Tuesday 6 June 2023

This Day in Math - June 6

Map of 2012 Transit of Venus visibility *Eclipse Maps

No mathematician can be
a complete mathematician unless
he is also something of a poet.
~Karl Weierstrass

The 157th day of the year; 2157 is the smallest "apocalyptic number," i.e., a number of the form 2n that contains '666'. *Prime Curios (Can you find an apocalyptic number of the form 3n)

157 is prime and it's reverse, 751 is also prime. 157 is also the middle value in a sexy triplet (three primes successively differing by six; 151, 157, 163). 751 is also a sexy prime with 757.

157 is also the largest solution I know for a prime, p, such that \( \frac{p^p-p!}{p} \) is prime.

The number 157 in base ten is equal to \(31_{[52]}\), but don't worry if you get that backwards,
\(52_{[31]}\) is also equal to 157 in decimal. Can you find other examples of reversible numeral/base that give the same decimal value?

157 is a prime which is the arithmetic average of the squares of four consecutive primes, (7^2 + 11^2 + 13^2 + 17^2)/4 = 157.  There are only two such year days that meet this definition. and this is the smaller.  

And from Fermat's Library @fermatslibrary In 1993 Don Zagier found the smallest rational right triangle with area 157. He used sophisticated techniques using elliptic curves paired with a lot of computational power. If he could do that, certainly you ought to be able to find the smallest rational right triangle with area of 1.... (OK trick question, ask your teacher to explain)


  Fermat writes to Digby to repeat challenges he had set in January.  1) Find a cube that when added to the sum of its aliquot parts is a square.   2)  Find a square that when increased by the sum of its aliquot parts is a cube.  He added that \(7^3\) is not the only solution.  Can you solve either, or both?

In 1799, the first definitive prototype meter bars (mètre des Archives) and kilograms were constructed in platinum. This followed the legal definition of the metric system by the French National Assembly on 7 Apr 1795, that was itself established during the famous measurements of the Earth's meridian between Dunkerque and Barcelona. The use of a metal bar to define the standard meter continued until replaced in 1960 by a definition based upon a number of wavelengths of light from a certain spectroscopic light source.*TIS

1902 Scottish chemist professor James Dewar exhibits air in the solid state and a jet of liquid air rising six feet above it with beautiful effects, before the Prince and Princess of Wales. *Great Geek Manual
James Dewar lecturing at the Royal Institution, painting by Henry J. Brooks, 1904 (Royal Institution)

1944, Supreme Allied Commander General Dwight D. Eisenhower gives the go-ahead for largest amphibious military operation in history: Operation Overlord, code named D-Day, the Allied invasion of northern France.

1984 Sweden issued a series of stamps celebrating the centenary of their Patent System. One shows a tetrahedral container patented in 1948. [Scott #1501]. *VFR 

1984 Tetris is a Soviet tile-matching puzzle video game originally designed and programmed by Alexey Pajitnov. It was released on June 6, 1984. A nice post with ten things you did't know about about Tetris is at this blog from Wallifaction, a very good history blog by Adam Richter

2012 Last Chance. The most recent transit of Venus when observed from Earth took place on June 8, 2004. The event received significant attention, since it was the first Venus transit to take place after the invention of broadcast media. No human alive at the time had witnessed a previous Venus transit, since the previous Venus transit took place on December 6, 1882. The next transit of Venus will occur on June 5–June 6 in 2012. After 2012, the next transits of Venus will be in December 2117 and December 2125.


1436 Johann Mueller (6 June 1436 – 6 July 1476) , AKA Johannes Regiomontanus after the Latinization of his hometown, Konigsburg. He is the founder of trigonometry as an independent science. The spherical law of sines was first presented by Johann Muller, in his De Triangulis Omnimodis in 1464. This was the first book devoted wholly to trigonometry (a word not then invented). David E. Smith suggests that the theorem was Muller's creation.
The ideas behind the law of sines, like those of the law of cosines, predate the word sine by over a thousand years. Theorems in Euclid on lengths of chords are essentially the same ideas we now call the law of sines. What we now call the law of sines for plane triangles was known to Ptolemy. By the tenth century Abu'l Wefa had clearly expounded the spherical law of sines. It seems that the term "law of sines" was applied sometime near 1850, but I am unsure of the origin of the phrase.

