraic curves and twisted curves. *SAU He matriculated at Wesleyan University in Connecticut and graduated with honors in 1882 at the age of twenty-one. White excelled at Wesleyan in astronomy, ethics, Latin, logic, mathematics, and philosophy. At the university, John Monroe Van Vleck taught White mathematics and astronomy. Later, Van Vleck persuaded White to continue to study mathematics at the graduate level.[1] Subsequently, White studied at the University of Göttingen under Klein, and received his doctorate in 1891.
White was Mathematics Department Chair at Northwestern University. He left Northwestern to be near his ill mother and became Chairman of the Mathematics Department at Vassar College. He "attributed his interest in geometry both to his work at Wesleyan and Goettingen and to summers spent working on his grandfather’s farm."[2] His particular interests were in the fields of the geometry of curves and surfaces (Curves, Differential geometry of surfaces), algebraic planes and twisted curves (Algebraic Geometry, Algebraic curves, Twisted curves), homeomorphic sets of lines in a plane (line coordinates), the theory of invariants, relativity in mechanics, and correspondences.
In 1915 Seely was elected a Fellow of the United States National Academy of Sciences. Northwestern conferred upon him an LL.D. in the same year. At the time of its 100th anniversary in 1932, Wesleyan conferred upon him an D.Sc. *Wik
Died on his birthday in 1943

1874 Friedrich Moritz Hartogs (20 May 1874, Brussels–18 August 1943, Munich) was a German-Jewish mathematician, known for work on set theory and foundational results on several complex variables. *Wik

1901 Machgielis (Max) Euwe (last name is pronounced [ˈøːwə]) (May 20, 1901 – November 26, 1981) was a Dutch chess Grandmaster, mathematician, and author. He was the fifth player to become World Chess Champion (1935–37). Euwe also served as President of FIDE, the World Chess Federation, from 1970 to 1978. *Wik

1901 Hannes Olof Gösta Alfvén (born 30 May 1908 in Norrköping, Sweden; died 2 April 1995 in Djursholm, Sweden) was a Swedish electrical engineer, plasma physicist and winner of the 1970 Nobel Prize in Physics for his work on magnetohydrodynamics (MHD). He described the class of MHD waves now known as Alfvén waves. He was originally trained as an electrical power engineer and later moved to research and teaching in the fields of plasma physics and electrical engineering. Alfvén made many contributions to plasma physics, including theories describing the behavior of aurorae, the Van Allen radiation belts, the effect of magnetic storms on the Earth's magnetic field, the terrestrial magnetosphere, and the dynamics of plasmas in the Milky Way galaxy.*Wik

1913 William Redington Hewlett (20 May 1913; Ann Arbor, Michigan - 12 Jan 2001 at age 87) was an American electrical engineer who co-founded the Hewlett-Packard Company, a leading manufacturer computers, computer printers, and analytic and measuring equipment. In 1939, he formed a partnership known as Hewlett-Packard Company with David Packard, a friend and Stanford classmate. (The order of their names was determined by a coin toss.) HP's first product was an audio oscillator based on a design developed by Hewlett when he was in graduate school. Eight were sold to Walt Disney for Fantasia. Lesser-known early products were: bowling alley foul-line indicator, automatic urinal flusher, weight-loss shock machine. The company began with \$538 intial capital, and its first production facility was a small garage in Palo Alto. *TIS

DEATHS

1798 Erland Bring (19 August 1736 – 20 May 1798) was a Swedish mathemaician who made contributions to the algebraic solution of equations.*SAU
At Lund he wrote eight volumes of mathematical work in the fields of algebra, geometry, analysis and astronomy, including Meletemata quaedam mathematematica circa transformationem aequationum algebraicarum (1786). This work describes Bring's contribution to the algebraic solution of equations. *Wik

1943 Henry Seely White, died on his birthday. (see 1861 above)

1947 Philipp Eduard Anton von Lenard (7 June 1862 – 20 May 1947), was a German physicist and the winner of the Nobel Prize for Physics in 1905 for his research on cathode rays and the discovery of many of their properties. He was a nationalist and anti-Semite; as an active proponent of the Nazi ideology, he had supported Adolf Hitler in the 1920s and was an important role model for the "Deutsche Physik" movement during the Nazi period.
Lenard is remembered today as a strong German nationalist who despised "English physics", which he considered to have stolen its ideas from Germany. He joined the National Socialist Party before it became politically necessary or popular to do so. During the Nazi regime, he was the outspoken proponent of the idea that Germany should rely on "Deutsche Physik" and ignore what he considered the fallacious and deliberately misleading ideas of "Jewish physics", by which he meant chiefly the theories of Albert Einstein, including "the Jewish fraud" of relativity. An advisor to Adolf Hitler, Lenard became Chief of Aryan physics under the Nazis. *Wik

Credits :
*CHM=Computer History Museum
*FFF=Kane, Famous First Facts
*NSEC= NASA Solar Eclipse Calendar
*RMAT= The Renaissance Mathematicus, Thony Christie
*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