Pat'sBlog

Thursday, 3 December 2020

A Curious Property of Vulgar Fractions

Another from 2008, hope you enjoy:

John Farey was a geologist, not a mathematician, but he is better remembered today for a single short (four paragraphs) paper he wrote in 1816 than for all his good works in geology. In that year he sent a paper to the Philosophical Magazine (or at least it was published in that year) called On a curious property of vulgar fractions .and described a pattern that appears in sequences of what we would today call common (vulgar is the Latin term for common) fractions, like 3/8 etc, which are in simplest terms. They observation has almost NO practical use, not even to prove other things mathematically, and yet, it seems to have all kinds of interesting properties that tend to keep us fascinated with it. If you have never been introduced to it, here is a brief description, and some novel relationships that I think are interesting, with some links to places where they are made clearer than I could do in this brief space.

So First... What is it we are talking about? If you take ALL the fractions that could be written in simplest terms with a denominator less or equal to some number n, say n=5 (since that is the one Farey used in his paper), and put them in order from lowest to highest... you get

The "curious" thing that Farey noticed is that if you ignore the way your fifth grade teacher taught you to add fractions, and do it the way YOU would have added them, "add the tops, add the bottoms", then each number in the sequence is the sum of the terms on each side of it... for example ¹/₄ and ²/₅ are on each side of ¹/₃, and if you add by this approach you get ⁽¹⁺²⁾/₍₄₊₅₎ = ³/₉ and that simplifies to ¹/₃ . The number obtained by adding two fractions in this fashion is often called the mediant .

OK, so that is how you make them. Our first question might be, how many of them are there? F(5) obviously has eleven terms (I counted). If we picked a value of N, what would be the number of fractions in the set F(N). A little investigation would show that F(1) = 2 (0 and 1); and F(2) = 3 (0, 1/2, and 1). So how many would be added to the next set... and the next... it turns out that each new set will have all the values of the previous set (of course) and will add one for every value of one through n that is co-prime (has no common divisors) to n. So the set for N=6 will have the eleven terms of F(5) plus 1/6 (one has no common factor with six), and 5/6. (notice that 2/6, 3/6, and 4/6 are already in the sequence in F(5) in simplified form)... thus F(6) has thirteen terms.. and in general we get a recursive formula that say the Order of F(N) = Order of F(n-1) + φ(n).
A second nice curiosity related to Farey Sequences are the Ford Circles. "Ford circles are named after American mathematician Lester R. Ford, Sr., who described them in an article in American Mathematical Monthly in 1938, volume 45, number 9, pages 586-601" (from Wikipedia). In fact, the wikipedia article is a nice place to see how they work, and I need say no more as it shows the circles for F(5). What is amazing is that each circle is tangent to every other circle for a fraction it will be adjacent to in ANY sequence.... ahhh, go on..say "cool".

I decided to mention this when I came across another curious property of Farey sequences that relates them to lines on the plane and Pick's Theorem. If you treated each fraction a/b as a point (b,a) then none of the lines cross. If you make a triangle with the origin and any two adjacent Farey fractions, since each of the triangles have a determinant of one (meaning the area is 1/2) and therefore, by PIck's theorem, they cannot contain any other lattice points in their interior. A nice explanation of this, including the photo below, is at the Cut-The-Knot web site

On This Day in Math - December 3

Symmetry, as wide or narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty, and perfection.
~Hermann Weyl

The 338th day of the year; the last day of the year which will be twice a perfect square.

338 is the arithmetic mean of two triangular numbers.

338 is the smallest number for which the number of divisors (6) and the sum of its prime factors (28) are both perfect numbers. Are there others?

EVENTS

1610 Galileo dedicates his Sidereus nuncius to Grandduke Cosmos II. *VFR I am not sure what event Professor Rickey is referring here. According to Albert Van Helden in his introduction to his translation, "The Dedicatory letter of Sidereus nuncius is dated 12 March 1610, and on the next day Galileo sent an advance, unbound copy, accompanied by a letter, to the Tuscan court."
Thony Christie sent this translation from page 33 of the same book, "Written in Padua on the fourth day before the Ides of March 1610. Your Highnesses's most loyal servant, Galileo Galilei."