"In Jan 1472 he made observations of a comet which were accurate enough to allow it to be identified with Halley's comet 210 years later (being three returns of the 70 year period comet). He also observed several eclipses of the Moon. His interest in the motion of the Moon led him to make the important observation that the method of lunar distances could be used to determine longitude at sea. However, instruments of the time lacked the necessary accuracy to use the method at sea. " *TIS {There is a nice blog at The Renaissance Mathematicus about the important role Regiomontanus played in scientific publishing.}

1553 Bernardino Baldi (6 June 1553 – 10 October 1617) was an Italian mathematician and writer.
Baldi descended from a noble family from Urbino, Marche, where he was born. He pursued his studies at Padua, and is said to have spoken about sixteen languages during his lifetime, though according to Tiraboschi the inscription on his tomb limits the number to twelve.
The appearance of the plague at Padua forced him to return to his native city. Shortly afterwards he was called to act as tutor to Ferrante Gonzaga, from whom he received the rich abbey of Guastalla. The oldest biography of Nicolaus Copernicus was completed on 7 October 1588 by him. He held office as abbot for 25 years, and then returned once again to Urbino. In 1612 he was employed by the duke as his envoy to Venice. Baldi died at Urbino on 12 October 1617.
He is said to have written upwards of a hundred different works, the chief part of which have remained unpublished. His various works show his abilities as a theologian, mathematician, geographer, antiquary, historian and poet. One of these has been recently found and is now at the Univ. of Oklahoma.
"Baldi is known to have written a treatise on sun dials and timekeeping. However, this treatise was never published and, since 1783, it has been considered lost. Now we are happy to announce that it has been recently acquired by the History of Science Collections, digitized in high resolution, and made available for study in the Collections’ Online Galleries." The Cronica dei Matematici (published at Urbino in 1707) is an abridgment of a larger work on which he had written for twelve years, and was intended to contain the lives of more than two hundred mathematicians. His life has been written of by Affò, Mazzucchelli and others. *Wik

1580 Govaert Wendelen (6 June 1580 – 24 October 1667) was a Flemish astronomer who was born in Herk-de-Stad. He is also known by the Latin name Vendelinus. His name is sometimes given as Godefroy Wendelin; his first name spelt Godefroid or Gottfried.
Around 1630 he measured the distance between the Earth and the Sun using the method of Aristarchus of Samos. The value he calculated was 60% of the true value (243 times the distance to the Moon; the true value is about 384 times; Aristarchus calculated about 20 times).
In 1643 he recognized that Kepler's third law applied to the satellites of Jupiter.
Wendelin corresponded with Mersenne, Gassendi and Constantijn Huygens.
The crater Vendelinus on the Moon is named after him.
Wendelin died in Ghent on 24 October 1667. *Wik

1842 Henry Martyn Taylor (6 June 1842, Bristol – 16 October 1927, Cambridge) born in Bristol, England. He was a fellow at Trinity College, Cambridge, and is most remembered because he devised a Braille notation when he was overtaken by blindness in 1894, when engaged in the preparation of an edition of Euclid for the Cambridge University Press. By means of his ingenious and well thought out Braille notation he was enabled to transcribe many advanced scientific and mathematical works, and in 1917, with the assistance of Mr. Emblen, a blind member of the staff of the National Institute for the Blind, he perfected it. It was recognised as so comprehensive that it was soon adopted as the standard mathematical and chemical notation. It seems that in the US it is more common to use the Nemeth code for mathematics and science symbols, first developed around 1947. I am not sure about usage at the present time in the rest of the world.

