## Wednesday, 31 January 2018

### On This Day in Math - January 31

Joost Bürgi nich at Kepler  monument
on the market-place in the city Weil der Stadt in  Baden-Württemberg

The capacity to blunder slightly is the real marvel of DNA.
Without this special attribute, we would still be anaerobic bacteria
and there would be no music.

The 31st day of the year; 31 = 22 + 33, i.e., The eleventh prime, and third Mersenne prime, it is also the sum of the first two primes raised to themselves. *Number Gossip  (Is there another prime which is the sum of consecutive primes raised to themselves?  A note from Andy Pepperdine of Bath who informed me that $2^2 + 3^3 +5^5 + 7^7 = 826699$, a prime.﻿)

There are exactly 31 positive integers which cannot be written as the sum of two distinct squares; and one of them is the number 31. Finding the 31 is not so hard, they all occur in the first six months of the year.  Proving there are no more is a little tougher.

Jim Wilder ‏@wilderlab offered, The sum of digits of the 31st Fibonacci number (1346269) is 31.

If you like unusual speed limits, the speed limit in downtown Trenton, a small city in northwestern Tennessee, is 31 miles per hour.
And the little teapot on the sign? Well, Trenton also bills itself as the teapot capital of the nation. The 31 mph road sign seems to come from a conflict between Trenton and a neighboring town which I will not name ,...but I will tell you they think of themselves as the white squirrel capital.

31 is also the smallest integer that can be written as the sum of four positive squares in two ways 1+1+4+25; 4+9+9+9.

31 is an evil math teacher number. The sequence of  the maximum number of regions obtained by joining n points around a circle by straight lines begins 2, 4, 8, 16... but for five points, it is 31.

And 31 is also the minimal number of moves to solve the Towers of Hanoi problem with five disks.  (now wondering if there is a mathematical connection between these two ideas other than coincidence)

EVENTS
1599 During an observation of the lunar eclipse, Tycho Brahe discovers that his predictive theory about the movement of the Moon is wrong since the eclipse started 24 minutes before his calculations predicted: he improves on his theory. On March 21 he sent a letter to Longomontanus, in which he reports his revised theory.*Wik

1802 Gauss elected a corresponding member of the St. Petersburg Academy of Science. *VFR

1834 Felix Klein declines to be the successor of J. J. Sylvester at John's Hopkins.  Klein had been offered the position on December 13th of the previous year, but had demanded a salary equal to the departing Sylvester and some form of security for his family which Johns Hopkins did not meet. By October he would send notes to his family, "Gottingen is beginning to make noises."  In the spring of 1836 he took over as Professor at Gottingen (he had been their second choice). *Constance Reid, The Road Not Taken, Mathematical Intelligencer, 1978

1839, Fox Talbot read a paper before the Royal Society, London, to describe his photographic process using solar light, with an exposure time of about 20 minutes: Some Account of the Art of Photogenic Drawing or the Process by which Natural Objects may be made to Delineate Themselves without the Aid of the Artist's Pencil. He had heard that Daguerre of Paris was working on a similar process. To establish his own priority, Fox Talbot had exhibited "such specimens of my process as I had with me in town," the previous week at a meeting of the Royal Institution, before he had this more detailed paper ready to present.*TIS

1939 Hewlett-Packard founded. Their calculators use the “reverse Polish notation” devised by Jan L Lukasiewicz (see here, 1878). *VFR

1939 Joseph Ehrenfried Hofmann began his academic career as a professor of the history of mathematics at the University of Berlin. He is noted for his work on Leibniz, especially the book Leibniz in Paris, 1672–1676: His Growth to Mathematical Maturity. *VFR Leibniz in Paris 1672-1676: His Growth to Mathematical Maturity

1958 Explorer 1 was launched on January 31, 1958 at 22:48 Eastern Time (equal to February 1, 03:48 UTC because the time change goes past midnight). It was the first spacecraft to detect the Van Allen radiation belt, returning data until its batteries were exhausted after nearly four months. It remained in orbit until 1970, and has been followed by more than 90 scientific spacecraft in the Explorer series. *Wik
Actually the Van Allen radiation was detectable by the Russian’s first satellite, Sputnik.  Because the signals were sent in a secret code, it’s signal could not be received by the Russians when it was detecting the radiation of the belt.  *Frederich Pohl, Chasing Science, pg 85