1833 Oberlin College, the ﬁrst truly coeducational institution of higher education in the U.S., opens with 29 men, 15 women. *VFR

1836 Adolphe Quetelet presents the average number of meteors per hour.
The 1833 Leonid storm had galvanized interest in meteors, and the time was ripe. Adolphe Quetelet, a Belgian statistician and founder and director of the Brussels Observatory, had mentioned mid-August meteors very tentatively six months earlier. His attention had been called to meteors by François Arago of France, who dominated European science at the time with his skill in discerning important scientific problems and suggesting experiments to solve them. What, asked Arago in the wake of the 1833 display, constituted a shower of meteors, and what was the rate of the ordinary, everynight drizzle?
The problem was ideal for Quetelet, whose passion was statistics. In a speech to the Royal Academy of Sciences and Arts of Brussels on December 3, 1836, Quetelet gave his answer: averaged over the night and year, a single observer should expect to see eight sporadic (nonshower) meteors per hour. That figure is still good today. After his speech Quetelet made a brief mention of unusual August meteors, and in his 1836 annual report of the Brussels Observatory he presented the idea timidly and almost in passing: "I thought I also noticed a greater frequency of these meteors in the month of August (from the 8th to the 15th)."
By the following year, Quetelet had accidentally found records in his observatory of exceptional meteor displays on August 10th of 1834 and 1835 to accompany the increase he had seen in 1836. He called for scientists at the March 4, 1837, session of the Royal Academy of Brussel to watch the sky on August 10, 1837. *Sky and Telescope

1880 The Railroad Gazette credits Ellis Holbrook with being the first to give an accurate description of the so-called track transition curve, a method of making the transition from a straight track to a circular curve without the sudden lateral "jerk" caused by the change in curvature.

Unknown to them, the curve had been described {in part} by James Bernoulli in 1694 and known then as the Cantilever problem., to find the curvature a beam must have in order to be straightened by a given weight. Bernoulli gave the solution a = s^2 r where a is a constant r is the radius of curvature, thus the curvature was proportional to the arc length.
Euler, in the same year, 1744, extended the curve to the shape now called the Euler Spiral.

*Wik

The curve emerged again in 1874 as the solution to a problem in diffraction, now known as the spiral of [Alfred] Cornu.
By 1886 the spiral attracted the attention of Italian Ernesto Cesaro, who thought the curve looked like the shape of thread wrapped on a spindle, and called the curve the clothoid, after Clotho, one of the three fates who was a spinner. HT to Julian Havel

1958 Germany issued a stamp to commemorate the 500th anniversary of the Cusanus Hospice at Kues, founded by Cardinal Nicolaus (1401-1464), Nicolaus Cusanus (Nickolaus Krebs). *VFR

1968 CDC Announces 7600 Supercomputer: Control Data Corporation announces its 7600 model, considered by some to be the first true supercomputer. The CDC 7600 calculated at a speed of nearly 40 megaflops. Seymour Cray designed this computer, as well as its predecessor, the 6600 that was popular with scientific researchers, and a successor, the 8600, which the company never marketed. *CHM

BIRTHS

1616 John Wallis (3 Dec 1616; 8 Nov 1703) English mathematician. Wallis was skilled in cryptography and decoded Royalist messages for the Parliamentarians during the Civil War. Wallis was part of a group interested in natural and experimental science who started to meet in London. This group became the Royal Society (1663), with Wallis as a founder member and one of its first Fellows. He contributed substantially to the origins of calculus and was the most influential English mathematician before Newton. Wallis introduced our symbol for infinity (1656), and exponents using negative or fractional numbers (such as 1/x² = x^-2 or square root of x = x^-1/2). In 1668, he was the first to suggest the law of conservation of momentum for colliding bodies, the first of all-important conservation laws.*TIS

1903 Sydney Goldstein (3 Dec 1903 in Hull, England - 22 Jan 1989 in Belmont, Massachusetts, USA) Goldstein's work in fluid dynamics is of major importance. He is described as, "... one of those who most influenced progress in fluid dynamics during the 20th century." He studied numerical solutions to steady-flow laminar boundary-layer equations in 1930. In 1935 he published work on the turbulent resistance to rotation of a disk in a fluid. His work was important in aerodynamics, a subject in which Goldstein was extremely knowledgeable. *SAU

1938 Cleveland Abbe (3 Dec 1838; 28 Oct 1916) U.S. astronomer and first meteorologist, born in New York City, the "father of the U.S. Weather Bureau," which was later renamed the National Weather Service. Abbe inaugurated a private weather reporting and warning service at Cincinnati. His weather reports or bulletins began to be issued on Sept. 1, 1869. The Weather Service of the United States was authorized by Congress on 9 Feb 1870, and placed under the direction of the Signal Service. Abbe was the only person in the country who was already experienced in drawing weather maps from telegraphic reports and forecasting from them. Naturally, he was offered an important position in this new service which he accepted, beginning 3 Jan 1871, and was often the official forecaster of the weather.*TIS

1924 John Backus (3 Dec 1924; 28 Oct 1988) American computer scientist who invented the FORTRAN (FORmula TRANslation) programming language in the mid 1950s. He had previously developed an assembly language for IBM's 701 computer when he suggested the development of a compiler and higher level language for the IBM 704. As the first high-level computer programming language, FORTRAN was able to convert standard mathematical formulas and expressions into the binary code used by computers. Thus a non-specialist could write a program in familiar words and symbols, and different computers could use programs generated in the same language. This paved the way for other computer languages such as COBOL, ALGOL and BASIC. *TIS