1850 Karl Ferdinand Braun (6 June 1850 – 20 April 1918) was a German inventor, physicist and Nobel laureate in physics. Braun contributed significantly to the development of the radio and television technology: he shared with Guglielmo Marconi the 1909 Nobel Prize in Physics.
Braun was born in Fulda, Germany, and educated at the University of Marburg and received a Ph.D. from the University of Berlin in 1872. In 1874 he discovered that a point-contact semiconductor rectifies alternating current. He became director of the Physical Institute and professor of physics at the University of Strassburg in 1895.
In 1897 he built the first cathode-ray tube (CRT) and cathode ray tube oscilloscope. CRT technology has been replaced by flat screen technologies (such as liquid crystal display (LCD), light emitting diode (LED) and plasma displays) on television sets and computer monitors. The CRT is still called the "Braun tube" in German-speaking countries (Braunsche Röhre) and in Japan (ブラウン管: Buraun-kan). *Wik

1857 Aleksandr Mikhailovich Lyapunov (June 6[O.S. May 25] 1857 – November 3, 1918) born in Yaroslavl, Russia. He was the creator of the modern theory of stability of differential equations especially as applied to mechanical systems. He also proved the Central Limit Theorem under weaker hypotheses than his predecessors. *VFR He was a student of Chebyshev. In 1901, Lyapunov gave the first prominent proof of the Central Limit Theorem, which made the CLT one of the foundations of probability theory today. (Unlike the classical CLT, Lyapunov’s condition only requires the random variables in question to be independent instead of both independent and identically distributed.)

1882 Clement Vavasor Durell (6 June 1882 in Fulbourn, near Cambridge, England, -10 December 1968 in South Africa) Durell was educated at Felsted School and, while still at school, he published his first note in the Mathematical Gazette, the journal of the Mathematical Association. The note was A geometrical method of trisecting any angle with the aid of a rectangular hyperbola written jointly with W F Beard.
Durell joined the Mathematical Association in 1900, the year in which he entered Clare College, Cambridge, to study mathematics. He was a First Class student in the Mathematical Tripos examinations, graduating in 1904. He was appointed as a mathematics teacher at Gresham's School immediately after graduating, and in the following year of 1905 he moved to take up the post of mathematics master at Winchester College.
Soon after taking up this post Durell's first textbook Elementary Problem Papers (1906) was published. He was promoted to senior mathematics master at Winchester College in 1910 and began publishing a series of articles in the Mathematical Gazette. Before the outbreak of World War I, Durell published The arithmetic syllabus in secondary schools (1911) and Analysis and projective geometry (1911) in the Mathematical Gazette. During World War I, Durell served in the Royal Garrison Artillery as a lieutenant. After the end of the war he returned to Winchester College and began publishing a series of articles in the Mathematical Gazette and a remarkable series of textbooks which would make him the best known writer of English school mathematics texts.

As well as writing articles for the Mathematical Gazette such as The use of limits in elementary geometry (1925) and The teaching of loci in the elementary geometry course to school certificate stage (1936), he was also actively involved with the committee work of the Mathematical Association and its report production. He wrote reports The teaching of geometry in schools (1925), Memo from the Girls' Schools' Committee: Mathematics for girls (1926), and Questionnaire on the teaching of mathematics in evening continuation schools (1926). Among the books he wrote around this time were: Readable relativity (1926), A Concise Geometry (1928), Matriculation Algebra (1929), Arithmetic (1929), Advanced Trigonometry (1930), A shorter geometry (1931), The Teaching of Elementary Algebra (1931), Elementary Calculus (1934), A School Mechanics (1935), and General Arithmetic (1936). In a catalogue produced by the Mathematical Association's publishers G Bell & Sons in 1934, they listed 20 textbooks by Durell and write
There can indeed be few secondary schools in the English-speaking world in which some at least of Mr Durell's books are not now employed in the teaching of mathematics.

1906 Max Zorn (June 6, 1906 in Krefeld, Germany – March 9, 1993 in Bloomington, Indiana, United States) To his chagrin, he is most famous for discovering something yellow and equivalent to the Axiom of Choice. *VFR (with a smile, I'm sure) He was an algebraist, group theorist, and numerical analyst. He is best known for Zorn's lemma, a powerful tool in set theory that is applicable to a wide range of mathematical constructs such as vector spaces, ordered sets, etc. Zorn's lemma was first discovered by K. Kuratowski (see June 18) in 1922, and then independently by Zorn in 1935.*Wik  Interesting that he was born on 6/6/6. 