1995 AT&T Bell Laboratories and VLSI Technology announce plans to develop strategies for protecting communications devices from eavesdroppers. The goal would be to prevent problems such as insecure cellular phone lines and Internet transmissions by including security chips in devices. *CHM

2016, Since the year is a leap year beginning on a Friday, the typical calendar page for January takes six lines.  Such months are called perverse months. (The same months will be perverse in a year starting on Saturday.  2016 has three such months, Jan, July and October. 2012 had only two. It is possible for there to be four in a single year. When will that year be?  Is it possible for there to be a year with no perverse months?
There is an inverse relationship between Friday-the-thirteenths and perverse months; so what is good for the calendar makers is bad for the superstitious., so 2016 has only one Friday the 13th

 Image credit: NASA/JPL-Caltech/MSSS/TAMU
2014 The Mars rover's view of its original home planet even includes our moon, just below Earth.
The images, taken about 80 minutes after sunset during the rover's 529th Martian day (Jan. 31, 2014) are available for a broad scene of the evening sky, and a zoomed-in view of Earth and the moon.
The distance between Earth and Mars when Curiosity took the photo was about 99 million miles (160 million kilometers). * NASA

2018 The rare combination of a blue moon (generally the second full moon of a month), a Supermoon (the full moon occurring nearest to perigee when moon is closer to earth), and a total lunar eclipse occurs early in the morning (8:37 am EST). Unfortunately it was only total in the western US. It is the first such triple treat in the US since 1866. *USA Today

BIRTHS
1715 Giovanni Francesco Fagnano dei Toschi (31 Jan 1715 in Sinigaglia, Italy - 14 May 1797 in Sinigaglia, Italy) He proved that the triangle which has as its vertices the bases of the altitudes of any triangle has those altitudes as its bisectors. *VFR  Of all the triangles that could be inscribed in a given triangle, the one with the smallest perimeter is the orthic triangle. This has sometimes been called Fagnano's Problem since it was first posed and answered by Giovanni Francesco Fagnano dei Toschi. Fagnano also was the first to show that the altitudes of the original triangle are the angle bisectors of the orhtic triangle, so the incenter of the orthic triangle is the orthocenter of the original triangle.*pb
He was the son of the mathematician
Giulio Carlo Fagnano. He calculated the integral of the tangent and also proved the reduction  formula $\int x^n \sin {x} dx = -x^n \cos{x}+n\int x^{n-1} \cos{x} dx$

*VFR

1763 The Rt. Rev. John Mortimer Brinkley D.D. (ca. 1763 (Baptized 31 Jan,1763, Woodbridge, Suffolk – 14 September 1835, Dublin) was the first Royal Astronomer of Ireland and later Bishop of Cloyne.
He graduated B.A. in 1788 as senior wrangler and Smith's Prizeman, was elected a fellow of the college and was awarded M.A. in 1791. He was ordained at Lincoln Cathedral in the same year, and in 1792 became the second Andrews Professor of Astronomy in the University of Dublin, which carried the new title of Royal Astronomer of Ireland. Together with John Law, Bishop of Elphin, he drafted the chapter on "Astronomy" in William Paley's Natural Theology. His main work concerned stellar astronomy and he published his Elements of Plane Astronomy in 1808. In 1822 he was elected a Foreign Honorary Member of the American Academy of Arts and Sciences. He was awarded the Copley Medal by the Royal Society in 1824. Brinkley's observations that several stars shifted their apparent place in the sky in the course of a year were disproved at Greenwich by his contemporary John Pond, the Astronomer Royal. In 1826, he was appointed Bishop of Cloyne in County Cork, a position he held for the remaining nine years of his life. Brinkley was elected President of the Royal Astronomical Society in 1831, serving in that position for two years.
He died in 1835 at Leeson Street, Dublin and was buried in Trinity College chapel. He was succeeded at Dunsink Observatory by Sir William Rowan Hamilton. *Wik