1942 Joseph Ivor Silk FRS (3 December 1942-) was the Savilian Chair of Astronomy at the University of Oxford from 1999 to September 2011. He was educated at Tottenham County School (1954-1960) and went on to study Mathematics at the University of Cambridge (1960-1963). He gained his PhD in Astronomy from Harvard in 1968. Silk took up his first post at Berkeley in 1970, and the Chair in Astronomy in 1978. Following a career of nearly 30 years there, Silk returned to the UK in 1999 to take up the Savilian Chair at the University of Oxford. He is currently Professor of Physics at the Institut d’Astrophysique de Paris, Université Pierre et Marie Curie, and he joined Johns Hopkins University in 2010 as Homewood Professor of Physics and Astronomy.

He is an Emeritus Fellow of New College, Oxford and a Fellow of the Royal Society (elected May 1999). He was awarded the 2011 Balzan Prize for his works on the early Universe. Silk has given more than two hundred invited conference lectures, primarily on galaxy formation and cosmology.
In 2015 he was selected the Gresham Professor of Astronomy. *Wik

DEATHS

1882 James Challis (12 Dec 1803, 3 Dec 1882) British clergyman and astronomer, famous in the history of astronomy for his failure to discover the planet Neptune. Astronomer and mathematician John Couch Adams had studied the known deviations in the orbit of the planet Uranus which indicated a planet even further out. In 1845, Adams gave Astronomer Royal George Airy a calculated orbital path for the unknown planet. But Airy was more interested in the primary job of navigation and timekeeping observations. Airy informed Challis, who did not begin until July 1846, and actually sighted the new planet four times without recognizing it. On 23 Sep 1845, the new planet was instead discovered from Berlin Observatory. Challis admitted that Adam's prediction was within 2° of the planet's position.*TIS

1956 Felix Bernstein (24 Feb 1878 in Halle, Germany - 3 Dec 1956 in Zurich, Switzerland) established his famous theorem on the equivalence of sets while in Cantor's seminar at Halle in 1897. He also worked on transfinite ordinal numbers.Bernstein is best remembered by mathematicians for the Schröder-Bernstein Theorem. This theorem states:
If each of two sets A and B are equivalent to a subset of the other, then A is equivalent to B. *SAU

1983 Elliott Waters Montroll (May 4, 1916 in Pittsburgh, Pennsylvania, USA - December 3, 1983 in Chevy Chase, Maryland, USA) was an American scientist and mathematician.Montroll had an exceptionally varied career: was a Sterling Research Fellow at Yale University where his work on the Ising model of a ferromagnet led him to solve certain Markov chain problems. Following this he was a Research Associate at Cornell University in 1941-42 where he began his studies of the problem of finding the frequency spectrum of elastic vibrations in crystal lattices. He was elected to the National Academy of Sciences (United States) in 1969, and to the American Academy of Arts and Sciences in 1973. His work on traffic flow led to him winning (jointly) the Lanchester Prize of the Operations Research Society of America in 1959. *Wik

2004 Shiing-shen Chern (26 Oct 1911, 3 Dec 2004) Chinese-American mathematician and educator whose researches in differential geometry include the development of the Chern characteristic classes in fibre spaces, which play a major role in mathematics and in mathematical physics. "When Chern was working on differential geometry in the 1940s, this area of mathematics was at a low point. Global differential geometry was only beginning, even Morse theory was understood and used by a very small number of people. Today, differential geometry is a major subject in mathematics and a large share of the credit for this transformation goes to Professor Chern." *TIS

2008 Oliver Gordon Selfridge (May 10, 1926 – December 3, 2008), grandson of Harry Gordon Selfridge, the founder of Selfridges' department stores, was a pioneer of artificial intelligence. He has been called the "Father of Machine Perception."
Selfridge was born in England, educated at Malvern College and Middlesex School and then earned an S.B. from MIT in mathematics in 1945. He then became a graduate student of Norbert Wiener's at MIT, but did not write up his doctoral research and never earned a Ph.D. While at MIT, he acted as one of the earlier reviewers for Wiener's Cybernetics book in 1949. He was also technically a supervisor of Marvin Minsky, and helped organize the first ever public meeting on Artificial Intelligence (AI) with Minsky in 1955.
Selfridge wrote important early papers on neural networks and pattern recognition and machine learning, and his "Pandemonium" paper (1959) is generally recognized as a classic in artificial intelligence. In it, Selfridge introduced the notion of "demons" that record events as they occur, recognize patterns in those events, and may trigger subsequent events according to patterns they recognize. Over time, this idea gave rise to Aspect-oriented programming.
In 1968, in their formative paper "The Computer as a Communication Device", J. C. R. Licklider and Robert Taylor introduced a concept known as an OLIVER (Online Interactive Expediter and Responder) which was named in honor of Selfridge.
Selfridge spent his career at Lincoln Laboratory, MIT (where he was Associate Director of Project MAC), Bolt, Beranek and Newman, and GTE Laboratories where he became Chief Scientist. He served on the NSA Advisory Board for 20 years, chairing the Data Processing Panel. Selfridge retired in 1993.
Selfridge also authored four children's books, "Sticks", "Fingers Come In Fives", "All About Mud", and "Trouble With Dragons". *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

