Wednesday 31 May 2023

On This Day in Math - May 31


Geometry is the science of 
correct reasoning on incorrect figures.
~George Polya

The 151st Day of the Year

151 is the larger of a pair of twin primes

The 151st day of the year; The smallest prime that begins a 3-run of sums of 5 consecutive primes: 151 + 157 + 163 + 167 + 173 = 811; and 811 + 821 + 823 + 827 + 829 = 4111; and 4111 + 4127 + 4129 + 4133 + 4139 = 20639. *Prime Curios... Can you find the smallest 4-run example?

151 is the 36th prime number, and a Palindromic Prime. Did I ever mention that palindrome is drawn almost directly from an Ancient Greek word that literally means "running back again." First used in English in 1636 in "Camdon's Remains Epitaths".

151 is also the mean (and median) of the first five three digit palindromic primes, 101, 131, 151, 181, 191

Thanks to Derek Orr, who also pointed out that any day in May (in non-leap year) 5/d is such that 5! + d = year day, so today, 5/31, is 5!+31=151.

151 is an undulating palindrome in base 3 (12121)

Thanks to Derek Orr, who also pointed out that any day in May (in non-leap year) 5/d is such that 5! + d = year day

In 1927 Babe Ruth hit 60 Home Runs, a long lasting record. He hit them in 151 games.

And from base to torch, Lady Liberty is 151 ft tall.

$151 is the largest prime amount you can make with three distinct US bank notes.


1503 Copernicus received a doctoral degree in canon law from the University of Ferrara. *VFR

1676 Antonie van Leeuwenhoek describes the little animals he sees through a microscope. "The 31th of May, I perceived in the same water more of those Animals, as also some that were somewhat bigger. And I imagine, that [ten hundred thousand] of these little Creatures do not equal an ordinary grain of Sand in bigness: And comparing them with a Cheese-mite (which may be seen to move with the naked eye) I make the proportion of one of these small Water-creatures to a Cheese-mite, to be like that of a Bee to a Horse: For, the circumference of one of these little Animals in water, is not so big as the thickness of a hair in a Cheese-mite." *The Collected Letters of Antoni van Leeuwenhoek (1957), Vol. 2, 75.

1753 A View of the Relation between the Celebrated. Dr. Halley's Tables, and the Notions of Mr. De Buffon, for Establishing a Rule for the Probable Duration of the Life of Man; By Mr. William Kersseboom, of the Hague. Translated from the French, by James Parsons, M. D. and F. R. S. read by the Royal Society on May 31.  

1764 “I went this far with him: ‘Sir, allow me to ask you one question. If the Church should say to you, ‘two and three make ten,’ what would you do? ‘Sir,’ said he, ‘I should believe it, and I should count like this: one, two, three, four, ten.’ I was now fully satisfied.” From Boswell’s Journal as quoted by J. Gallian, Contemporary Abstract Algebra, p. 43. *VFR  (Now you know, It was Boswell who invented Base Five... )

1780 Laplace’s “Memoir on Probality” read to the Academy of Sciences on this date.  This and his “Memoir on the Probability of events” are among the most important and the most difficult works in the early history of mathematical probability, and together they are the are the most influential 18th Century on the use of probability in inference. The History of Statistics, S. M. Stigler

1790 US Copyright law passed. *VFR  the first copyright law is enacted under the new United States Constitution. Modeled off Britain's Statute of Anne, the new law is relatively limited in scope, protecting books, maps, and charts for only fourteen years with a renewal period of another fourteen years.