1841 Samuel Loyd (31 Jan 1841 ; died 10 Apr 1911)  was an American puzzlemaker who was best known for composing chess problems and games, including Parcheesi, in addition to other mathematically based games and puzzles. He studied engineering and intended to become a steam and mechanical engineer but he soon made his living from his puzzles and chess problems. Loyd's most famous puzzle was the 14-15 Puzzle which he produced in 1878. The craze swept America where employers put up notices prohibiting playing the puzzle during office hours. Loyd's 15 puzzle is the familiar 4x4 arrangement of 15 square numbered tiles in a tray that must be reordered by sliding one tile at a time into the vacant space. *TIS When he offered a cash prize to anyone who could solve the puzzle with 14&15 reversed, it swept the country.  To show it impossible requires only a little group theory; see W. E. Story, “Note on the ‘15’ puzzle,” American Journal of Mathematics, 2, 399–404. For samples of Loyd’s many puzzles, see Mathematical Puzzles of Sam Loyd, edited by Martin Gardner, Dover 1959 [p. xi]. *VFR
Although Lloyd popularized the puzzle in his books and articles, he most certainly did not invent it. Loyd's first article about the puzzle was published in 1886 and it wasn't until 1891 that he first claimed to have been the inventor.  The article mentioned by Story(1878) was dated prior to Loyd's first mention of the puzzle) Here is the history of the puzzle as related by Wikipedia:The puzzle was "invented" by Noyes Palmer Chapman, a postmaster in Canastota, New York, who is said to have shown friends, as early as 1874, a precursor puzzle consisting of 16 numbered blocks that were to be put together in rows of four, each summing to 34. Copies of the improved Fifteen Puzzle made their way to Syracuse, New York by way of Noyes' son, Frank, and from there, via sundry connections, to Watch Hill, RI, and finally to Hartford (Connecticut), where students in the American School for the Deaf started manufacturing the puzzle and, by December 1879, selling them both locally and in Boston, Massachusetts. Shown one of these, Matthias Rice, who ran a fancy woodworking business in Boston, started manufacturing the puzzle sometime in December 1879 and convinced a "Yankee Notions" fancy goods dealer to sell them under the name of "Gem Puzzle". In late-January 1880, Dr. Charles Pevey, a dentist in Worcester, Massachusetts, garnered some attention by offering a cash reward for a solution to the Fifteen Puzzle.
The game became a craze in the U.S. in February 1880, Canada in March, Europe in April, but that craze had pretty much dissipated by July. Apparently the puzzle was not introduced to Japan until 1889.
Noyes Chapman had applied for a patent on his "Block Solitaire Puzzle" on February 21, 1880. However, that patent was rejected, likely because it was not sufficiently different from the August 20, 1878 "Puzzle-Blocks" patent (US 207124) granted to Ernest U. Kinsey.*Wik
Play with an online version here.

1886 George Neville Watson (31 Jan 1886 in Westward Ho!, Devon, England - 2 Feb 1965 in Leamington Spa, Warwickshire, England) studied at Cambridge, and then taught at Cambridge and University College London before becoming Professor at Birmingham. He is best known as the joint author with Whittaker of one of the standard text-books on Analysis. Titchmarsh wrote of Watson's books, "Here one felt was mathematics really happening before one's eyes. ... the older mathematical books were full of mystery and wonder. With Professor Watson we reached the period when the mystery is dispelled though the wonder remains." *SAU

1914 Lev Arkad'evich Kaluznin (31 Jan 1914 in Moscow, Russia - 6 Dec 1990 in Moscow, Russia) Kaluznin is best known for his work in group theory and in particular permutation groups. He studied the Sylow p-subgroups of symmetric groups and their generalisations. In the case of symmetric groups of degree pn, these subgroups were constructed from cyclic groups of order p by taking their wreath product. His work allowed computations in groups to be replaced by computations in certain polynomial algebras over the field of p elements. Despite the fact that the earliest applications of wreath products of permutation groups was due to C Jordan, W Specht and G Polya, it was Kaluznin who first developed special computational tools for this purpose. Using his techniques, he was able to describe the characteristic subgroups of the Sylow p-subgroups, their derived series, their upper and lower central series, and more. These results have been included in many textbooks on group theory. *SAU