Wednesday, 2 December 2020

On This Day in Math - December 2

The saddest aspect of life right now is that science gathers knowledge faster than society gathers wisdom.
~Isaac Asimov

The 337th day of the year, 337 is a Pythagorean prime number, and when its digits are reversed, that is also prime. (A Pythagorean prime is a prime number of the form 4n + 1. Pythagorean primes are exactly the primes that are the sum of two squares (and from this derives the name in reference to the famous Pythagorean theorem.)

The mean of the first 337 square numbers is itself a square. This is the smallest number for which this is true.

The famous Fibonacci area paradox shows a 13x13 square converted to an 8x21 rectangle. The areas of the two figures, 13x13 + 8x21 = 337 (this illusion works with any Fibonacci number F(n) squared and a rectangle that is F(n-1) by F(n+1) ) Students must be aware that 13 x 13 = 169 is NOT equal to 8 x 21 = 168, so where is the flaw. Here is a post for a little history of these geometric vanishes.

EVENTS
1697 St Paul's Cathedral, reconstructed after the Great Fire of 1666 as redesigned by Christopher Wren, was officially opened on December 2nd, 1697. *History Today

*Wik St Paul's Cathedral in 1896

In 1895, James Dewar exhibited his new apparatus for the production of liquid air at the Royal Institution.*TIS

In 1934, the molten glass was poured in the Corning, N.Y. for the first 200-inch diameter telescope mirror. Pyrex glass at 2,700 degrees Fahrenheit was poured into a ceramic mold. The mold had been constructed over a period of several months. The temperature of the glass was lowered during 11 months, a degree or two a day. It was then allowed to cool to room temperature. The 20-ton disk was shipped 26 Mar 1936 for grinding and polishing at the California Institute of Technology, which spanned 11 years, completed on 3 Oct 1947. It was installed in a telescope at the Mount Palomar Observatory on Palomar Mountain, San Diego County, California, which was named the Hale telescope in honour of Dr George Hale who had conceived and promoted it.*TIS

1942 At 3:36 p.m. in a squash court (Actually it was a Racketball court *James Zug Squash, A History of the Game pgs. 135–136.) under the West Stands of Stagg Field (the abandoned football stadium) at the University of Chicago, the ﬁrst self-sustaining nuclear (ﬁssion) reaction took place. Enrico Fermi (1901–1954) was leader of the Manhattan Project. [DSB 4, 582]. *VFR This first run of the nuclear pile produced a single watt of power, Just enough to show that the process was feasible. One of the “about 40” people who watched was Leo Szilard who had conceived the idea of a chain reaction leading to power while stopped at a red light on Southampton Row in London only four years before. *Frederik Pohl, Chasing Science, Pg 20
One of Fermi's assistants ran from the test to the telephone to notify Openheimer, "The Italian navigator has just landed in the New World." *Brody & Brody, The Science Class You Wished You Had

1954 The U.S. Navy dedicates its Naval Ordnance Research Calculator (NORC) at the Naval Surface Weapons Center in Dahlgren, Virginia. John von Neumann was the keynote speaker. The machine was built at the Watson Scientific Computing Laboratory under the direction of Wallace Eckert.
This computer was in demand by many organizations, including two different Navy facilities and Lawrence Livermore National Laboratory in California. Physicist Edward Teller had been trying to receive NORC arguing that the LLNL's nuclear calculations were more important than Dahlgren's ballistic calculations. The Navy won and NORC was delivered to Dahlgren, following the Mark II (1948) and the Mark III (1951).*CHM

1967 Italy issued a postage stamp to commemorate the 25th anniversary of the ﬁrst atomic chain reaction. Pictured is Enrico Fermi at Los Alamos and a model of the ﬁrst Atomic Reactor. *VFR

