Friday, 21 April 2017

On This Day in Math - April 21

A drunk man will find his way home,
but a drunk bird may get lost forever.

From Colloquial catchy statements encoding serious mathematics at Math Overflow.[ The serious math...A 2-dimensional random walk is recurrent (appropriately defined for either the discrete or continuous case) whereas in higher dimensions random walks are not. More details can be found for instance in this enjoyable blog post by Michael Lugo.  "This particular saying, by the way, is usually attributed to Shizuo Kakutani. (I don't want anybody thinking I came up with it!)"  ]

The 111th day of the year; 111 would be the magic constant for the smallest magic square composed only of prime numbers if 1 were counted as a prime (and we often used to)
It seems that Henry Ernest Dudeney may have been the first person to explore the use of primes to create a magic square. He gave the problem of constructing a magic in The Weekly Dispatch, 22nd July and 5th August 1900. At that time, 1 was sometimes (often?) considered as a prime number. His magic square gives the lowest possible sum for a 3x3 using primes (assuming one is prime)

The smallest magic square with true primes (not using one) has a magic constant of 177. Good luck.

A six-by-six magic square using the numbers 1 through 36 also has a magic constant of 111.
*Tanya Khovanova, Number Gossip

numbers like 111 that appear the same under 180o rotations are called strobograms. For numbers like the recent 109 which appears as a different number under rotation, but is still a number, I have created the term ambinumerals.

If you concatenated three copies of 111 and then squared the result, you get (111,111,111)2 = 12,345,678,987,654,321 *Cliff Pickover@pickover

Lagrange's theorem tells us that each positive integer can be written as a sum of squares with no more than four squares needed. Most numbers don't require the maximum four, but there are 58 year dates that can not be done with less than four. 111 and 112 are the smallest consecutive pair that both require the maximum. There is one other pair of consecutive year dates that also require four, seek them my children.

EVENTS

1547 In a dispute over the priority for solving cubics, Tartaglia sent Ferrari 31 challenge problems. They were no harder than those in Luca Pacioli’s Summa (1494). *VFR
[Here is the poem in which Niccolo Fontana (Tartaglia is a nickname meaning "stutterer") revealed the secret of solving the cubic to Cardan]
When the cube and the things together
Are equal to some discrete number, 10
Find two other numbers differing in this one.
Then you will keep this as a habit
That their product shall always be equal
Exactly to the cube of a third of the things. 2)
The remainder then as a general rule
Of their cube roots subtracted
Will be equal to your principal thing. 3)
1 [Solve x3 + cx = d]
2 [Find u, v such that u - v = d and uv = (c/3)3 ]
3 [Then x = 3√u - 3√v ]
*SAU
The Math DL site has digital copies of pages from Cardano's classic Practica Arithmetice.

Practica Arithmetice printed and published by Johannes Petreius in Nürnberg 1439,

1702 "Early in the morning (about 2 a.m.) ..... my wife, as I slept, ...found a comet in the sky, at which time she woke me..." thus Gottfried Kirch describes the first discovery of a comet by a woman, Maria Winkelmann. The official report would list Kirch as the discoverer, but eventually Winkelmann's credit would become known. Leibniz was an admirer of Winkelmann's talent with "quadrant and telescope." *Lisa Jardine, Ingenious Pursuits, pg 335

1692 David Gregory delivered his inaugural lecture as Savilian professor of astronomy at Oxford. He received his post on the recommendation of Newton. *VFR

1791 Benjamin Bannaker, the outstanding Black self-taught mathematical-astronomer, completed the outline of the boundaries of the federal district, Washington D. C. *VFR

1826 Thomas Jefferson spent his last years actively engaged in managing the University of Virginia. On this day he writes to Charles Bonnycastle, Professor of Natl Philosophy (later mathematics) . "I omitted, in conversn with you yesterday to observe on the arrangement of the Elliptical lecturing room that one third of the whole Area may be saved by the use of lap boards for writing on instead of tables, the room will hold half as many again, and the expence & lumber of tables be spared. a bit of thin board 12. I. square covered or not with cloth to every person is really a more convenient way of writing than a table I am now writing on such an one, and often use it of preference it may be left always on the sitting bench so as to be ready at hand when wanted. a bit of pasteboard, if preferred, might be furnished. I pray you to think on this for the economy of room, and as equivalent to the enlargemt of the room by one half. I salute you with frdshp & esteem *Letters of Thomas Jefferson, http://etext.lib.virginia.edu

1910 Halley’s comet passed perihelion. *VFR The New York Times reported, “Observatories report comet closer; is visible to naked eye in Curacao.” It would reach its maximum viewing brillance in May, with rooftop parties and predictions of doom.