1928 Heinz Bauer (31 January 1928 – 15 August 2002) was a German mathematician.
Bauer studied at the University of Erlangen-Nuremberg and received his PhD there in 1953 under the supervision of Otto Haupt and finished his habilitation in 1956, both for work with Otto Haupt. After a short time from 1961 to 1965 as professor at the University of Hamburg he stayed his whole career at the University of Erlangen-Nuremberg. His research focus was the Potential theory, Probability theory and Functional analysis
Bauer received the Chauvenet Prize in 1980 and became a member of the German Academy of Sciences Leopoldina in 1986. Bauer died in Erlangen. *Wik

1929 Rudolf Ludwig Mössbauer (31 Jan 1929 -  14 September 2011) German physicist and co-winner (with American Robert Hofstadter) of the Nobel Prize for Physics in 1961 for his researches concerning the resonance absorption of gamma-rays and his discovery in this connection of the Mössbauer effect. The Mössbauer effect occurs when gamma rays emitted from nuclei of radioactive isotopes have an unvarying wavelength and frequency. This occurs if the emitting nuclei are tightly held in a crystal. Normally, the energy of the gamma rays would be changed because of the recoil of the radiating nucleus. Mössbauer's discoveries helped to prove Einstein's general theory of relativity. His discoveries are also used to measure the magnetic field of atomic nuclei and to study other properties of solid materials. *TIS
Rudolf Mössbauer was an excellent teacher. He gave highly specialized lectures on numerous courses, including Neutrino Physics, Neutrino Oscillations, The Unification of the Electromagnetic and Weak Interactions and The Interaction of Photons and Neutrons With Matter. In 1984, he gave undergraduate lectures to 350 people taking the physics course. He told his students: “Explain it! The most important thing is, that you are able to explain it! You will have exams, there you have to explain it. Eventually, you pass them, you get your diploma and you think, that's it! – No, the whole life is an exam, you'll have to write applications, you'll have to discuss with peers... So learn to explain it! You can train this by explaining to another student, a colleague. If they are not available, explain it to your mother – or to your cat!” *Wik

1945 Persi Warren Diaconis (January 31, 1945;  ) is an American mathematician and former professional magician. He is the Mary V. Sunseri Professor of Statistics and Mathematics at Stanford University. He is particularly known for tackling mathematical problems involving randomness and randomization, such as coin flipping and shuffling playing cards.
Diaconis left home at 14 to travel with sleight-of-hand legend Dai Vernon, and dropped out of high school, promising himself that he would return one day so that he could learn all of the math necessary to read William Feller's famous two-volume treatise on probability theory, An Introduction to Probability Theory and Its Applications. He returned to school (City College of New York for his undergraduate work graduating in 1971 and then a Ph.D. in Mathematical Statistics from Harvard University in 1974), and became a mathematical probabilist.
According to Martin Gardner, at school Diaconis supported himself by playing poker on ships between New York and South America. Gardner recalls that Diaconis had "fantastic second deal and bottom deal".
Diaconis is married to Stanford statistics professor Susan Holmes. *Wik

DEATHS
1632 Joost Bürgi (28 Feb 1552, 31 Jan 1632) Swiss watchmaker and mathematician who invented logarithms independently of the Scottish mathematician John Napier. He was the most skilful, and the most famous, clockmaker of his day. He also made astronomical and practical geometry instruments (notably the proportional compass and a triangulation instrument useful in surveying). This led to becoming an assistant to the German astronomer Johannes Kepler. Bürgi was a major contributor to the development of decimal fractions and exponential notation, but his most notable contribution was published in 1620 as a table of antilogarithms. Napier published his table of logarithms in 1614, but Bürgi had already compiled his table of logarithms at least 10 years before that, and perhaps as early as 1588.
*TIS  I posted about Burgi and his work w/ "proto" logarithms here if you would like more detail.