1978 Science News reports, p. 390, that 2^21,701 − 1 is prime.

BIRTHS
1831 Paul David Gustav du Bois-Reymond (2 Dec 1831 in Berlin, Germany - 7 April 1889 in Freiburg, Germany) Du Bois-Reymond's work is almost exclusively on calculus, in particular partial differential equations and functions of a real variable. The standard technique to solve partial differential equations used Fourier series but Cauchy, Abel and Dirichlet had all pointed out problems associated with the convergence of the Fourier series of an arbitrary function. In 1873 du Bois-Reymond was the first person to give an example of a continuous function whose Fourier series diverges at a point. Perhaps what was even more surprising, the Fourier series of du Bois-Reymond function diverged at a dense set of points. The important work Eine neue Theorie der Convergenz und Divergenz von Reihen mit positiven Gliedern ("A new theory of convergence and divergence of series with positive terms") led to an increasing understanding of the whole concept of a function.
Du Bois-Reymond published an example of a continuous function which is nowhere differentiable in 1875. It was inspired by a similar function found by Weierstrass in 1872 but not published by him until much later. This example contradicted most mathematicians' intuition, for it was generally believed that a continuous function was differentiable everywhere except in special points. *SAU

1865 Niels Nielsen (2 Dec 1865 in Orslev, Denmark - 16 Sept 1931 in Copenhagen, Denmark) was a Danish mathematician who worked on special functions and number theory. *SAU

1901 Dom George Frederick James Temple FRS(born 2 December 1901, London; died 30 January 1992, Isle of Wight) was an English mathematician, recipient of the Sylvester Medal in 1969. He was President of the London Mathematical Society in the years 1951-1953.[2]
Temple took his first degree as an evening student at Birkbeck College, London, between 1918 and 1922, and also worked there as a research assistant. In 1924 he moved to Imperial College as a demonstrator, where he worked with Professor Sydney Chapman. After a period spent with Eddington at Cambridge, he returned to Imperial as reader in mathematics. He was appointed professor of mathematics at King's College London in 1932, where he returned after war service with the Royal Aircraft Establishment at Farnborough. In 1953 he was appointed Sedleian Professor of Natural Philosophy at the University of Oxford, a chair which he held until 1968, and in which he succeeded Chapman. He was also an honorary Fellow of Queen's College, Oxford.
After the death of his wife in 1980, Temple, a devout Christian, took monastic vows in the Benedictine order and entered Quarr Abbey on the Isle of Wight, where he remained until his death. *Wik

1914 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

DEATHS
1594 Gerardus Mercator (5 Mar 1512- 2 Dec 1594) Flemish cartographer whose most important innovation was a map, embodying what was later known as the Mercator projection, on which parallels and meridians are rendered as straight lines spaced so as to produce at any point an accurate ratio of latitude to longitude. He also introduced the term atlas for a collection of maps. *TIS

1873 Karl Gräffe Gräffe (7 Nov 1799 in Brunswick, Germany - 2 Dec 1873 in Zurich, Switzerland) is best remembered for his method of numerical solution of algebraic equations, developed to answer a prize question of the Berlin Academy of Sciences. It is particularly suitable for methods developed for using computers to solve mathematical problems. This method is today called the Dandelin-Gräffe method after the two mathematicians who independently investigated it. The history of the Dandelin-Gräffe method is discussed in and . Lobachevsky is also credited with the independent discovery of the method which appears in his little-known book on algebra.*SAU

1966 L(uitzen) E(gbertus) J(an) Brouwer (27 Feb 1881, 2 Dec 1966) was a Dutch mathematician who founded mathematical Intuitionism (a doctrine that views the nature of mathematics as mental constructions governed by self-evident laws). He founded modern topology by establishing, for example, the topological invariance of dimension and the fixpoint theorem. (Topology is the study of the most basic properties of geometric surfaces and configurations.) The Brouwer fixed point theorem is named in his honor. He proved the simplicial approximation theorem in the foundations of algebraic topology, which justifies the reduction to combinatorial terms, after sufficient subdivision of simplicial complexes, the treatment of general continuous mappings. *TIS

1982 Geoffrey Timms studied at Leeds and Cambridge and then took up a post at St Andrews. During World War II he served at Bletchley Park and Cheltenham and joined the Foreign Office afterwards. He became President of the EMS in 1941.*SAU

2006 Dikran "Dick" Tahta (7 August 1928 – 2 December 2006) was a British-Armenian mathematician, teacher and author.
Dikran Tahta is a descendant of an Ottoman Armenian family who settled in Manchester after the First World War. Much of his childhood, and the influence of his Armenian religious upbringing, is reflected upon in his penultimate book Ararat Associations, in which he notes how his parents were keen for their children to have an English education, yet made sure that they spoke Armenian at home. He was christened by Bishop Tourian in the Armenian Church in Manchester, and his name Dikran was shortened to Dick, but he never forgot his Armenian roots.
From Rossall School, in Fleetwood, Lancashire, he gained a scholarship to Christ Church, Oxford, in 1946. His main subject was Mathematics, but he also read widely in English literature, philosophy and history.
In the 1970s he was involved in the ATV television programme of mathematics for schools entitled 'Leapfrogs' (produced and directed by Paul Martin) and promoted visual approaches to mathematics. His paper "On Geometry" argued that geometrical approaches to mathematics could not be reduced to algebraic approaches. In line with this thinking, he produced the ATM book Geometric Images, and co-authored Images of Infinity with Ray Hemmings. The Leapfrogs group of Tahta and Hemmings, together with David Sturgess, Leo Rogers and Derick Last also produced hands-on teaching materials including workbooks for the polycube. He also drew upon insights into pedagogy in the writings of Mary Boole on mathematics education.
After retirement, he went to teach in the United States and South Africa, and became a tutor for the Open University.
His last book was The Fifteen Schoolgirls about Thomas Kirkman, known for the Kirkman's schoolgirl problem, a problem in combinatorics, which also delved into the byways of Victorian amateur mathematics.
In his obituary, The Guardian newspaper described Dick as "one of the outstanding mathematics teachers of his generation", who was notable for having inspired physicist Stephen Hawking. The Guardian commented on his death that "He was a wise and generous man who inspired love and an increase of intellectual energy in everyone who came within his ambit." *Wik