2011 April 21st is when computers take over the world in Terminator. *@imranghory on Twitter

2012 The Lyrid meteor shower is expected to reach maximum intensity overnight from Saturday to Sunday. Meteor showers are generated when Earth plows through streams of debris shed by comets on their path around the sun. These icy, dusty chunks burn up in our planet's atmosphere, leaving behind bright streaks in the sky to commemorate their passing.
The Lyrids' parent comet is called C/1861 G1 Thatcher (Comet Thatcher for short). The Lyrids take their name from the constellation Lyra (The Lyre), because they appear to emanate from this part of the sky. Lyra is a northern constellation, so skywatchers in the Northern Hemisphere generally get much better looks at the Lyrids every year than do folks who live south of the equator. *Mike Wall, SPACE.com

BIRTHS

1652 Michel Rolle (April 21, 1652 – November 8, 1719) was a French mathematician. He is best known for Rolle's theorem (1691), and he deserves to be known as the co-inventor in Europe of Gaussian elimination (1690).*Wik His favorite area of research was the theory of equations. He introduced the symbol we use for nth roots. *VFR (famous to Calculus I students for Rolle's Theorem... and I always tell my students he had a daughter named Tootsie) .

1759 William Farish (baptized on 21 April 1758 {1759 NS}–1837) was a British scientist who was a professor of Chemistry and Natural Philosophy at the University of Cambridge, known for the development of the method of isometric projection and development of the first written university examination.
Farish's father was the Reverend James Farish (1714–1783), vicar of Stanwix near Carlisle. Farish himself was educated at Carlisle Grammar School, entered Magdalene College, Cambridge, as a sizar in 1774, and graduated Senior Wrangler and first in Smith's Prize in 1778. As tutor in 1792, Farish developed the concept of grading students' work quantitatively.
He was Professor of Chemistry at Cambridge from 1794 to 1813, lecturing on chemistry's practical application. Farish's lectures as professor of chemistry, which were oriented towards natural philosophy while the professor of natural and experimental philosophy F. J. H. Wollaston (1762–1828) gave very chemically oriented lectures.
From 1813 to 1837 Farish was Jacksonian Professor of Natural Philosophy. In 1819 Professor Farish became the first president of the Cambridge Philosophical Society.
Farish was also Vicar of St. Giles' and St. Peter from 1800 to 1837.
At Cambridge University, according to Hilkens (1967), Farish was "the first man to teach the construction of machines as a subject in its own right instead of merely using mechanisms as examples to illustrate the principles of theoretical physics or applied mathematics." He further became "famous for his work in applying chemistry and mechanical science to arts and manufactures".
In his lectures on the mechanical principles of machinery used in manufacturing industries, Farish often used models to illustrated particular principles. This models were often especially assembled for these lectures and disassembled for storage afterwards. In order to explain how these models were to be assembled he had developed a drawing technique, which he called "Isometrical Perspective".
Although the concept of an isometric had existed in a rough way for centuries, Farish is generally regarded as the first to provide rules for isometric drawing. In the 1822 paper "On Isometrical Perspective" Farish recognized the "need for accurate technical working drawings free of optical distortion. This would lead him to formulate isometry. Isometry means "equal measures" because the same scale is used for height, width, and depth".
From the middle of the 19th century, according to Jan Krikke (2006) isometry became an "invaluable tool for engineers, and soon thereafter axonometry and isometry were incorporated in the curriculum of architectural training courses in Europe and the U.S. The popular acceptance of axonometry came in the 1920s, when modernist architects from the Bauhaus and De Stijl embraced it". De Stijl architects like Theo van Doesburg used "axonometry for their architectural designs, which caused a sensation when exhibited in Paris in 1923". *Wik

1774 Jean-Baptiste Biot (21 April 1774 – 3 February 1862) was a French physicist, astronomer, and mathematician who established the reality of meteorites, made an early balloon flight, and studied the polarization of light.*Wik He co-developed the Biot-Savart law, that the intensity of the magnetic field produced by current flow through a wire varies inversely with the distance from the wire. He did work in astronomy, elasticity, heat, optics, electricity and magnetism. In pure mathematics, he contibuted to geometry. In 1804 he made a 13,000-feet (5-km) high hot-air balloon ascent with Joseph Gay-Lussac to investigate the atmosphere. In 1806, he accompanied Arago to Spain to complete earlier work there to measure of the arc of the meridian. Biot discovered optical activity in 1815, the ability of a substance to rotate the plane of polarization of light, which laid the basis for saccharimetry, a useful technique of analyzing sugar solutions. *TIS