1933  Heinrich Rohrer (6 June 1933 – 16 May 2013) was a Swiss physicist who shared half of the 1986 Nobel Prize in Physics with Gerd Binnig for the design of the scanning tunneling microscope (STM). The other half of the Prize was awarded to Ernst Ruska. Ruska's electron microscope of the 1930s was unable to show surface structure at the atomic level. Rohrer and Binnig began work in 1978 on a scanning tunneling microscope in which a fine probe passes within a few angstroms of the surface of the sample. A positive voltage on the probe enables electrons to move from the sample to the probe by the tunnel effect, and the detected current can used to keep the probe at a constant distance from the surface. As the probe moves in parallel lines, a 3D image of the surface can be constructed.


1834 Erastus Lyman De Forest (27 June 1834 in Watertown, Connecticut, USA - 6 June 1888 in Watertown, Connecticut, USA) His parents were Lucy Starr Lyman and Dr John De Forest. He was named after his mother's father, Erastus Lyman, who was from Litchfield, Connecticut. Both sides of the family were well off and Erastus was born into a privileged place in society. John De Forest graduated from Yale College and wished his son to follow in his footsteps as indeed he did, entering Yale at the age of sixteen to study mathematics. He was awarded his B.A. in 1854 and his father celebrated the occasion by endowing the De Forest Mathematical Prize at Yale. Erastus's maternal grandfather celebrated the occasion by making him a large bequest.

De Forest remained at Yale to study engineering and at this time was a fellow student with J Willard Gibbs who entered Yale in the year that De Forest was awarded his B.A. In 1856 De Forest was awarded a Ph.B. by Yale and then in February of the following year he set off with his aunt for New York to begin a journey with her to Havana. However, before the ship was due to depart De Forest vanished leaving his luggage. When his family could find no trace of him they put an advertisement in the New York Times asking for information. They received a reply which told them his body was in East River but a search revealed nothing.

For two years De Forest's family continued to make desperate efforts to locate him but receiving not a shred of information they came to believe that he must have been murdered. It was more than two years after he vanished that John De Forest received a letter from his son, posted in Australia. De Forest, depressed with his privileged life, had travelled to California where he had got a job at a mine. After a while he was appointed as a teacher in a private school where he taught for about a year before going to Australia where again he taught, this time at Melbourne Church of England Grammar School in South Yarra. After more than four years away, he returned to the United States in 1861 visiting India and England on his way. He returned to Europe in 1863 for a lengthy trip which lasted until 1865.

From his return to Connecticut in 1865 he devoted himself to the study of mathematics. After publishing papers interpolation and its applications, he was asked by his uncle, who was President of the Knickerbocker Life Insurance Company of New York, to examine the liabilities that the company's life policies involved. De Forest became deeply involved in improving mortality tables, publishing over 20 papers on the topic between 1870 and 1885.

The remarkable contributions of De Forest to statistics had little or no influence on the subject since those who later developed similar ideas were totally unaware of his contributions. This was for a number of reasons. De Forest was not associated with any institution so lacked the visibility that such a position would have meant. He worked in the United States at a time when little of mathematical significance was happening in that country. Also he published his work in somewhat obscure American journals. His contributions were recognized, however, by Pearson whose attention was drawn to De Forest's papers. Pearson acknowledged De Forest's priority in deriving the chi square distribution. The book contains reprints of four of De Forest's papers as well as a biographical article written by J Anderson. His life and work are both discussed by Stigler . Stigler uses information on De Forest available to him from a well researched but unpublished work on De Forest by H H Wolfenden.