Tuesday, 1 December 2020

On This Day in Math - December 1

Beauty is the first test:
there is no permanent place in the world for ugly mathematics.
~Godfrey Harold Hardy

The 336th day of the year; there are 336 dimples on an American golf ball. There are 336 ways to partition 41 into primes.

336 is the product of three consecutive integers, 6*7*8 = 336

336 = (3^1 + 3^1 + 6^1) + (3^2 + 3^2 + 6^2) + (3^3 + 3^3 + 6^3)

Yesterday I mentioned LaGrange's theorem that every number can be written as the sum of four integral squares. Some can be written as the sum of four squares in many different ways which include the use of 0². The number 26 can be partitioned as the sum of four squares in 336 different ways. If that sounds too trivial, tell folks 336 is a Lipschitz integer quaternion.

EVENTS
1729 Euler/Goldbach correspondence begins: Goldbach was also a kind of mentor to Leonhard Euler. For over 25 years they exchanged letters, 196 of which survive. These letters give us a window into Euler’s scientific and personal life. In Goldbach’s very first letter to Euler, dated December 1, 1729, Goldbach got Euler interested in number theory. Goldbach added note at the end of the letter: “P. S. Have you noticed the observation of Fermat that all numbers of the form 2^{2^x}+1, that is 3, 5, 17, etc., are prime numbers, but he did not dare to claim he could demonstrate it, nor, as far as I know, has anyone else been able to prove it.” Three years later, in a five-page paper that now bears the index number E26, Euler shows that the F(5) = 4,294,967,297 = 641× 6,700,417 . That is, Fermat was wrong. *Ed Sandifer, How Euler Did It (Like Fermat before him, Euler found most mathematicians less than excited about problems in Number Theory, but for most of his life, Goldbach would be his mentor, and his student in this area.)

1764 Alexander Small writes Benjamin Franklin from England, "My Namesake the Virginian Professor (William Small) is here; and desires to be most particularly remembered to you."
Small is known for being Thomas Jefferson's professor of Natural Philosophy at William and Mary, and for having an influence on the young Jefferson. (I could not determine if Alexander and William Small are related) *Natl. Archives

1783 J. A. C. Charles was the ﬁrst man to see the sun set twice in one day. He did it by making a ﬂight (to 9000 feet) in a hydrogen balloon. *VFR (Charles is often considered the inventor of the hydrogen balloon.) The first manned voyage of a hydrogen balloon left Paris carrying Professor Jacques Alexander Cesar Charles and Marie-Noel Robert to about 600 m and landed 43 km away after 2 hours in the air. Robert then left the balloon, and Charles continued the flight briefly to 2700 m altitude, measured by a barometer. This hydrogen-filled balloon was generally spherical and used a net, load ring, valve, open appendix and sand ballast, all of which were to be universally adopted later. His hydrogen generator mixed huge quantities of sulfuric acid with iron filings. On 27 Aug 1783, Charles had launched an unmanned hydrogen balloon, just before the Montgolfiers' flight. *TIS (One of these altitudes is obviously wrong. )

1851 On December the first, Louis-Napoleon Bonaparte, who had been instrumental in supporting Foucault in the demonstration of his pendulum, ordered that the pendulum demonstration cease and the Pantheon return to being used as a church (Louis Philippe had secularized the Pantheon in 1830 and stopped burials in the crypt). Why did he stop the popular demonstrations? We do not know, but on the next day citizens of France awoke to find notices posted on the major buildings, “The National Assembly is dissolved… “ Louis-Napoleon had taken his first step to becoming Emperor of France. *Amir D Aczel, Pendulum, pg 174

1890, after regular competition, Peano was named extraordinary
professor of infinitesimal calculus at the University of Turin. *Hubert Kennedy
Eight Mathematical Biographies Pg 23

1896 Frank Broaker of New York City received certificate No. 1 from the New York State Board of Certified Public Account Examiners thus becoming the first CPA in the US. *JN Kane, Famous First Facts,