1882 Percy Williams Bridgman (21 Apr 1882; 20 Aug 1961 at age 79) was an American experimental physicist noted for his studies of materials at high temperatures and pressures. He was awarded the Nobel Prize for Physics in 1946 for his “invention of an apparatus to produce extremely high pressures, and for the discoveries he made therewith in the field of high pressure physics.” He was the first Harvard physicist to receive a Nobel Prize in Physics. In 1908, he began his first experimental work with static high pressures of about 6,500 atmospheres. Eventually, he reached about 400,000 atmospheres. During studies of the phase changes of solids under pressure, he discovered several high-pressure forms of ice. Bridgman also wrote eloquently on matters of general interest in the physics of his day. *TIS

1882 Maurice Kraitchik (April 21, 1882, Minsk - August 19, 1957, Bruxelles) was a Belgian mathematician, author, and game designer. His main interests were the theory of numbers and recreational mathematics.
He is famous for having inspired the two envelopes problem in 1953, with the following puzzle in La mathématique des jeux:
Two people, equally rich, meet to compare the contents of their wallets. Each is ignorant of the contents of the two wallets. The game is as follows: whoever has the least money receives the contents of the wallet of the other (in the case where the amounts are equal, nothing happens). One of the two men can reason: "Suppose that I have the amount A in my wallet. That's the maximum that I could lose. If I win (probability 0.5), the amount that I'll have in my possession at the end of the game will be more than 2A. Therefore the game is favorable to me." The other man can reason in exactly the same way. In fact, by symmetry, the game is fair. Where is the mistake in the reasoning of each man?

Kraitchik wrote several books on number theory during 1922-1930 and after the war, and from 1931 to 1939 edited Sphinx, a periodical devoted to recreational mathematics.

During World War II, Kraïtchik emigrated to the United States, where he taught a course at the New School for Social Research in New York City on the general topic of "mathematical recreations." *Wik

1951 Michael H. Freedman (21 April 1951 in Los Angeles, California, ). In 1986 he received a Fields Medal for his proof of the four-dimensional Poincar´e conjecture. *VFR [The Poincaré conjecture, one of the famous problems of 20th-century mathematics, asserts that a simply connected closed 3-dimensional manifold is a 3-dimensional sphere. The higher dimensional Poincaré conjecture claims that any closed n-manifold which is homotopy equivalent to the n-sphere must be the n-sphere. When n = 3 this is equivalent to the Poincaré conjecture. Smale proved the higher dimensional Poincaré conjecture in 1961 for n at least 5. Freedman proved the conjecture for n = 4 in 1982 but the original conjecture remained open until settled by G Perelman who was offered the 2006 Fields medal for his proof. ] *Wik

DEATHS
1142 Peter Abelard (Petrus Abaelardus or Abailard) (1079 – April 21, 1142) was a medieval French scholastic philosopher, theologian and preeminent logician. The story of his affair with and love for Héloïse has become legendary. The Chambers Biographical Dictionary describes him as "the keenest thinker and boldest theologian of the 12th Century" *Wik

1552 Petrus Apianus (16 April 1495 – 21 April 1552), also known as Peter Apian, was a German humanist, known for his works in mathematics, astronomy and cartography.*Wik His Instrumentum sinuum sivi primi mobilis (1534),  gave tables of his calculations of sines for every minute, with a decimal division of the radius.*Tis  He published important popular works on astronomy and geography. *SAU [His arithmetic is shown in "The Ambasadors" by the younger Hans Holbein]

The book is the one closed on a ruler near the front left leg of the table as shown in the close-up.

1793 John Michell (? 1724, 21 Apr 1793). British geologist and astronomer who was first to devise a realistic estimate of the distance to the stars, discovered physical double stars, and is considered the father of seismology. After the Lisbon earthquake of 1755 (which killed 70,000 people), he suggested that earthquakes set up wave motion in the earth. He noted the increased frequency of earthquakes in volcanic areas. Michell realized that by comparing the time at which earthquakes are felt, the epicentre could be calculated. He invented a torsion balance, a device to measure very small forces, though died before carrying out its purpose to determine the density of the Earth. His rebuilt apparatus was used by Cavendish to make that measurement, which also gives the gravitational constant).*TIS

1825 Johann Friedrich Pfaff (22 December 1765,Stuttgart, - 21 April 1825,Halle) German mathematician who proposed the first general method of integrating partial differential equations of the first order. Pfaff did important work on special functions and the theory of series. He developed Taylor's Theorem using the form with remainder as given by Lagrange. In 1810 he contributed to the solution of a problem due to Gauss concerning the ellipse of greatest area which could be drawn inside a given quadrilateral. His most important work on Pfaffian forms was published in 1815 when he was nearly 50, but its importance was not recognised until 1827 when Jacobi published a paper on Pfaff's method. *TIS