1903 Norman Macleod Ferrers; (11 Aug 1829 in Prinknash Park, Upton St Leonards, Gloucestershire, England - 31 Jan 1903 in Cambridge, England)  John Venn wrote of him,.. ,
the Master, Dr Edwin Guest, invited Ferrers, who was by far the best mathematician amongst the fellows, to supply the place. His career was thus determined for the rest of his life. For many years head mathematical lecturer, he was one of the two tutors of the college from 1865. As lecturer he was extremely successful. Besides great natural powers in mathematics, he possessed an unusual capacity for vivid exposition. He was probably the best lecturer, in his subject, in the university of his day.
It was as a mathematician that Ferrers acquired fame outside the university. He made many contributions of importance to mathematical literature. His first book was "Solutions of the Cambridge Senate House Problems, 1848 - 51". In 1861 he published a treatise on "Trilinear Co-ordinates," of which subsequent editions appeared in 1866 and 1876. One of his early memoirs was on Sylvester's development of Poinsot's representation of the motion of a rigid body about a fixed point. The paper was read before the Royal Society in 1869, and published in their Transactions. In 1871 he edited at the request of the college the "Mathematical Writings of George Green" ... Ferrers's treatise on "Spherical Harmonics," published in 1877, presented many original features. His contributions to the "Quarterly Journal of Mathematics," of which he was an editor from 1855 to 1891, were numerous ... They range over such subjects as quadriplanar co-ordinates, Lagrange's equations and hydrodynamics. In 1881 he applied himself to study Kelvin's investigation of the law of distribution of electricity in equilibrium on an uninfluenced spherical bowl. In this he made the important addition of finding the potential at any point of space in zonal harmonics (1881).
Ferrers proved the proposition by Adams that "The number of modes of partitioning (n) into (m) parts is equal to the number of modes of partitioning (n) into parts, one of which is always m, and the others (m) or less than (m). " with a graphic transformation that is named for him. *SAU

1934 Duncan MacLaren Young Sommerville (24 Nov 1879 in Beawar, Rajasthan, India - 31 Jan 1934 in Wellington, New Zealand) Sommerville studied at St Andrews and then had a post as a lecturer there. He left to become Professor of Pure and Applied mathematics at Victoria College, Wellington New Zealand. He worked on non-Euclidean geometry and the History of Mathematics. He became President of the EMS in 1911. *SAU

1966 Dirk Brouwer (1 Sep 1902; 31 Jan 1966) Dutch-born U.S. astronomer and geophysicist known for his achievements in celestial mechanics, especially for his pioneering application of high-speed digital computers for astronomical computations. While still a student he determined the mass of Titan from its influence on other Saturnian moons. Brouwer developed general methods for finding orbits and computing errors and applied these methods to comets, asteroids, and planets. He computed the orbits of the first artificial satellites and from them obtained increased knowledge of the figure of the earth. His book, Methods of Celestial Mechanics, taught a generation of celestial mechanicians. He also redetermined astronomical constants.*TIS

1973 Noel Bryan Slater, often cited NB Slater, (29 July 1912 in Blackburn, Lancashire, England - January 31 1973 in Hull, England) was a British mathematician and physicist who worked on including statistical mechanics and physical chemistry, and probability theory.*Wik

1995 George Robert Stibitz (30 Apr 1904, 31 Jan 1995) U.S. mathematician who was regarded by many as the "father of the modern digital computer." While serving as a research mathematician at Bell Telephone Laboratories in New York City, Stibitz worked on relay switching equipment used in telephone networks. In 1937, Stibitz, a scientist at Bell Laboratories built a digital machine based on relays, flashlight bulbs, and metal strips cut from tin-cans. He called it the "Model K" because most of it was constructed on his kitchen table. It worked on the principle that if two relays were activated they caused a third relay to become active, where this third relay represented the sum of the operation. Also, in 1940, he gave a demonstration of the first remote operation of a computer.*TIS

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