In 1997 eight planets from our Solar System lined up from West to East beginning with Pluto, followed by Mercury, Mars, Venus, Neptune, Uranus, Jupiter, and Saturn, with a crescent moon alongside, in a rare alignment visible from Earth that lasted until Dec 8. Mercury, Mars, Venus, Jupiter and Saturn were visible to the naked eye, with Venus and Jupiter by far the brightest. A good pair of binoculars is needed to see the small blue dots that are Uranus and Neptune. Pluto is visible only by telescope. The planets also aligned in May 2000, but too close to the sun to be visible from Earth. It will be at least another 100 years before so many planets will be so close and so visible.*TIS

BIRTHS
1671 John Keill (1 Dec 1671; 31 Aug 1721) Scottish mathematician and natural philosopher, who was a major proponent of Newton’s theories. He began his university education at Edinburgh under David Gregory, whom he followed to Oxford, where Keill lectured on Newton's work, and eventually became professor of astronomy. In his book, An Examination of Dr. Burnett's Theory of the Earth (1698), Keill applied Newtonian principles challenging Burnett's unsupportable speculations on Earth's formation. In 1701, Keill published Introductio ad Veram Physicam, which was the first series of experimental lectures and provided a clear and influential introduction to Isaac Newton’s Principia. He supported Newton against priority claims by Leibnitz for the invention of calculus. *TIS

1792 Nikolay Ivanovich Lobachevsky (1 Dec 1792; 24 Feb 1856) Russian mathematician who, with János Bolyai of Hungary, is considered the founder of non-Euclidean geometry. Lobachevsky constructed and studied a type of geometry in which Euclid's parallel postulate is false (the postulate states that through a point not on a certain line only one line can be drawn not meeting the first line). This was not well received at first, but his greatest vindication came with the advent of Einstein's theory of relativity when it was demonstrated experimentally that the geometry of space is not described by Euclid's geometry. Apart from geometry, Lobachevsky also did important work in the theory of infinite series, algebraic equations, integral calculus, and probabilty. *TIS William Kingdon Clifford called Lobachevsky the "Copernicus of Geometry" due to the revolutionary character of his work. Lobachevsky is the subject of songwriter/mathematician Tom Lehrer's humorous song "Lobachevsky" from his Songs by Tom Lehrer album. In the song, Lehrer portrays a Russian mathematician who sings about how Lobachevsky influenced him: "And who made me a big success / and brought me wealth and fame? / Nikolai Ivanovich Lobachevsky is his name." Lobachevsky's secret to mathematical success is given as "Plagiarize!", as long as one is always careful to call it "research". According to Lehrer, the song is "not intended as a slur on [Lobachevsky's] character" and the name was chosen "solely for prosodic reasons".*Wik (The lyrics are here)

1847 Christine Ladd-Franklin (1 Dec 1847; 5 Mar 1930) American scientist and logician known for contributions to the theory of colour vision accounting for the development of man's color sense which countered the established views of Helmholtz, Young, and Hering. Her position was that color-sense developed in stages. Ladd- Franklin's conclusions were particularly useful in accounting for color-blindness in some individuals. In logic, she published an original method for reducing all syllogisms to a single formula *TIS Ladd-Franklin was the first woman to have a published paper in the Analyst (at this time, 1877, it was more of a recreational mathematics publication still edited by the self-educated Ohio farmboy, Joel E Hendricks. The article was simply titled "Quaternions." ). She was also the first woman to receive a Ph.D. in mathematics and logic. The majority of her publications were based on visual processes and logic. Her views on logic influenced Charles S. Peirce’s logic and she was highly praised by Prior. *Wik

1892 Krishnaswami Ayyangar (1 Dec 1892 in Attipattu, Chingleput district, Tamil Nadu, India - June 1953 in Mysore, India) was an Indian mathematician who worked in Mysore. He produced important work on the history of Hindu mathematics. *SAU

1913 Colossus' Team Member Chandler is Born W.W. Chandler was born in Bridport, England. He obtained his B.Sc. from London University in 1938 by private study while working as a telephone engineer at the British Post Office Research Department. During the war he was responsible for the installation and maintenance of the Colossus at Bletchley Park. The Colossus represented the first electronic computer, however it was programmed by a mechanical switchboard. Its was used to crack the German Fish codes which guarded the highest levels of German communication. Winston Churchill characterized the Bletchley Park team as the geese who laid the golden eggs but never cackled.
After the war Chandler participated in development and installation of the MOSAIC computer and worked on optical character recognition. He died on September 11, 1989. *CHM