1946 John Maynard Keynes, 1st Baron Keynes was a British economist whose ideas have profoundly affected the theory and practice of modern macroeconomics, as well as the economic policies of governments. He greatly refined earlier work on the causes of business cycles, and advocated the use of fiscal and monetary measures to mitigate the adverse effects of economic recessions and depressions. His ideas are the basis for the school of thought known as Keynesian economics, as well as its various offshoots. (WIkipedia) He once said, "The avoidance of taxes is the only intellectual pursuit that carries any reward. " (John A Paulos on twitter)

1954 Emil Leon Post (February 11, 1897, Augustów – April 21, 1954, New York City) was a mathematician and logician. He is best known for his work in the field that eventually became known as computability theory. In 1936, Post developed, independently of Alan Turing, a mathematical model of computation that was essentially equivalent to the Turing machine model. Intending this as the first of a series of models of equivalent power but increasing complexity, he titled his paper Formulation 1. This model is sometimes called "Post's machine" or a Post-Turing machine, but is not to be confused with Post's tag machines or other special kinds of Post canonical system, a computational model using string rewriting and developed by Post in the 1920s but first published in 1943. Post's rewrite technique is now ubiquitous in programming language specification and design, and so with Church's lambda-calculus is a salient influence of classical modern logic on practical computing. Post devised a method of 'auxiliary symbols' by which he could canonically represent any Post-generative language, and indeed any computable function or set at all.
The unsolvability of his Post correspondence problem turned out to be exactly what was needed to obtain unsolvability results in the theory of formal languages.
In an influential address to the American Mathematical Society in 1944, he raised the question of the existence of an uncomputable recursively enumerable set whose Turing degree is less than that of the halting problem. This question, which became known as Post's problem, stimulated much research. It was solved in the affirmative in the 1950s by the introduction of the powerful priority method in recursion theory.
Post made a fundamental and still influential contribution to the theory of polyadic, or n-ary, groups in a long paper published in 1940. His major theorem showed that a polyadic group is the iterated multiplication of elements of a normal subgroup of a group, such that the quotient group is cyclic of order n − 1. He also demonstrated that a polyadic group operation on a set can be expressed in terms of a group operation on the same set. The paper contains many other important results.*Wik

1965 Sir Edward Victor Appleton (6 Sep 1892, 21 Apr 1965 at age 72) was an English physicist who won the 1947 Nobel Prize for Physics for his discovery of the Appleton layer of the ionosphere. From 1919, he devoted himself to scientific problems in atmospheric physics, using mainly radio techniques. He proved the existence of the ionosphere, and found a layer 60 miles above the ground that reflected radio waves. In 1926, he found another layer 150 miles above ground, higher than the Heaviside Layer, electrically stronger, and able to reflect short waves round the earth. This Appleton layer is a dependable reflector of radio waves and more useful in communication than other ionospheric layers that reflect radio waves sporadically, depending upon temperature and time of day. *TIS

1967 André-Louis Danjon (6 Apr 1890, 21 Apr 1967 at age 76) French astronomer who devised a now standard five-point scale for rating the darkness and colour of a total lunar eclipse, which is known as the Danjon Luminosity Scale. He studied Earth's rotation, and developed astronomical instruments, including a photometer to measure Earthshine - the brightness of a dark moon due to light reflected from Earth. It consisted of a telescope in which a prism split the Moon's image into two identical side-by-side images. By adjusting a diaphragm to dim one of the images until the sunlit portion had the same apparent brightness as the earthlit portion on the unadjusted image, he could quantify the diaphragm adjustment, and thus had a real measurement for the brightness of Earthshine.*TIS

1990 Richard Bevan Braithwaite (15 Jan 1900, 21 Apr 1990 at age 90) was an English philosopher who trained in physics and mathematics, but turned to the philosophy of science. He examined the logical features common to all the sciences. Each science proceeds by inventing general principles from which are deduced the consequences to be tested by observation and experiment. Braithwaite was concerned with the impact of science on our beliefs about the world and the responses appropriate to that. He wrote on the statistical sciences, theories of belief and of probability, decision theory and games theory. He was interested in particular with the laws of probability as they apply to the physical and biological sciences. *TIS

2005 William H Kruskal (October 10, 1919 – April 21, 2005) was an American mathematician and statistician. He is best known for having formulated the Kruskal–Wallis one-way analysis of variance (together with W. Allen Wallis), a widely-used nonparametric statistical method. *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
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