De Forest never married and cared for his father for many years until his death in 1885, from which time his own health began to deteriorate. Shortly before he died he founded the Erastus L De Forest Professorship of Mathematics at Yale. *SAU

1898 Henry Perigal, Jr. FRAS MRI (1 April 1801 – 6 June 1898) was a British stockbroker and amateur mathematician, known for his dissection-based proof of the Pythagorean theorem and for his unorthodox belief that the moon does not rotate.
In his booklet Geometric Dissections and Transpositions (London: Bell & Sons, 1891) Perigal provided a proof of the Pythagorean theorem based on the idea of dissecting two smaller squares into a larger square. The five-piece dissection that he found may be generated by overlaying a regular square tiling whose prototile is the larger square with a Pythagorean tiling generated by the
two smaller squares. Perigal had the same dissection printed on his business cards, and it also appears on his tombstone.

As well as being interested in mathematics, Perigal was an accomplished lathe worker, and made models of mathematical curves for Augustus De Morgan. He believed (falsely) that the moon does not rotate with respect to the fixed stars, and used his knowledge of curvilinear motion in an attempt to demonstrate this belief to others. *Wik

1928 Luigi Bianchi (January 18, 1856 – June 6, 1928) He did fundamental work on Lie groups. *VFR He was a leading member of the vigorous geometric school which flourished in Italy during the later years of the 19th century and the early years of the twentieth century.
In 1898, Bianchi worked out the Bianchi classification of nine possible isometry classes of three-dimensional Lie groups of isometries of a (sufficiently symmetric) Riemannian manifold. As Bianchi knew, this is essentially the same thing as classifying, up to isomorphism, the three-dimensional real Lie algebras. This complements the earlier work of Lie himself, who had earlier classified the complex Lie algebras.
Through the influence of Luther P. Eisenhart and Abraham Haskel Taub, Bianchi's classification later came to play an important role in the development of the theory of general relativity. Bianchi's list of nine isometry classes, which can be regarded as Lie algebras, Lie groups, or as three dimensional homogeneous (possibly nonisotropic) Riemannian manifolds, are now often called collectively the Bianchi groups.
In 1902, Bianchi rediscovered what are now called the Bianchi identities for the Riemann tensor, which play an even more important role in general relativity. (They are essential for understanding the Einstein field equation.) According to Tullio Levi-Civita, these identities had first been discovered by Ricci in about 1880, but Ricci apparently forgot all about the matter, which led to Bianchi's rediscovery! *Wik

1943 Guido Fubini (19 January 1879 – 6 June 1943) He is best known for a theorem on the exchange of order of integration. his research focused primarily on topics in mathematical analysis, especially differential equations, functional analysis, and complex analysis; but he also studied the calculus of variations, group theory, non-Euclidean geometry, and projective geometry, among other topics. With the outbreak of World War I, he shifted his work towards more applied topics, studying the accuracy of artillery fire; after the war, he continued in an applied direction, applying results from this work to problems in electrical circuits and acoustics. *Wik

1972 Abraham Adrian Albert (9 November 1905 Chicago, Illinois, USA - 6 June 1972 Chicago, Illinois, USA) A Adrian Albert's parents were Russian. His father, Elias Albert, came to the United States from England and had set up a retail business. His mother, Fannie Fradkin, had come to the United States from Russia. Adrian was the second of Elias and Fannie's three children, but he also had both a half-brother and half-sister from his mother's side.
Albert completed his B.S. degree in 1926 and was awarded his Master's degree in the following year. He remained at the University of Chicago undertaking research under L E Dickson's supervision.
By the time that he received his doctorate Albert was a married man, having married Freda Davis on 18 December 1927.
In his doctoral thesis Albert had made considerable progress in classifying division algebras. It was an impressive piece of work and it led to him being awarded a National Research Council Fellowship to enable him to undertake postdoctoral study at Princeton. He spent nine months at Princeton in 1928-29 and this was an important period for Albert since during his time there Lefschetz suggested that he look at open problems in the theory of Riemann matrices. These matrices arise in the theory of complex manifolds and Albert went on to write an important series of papers on these questions over the following years.

Albert was then offered a post as an instructor at Columbia University and he worked there for two years from 1929 to 1931. His first paper A determination of all normal division algebras in sixteen units was published in 1929. It was based on the second half of his doctoral thesis but Albert had, by this time, pushed the ideas further classifying division algebras of dimension 16 over their centres. The case of dimension 9, the next smaller case, had been solved by Wedderburn.