1941 Stephen A. Benton (1 Dec 1941; 9 Nov 2003.) American physicist who was a pioneer in medical imaging and fine-arts holography. His fascination with optical phenomena began with the 3-D glasses he used as an 11-year-old to watch te 1953 movie "House of Wax." In 1968, he invented the "rainbow holograms" as seen on credit cards while working for Polaroid Corporation. He turned to academia as an assistant professor at Harvard (1968) and later a professor at Massachusetts Institute of Technology from 1985 where he helped set up the Spatial Imaging Group and headed the M.I.T. media art and sciences program. Benton was a pioneer in natural light holography as a artistic medium, and was a curator at the Museum of Holography in Manhattan until it closed in 1992.*TIS

DEATHS
1750 Johann Doppelmayr (27 Sept 1677 in Nuremberg, Germany - 1 Dec 1750 in Nuremberg, Germany)was a German mathematician who wrote on astronomy, spherical trigonometry, sundials and mathematical instruments. Doppelmayr also wrote a book of tremendous value giving biographical details of 360 mathematicians and instrument makers of Nuremberg from the 15th to the 18th century. This had the lengthy title Historische Nachricht von den Nürnbergischen Mathematicis und Künstlern, welche fast von dreyen Seculis her durch ihre Schriften und Kunst-Bemühungen die Mathematic und mehrere Künste in Nürnberg vor andern trefflich befördert und sich um solche sehr wohl verdient gemacht zu einem guten Exempel, und zur weitern rühmlichen Nachahmung and was published in 1730. *SAU

1866 Sir George Everest (1790, 1 Dec 1866) British military engineer and geodesist, born in Gwernvale, Powys, Wales, UK. He worked on the trigonometrical survey of India (1818-43), providing the accurate mapping of the subcontinent. For more than twenty-five years and despite numerous hardships, he surveyed the longest arc of the meridian ever accomplished at the time. Everest was relentless in his pursuit of accuracy. He made countless adaptations to the surveying equipment, methods, and mathematics in order to minimize problems specific to the Great Survey: immense size and scope, the terrain, weather conditions, and the desired accuracy. Mount Everest, formerly called Peak XV, was renamed in his honour in 1865. *TIS (Mary Boole, self-taught mathematician and wife of George Boole was his niece)

1935 Bernhard Voldemar Schmidt (30 Mar 1879, 1 Dec 1935) Astronomer and optical instrument maker who invented the telescope named for him. In 1929, he devised a new mirror system for reflecting telescopes which overcame previous problems of aberration of the image. He used a vacuum to suck the glass into a mold, polishing it flat, then allowing in to spring back into shape. The Schmidt telescope is now widely used in astronomy to photograph large sections of the sky because of its large field of view and its fine image definition. He lost his arm as a child while experimenting with explosives. Schmidt spent the last year of his life in a mental hospital.*TIS

1947 Godfrey Harold Hardy (1877, 1 Dec 1947)English mathematician known for his work in number theory and mathematical analysis. Hardy's interests covered many topics of pure mathematics - Diophantine analysis, summation of divergent series, Fourier series, the Riemann zeta function, and the distribution of primes. Although Hardy considered himself a pure mathematician, early in his career, he nevertheless worked in applied mathematics when he formulated a law that describes how proportions of dominant and recessive genetic traits will propagate in a large population (1908). Hardy considered it unimportant but it has proved of major importance in blood group distribution. As it was also independently discovered by Weinberg, it is known as the Hardy-Weinberg principle. *TIS G. H. Hardy died—on the same day that the Copley Medal was to be presented to him by the Royal Society of London. [Collected Papers of G. H. Hardy, vol. 1, p. 8].

1964 J.B.S.(John Burdon Sanderson) Haldane (5 Nov 1892, 1 Dec 1964) was a British geneticist and biometrician who opened new paths of research in population genetics and evolution. He began studying science at the age of eight, as assistant to his father (the noted physiologist John Scott Haldane). J.B.S. Haldane also worked in biochemistry, and on the effects of diving on human physiology. A Marxist from the 1930s, Haldane was well known for his outspoken Marxist views.He resigned from the Communist Party c. 1950 on the issue of Lysenko's claims to have manipulated the genetic structure of plants and "Stalin's interference with science". He became known to a large public as a witty popularizer of science with such works as Daedalus (1924), and Possible Worlds (1927).*TIS

1977 Kenneth O. May (July 8, 1915, Portland, Or. – December 1,1977) was an American mathematician and historian of mathematics, who developed May's theorem. The Kenneth O. May Prize is awarded for outstanding contributions to the history of mathematics. Ken May established Historia Mathematica, and preserved it by separating it from its creator, "The distinguished predecessors of HM were associated with their founders and died with them. If HM is to avoid this fate, we must prepare and carry through a prompt transfer of editorial responsibility to younger hands." His list of publications numbers above 300. *Henry S. Tropp, E'loge, Isis 70, Sept 1979, Pgs 419-422

1983 Leon Mirsky (19 Dec 1918 in Russia - 1 Dec 1983 in Sheffield, England)worked in Number Theory, Linear Algebra and Combinatorics.*SAU