Albert returned to the University of Chicago in 1931 where he was appointed as assistant professor. He remained on the staff there for the rest of his life being promoted to associate professor in 1937 and full professor in 1941. During the years 1958 to 1962 he was chairman of the Chicago Department.

Shortly after beginning his second three year term as Chairman of the Department Albert was asked to take on the post of Dean of Physical Sciences. He served Chicago for 9 year in the role until 1971.
His main work was on associative algebras, non-associative algebras, and Riemann matrices. He worked on classifying division algebras building on the work of Wedderburn but Brauer, Hasse and Emmy Noether got the main result first. Albert's major contribution is, however, detailed in a joint paper with Hasse. Albert's book Structure of Algebras, published in 1939, remains a classic. The content of this treatise was the basis of the Colloquium Lectures which he gave to the American Mathematical Society in 1939.
Albert's work on Riemann matrices was, as we mentioned above, a consequence of suggestions made by Lefschetz.
During the Second World War Albert contributed to the war effort as associate director of the Applied Mathematics Group at Northwestern University which tackled military problems. Another interest of Albert's, which appears to have been prompted by the War, was that of cryptography. He lectured to the American Mathematical Society on Some mathematical aspects of cryptography at the Society's meeting in November 1941.
Albert investigated just about every aspect of non-associative algebras.
Albert received many honours for his outstanding achievements. He was elected to the National Academy of Sciences in 1943, the Brazilian Academy of Sciences in 1952, and the Argentine Academy of Sciences in 1963. He served as chairman of the Mathematics Section of the National Research Council from 1958 to 1961, and President of the American Mathematical Society in 1965-66. *SAU

1977 Stefan Bergman (5 May 1895 in Częstochowa, Russian Empire (now Poland)- 6 June 1977 in Palo Alto, California, USA) Stefan Bergman (5 May 1895 – 6 June 1977) was a Polish-born American mathematician whose primary work was in complex analysis. He is best known for the kernel function he discovered while at Berlin University in 1922. This function is known today as the Bergman kernel. Bergman taught for many years at Stanford University, and served as an advisor to several students.
Bergman received his Ph.D. at Berlin University in 1921 for a dissertation on Fourier analysis. His adviser, Richard von Mises, had a strong influence on him, lasting for the rest of his career. In 1933, Bergman was forced to leave his post at the Berlin University because he was a Jew. He fled first to Russia, where he stayed until 1939, and then to Paris. In 1939, he emigrated to the United States, where he would remain for the rest of life. He was elected a Fellow of the American Academy of Arts and Sciences in 1951. In 1962 he was an invited speaker at the International Congress of Mathematicians in Stockholm (On meromorphic functions of several complex variables). He died in Palo Alto, California, aged 82.
The Stefan Bergman Prize in mathematics was initiated by Bergman's wife in her will, in memory of her husband's work. The American Mathematical Society supports the prize and selects the committee of judges. The prize is awarded for, "the theory of the kernel function and its applications in real and complex analysis; or function-theoretic methods in the theory of partial differential equations of elliptic type with a special attention to Bergman's and related operator methods." *Wik

1985 András P Huhn (Szeged, 26 January 1947 – Szeged, 6 June 1985) was a Hungarian mathematician. Huhn's theorem on the representation of distributive semilattices is named after him. At the height of his creative powers at the age of 38, Huhn was killed in a tragic accident. *Wik

1946 Jean-Louis Loday (12 January 1946 in Le Pouliguen, Pays de la Loire, France
- 6 June 2012 in Les Sables-d'Olonne, France) was a French mathematician who worked on cyclic homology and who introduced Leibniz algebras (sometimes called Loday algebras) and Zinbiel algebras. He occasionally used the pseudonym Guillaume William Zinbiel, formed by reversing the last name of Gottfried Wilhelm Leibniz.
Loday died in a tragic boating accident, falling from his boat off Les Sables-d'Olonne. *Wik *SAU

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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