1796 Gauss records in his diary a prime number theorem conjecture. Clifford Pickover, in “The Math Book”, points out that 1796 was “an auspicious year for Gauss, when his ideas gushed like a fountain from a fire hose.” In addition to the construction of the 17-gon in March, and the prime number theorem conjecture, he proved that every positive number could be expressed as the sum of (at most) three triangular numbers in July, and another about solutions of polynomials in October.
On May 31 he conjectured that π(n), the number of primes less than n is approximated (for large n) by the area under the logarithmic integral (from 2 to n I assume).
Based on the tables by Anton Felkel and Jurij Vega, Adrien-Marie Legendre conjectured in the same year that π(x) is approximated by the function x/(ln(x)-1.08),. Gauss considered the same question and he came up with his own approximating function, the logarithmic integral li(x), although he did not publish his results. Both Legendre's and Gauss' formulas imply the same conjectured asymptotic equivalence of π(x) = x / ln(x), although Gauss' approximation is closer in terms of the differences instead of quotients.
Most teachers tell the story of Gauss as a nine-year old summing the digits from 1 to 100 in his head. Here is another nice Gauss anecdote about his ability to do mental calculations: Once, when asked how he had been able to predict the trajectory of Ceres with such accuracy he replied, "I used logarithms." The questioner then wanted to know how he had been able to look up so many numbers from the tables so quickly. "Look them up?" Gauss responded. "Who needs to look them up? I just calculate them in my head!"

1813 Louis Poinsot elected to the mathematics section of of the French Acad´emie des Sciences, replacing Lagrange. [DSB 11, 61] *VFR Although little known today, he was a French mathematician and physicist. Poinsot was the inventor of geometrical mechanics, showing how a system of forces acting on a rigid body could be resolved into a single force and a couple. When Gustave Eiffel built the famous tower, he included the names of 72 prominent French scientists on plaques around the first stage, Poinsot was included.*Wik

1823 In a letter to a cousin, William Rowan Hamilton disclosed that he had made a “very curious discovery.” It is believed that he was referring to the characteristic function. [Thanks to Howard Eves] *VFR

clockworks for Elizabeth Tower Clock
The Elizabeth Tower, which stands at the north end of the Houses of Parliament, was completed in 1859 and the Great Clock started on 31 May, with the Great Bell's strikes heard for the first time on 11 July and the quarter bells first chimed on 7 September.
The name Big Ben is often used to describe the tower, the clock and the bell but the name was first given to the Great Bell. *

1868 During the eclipse of 18 August 1868 from the Red Sea through India to Malaysia and New
Guinea, prominences are first studied with spectroscopes and shown to be composed primarily of hydrogen by James Francis Tennant, John Herschel, George Rayet, Norman Pogson
and others. *NSEC
 1941, Hugo Steinhaus, who sometimes joined his colleagues in the Scottish Café, contributed the final question to the Scottish Book, only days before the Nazi troops entered the town. In 1935, the first entry was made in The Scottish Book.  
The last item, Item Number 193, contains a rather cryptic set of numerical results, signed by Steinhaus, dealing with the distribution of the number of matches in a box! After the start of war between Germany and Russia, the city was occupied by German troops that same summer and the inscriptions ceased.

The original Szkocka Café (Scottish Café) in Akamemichna in Lwów, Poland
now the Desertniy Bar at T. Shevchenko Prospekt 27, Lviv, Ukraine.

The café was a meeting place for many mathematicians including Banach, Steinhaus, Ulam, Mazur, Kac, Schauder, Kaczmarz and others. Problems were written in a book kept by the landlord and often prizes were offered for their solution.

A collection of these problems appeared later as the Scottish Book.

R D Mauldin, The Scottish Book, Mathematics from the Scottish Café (1981) contains the problems as well as some solutions and commentaries.


1975 “I had today my virginal experience with the HP [Hewlett-Packard 65 calculator] as a celestial triangle-breaker ... it worked! But I’ll keep plotting the sun to make sure.” William F. Buckley Jr. discussing celestial navigation in his delightful book, Airborn, a Sentimental Journey, about sailing. His caution was justified, for later he learned that the prepackaged program contained errors. *VFR

1985 Marion Tinsley retains the world checker championship by defeating Asa Long 6–1. The one game Long won was the first time in nearly 25 years that anyone has beaten Tinsley in a checkers game. But then perhaps Tinsley had an unfair advantage—a Ph.D. in mathematics from Ohio State with a dissertation in combinatorics directed by Herbert Ryser. [Clipping of June 2, 1985] *VFR
He is considered the greatest checkers player who ever lived. He was world champion from 1955–1958 and 1975–1991. Tinsley never lost a World Championship match, and lost only seven games (two of them to the Chinook computer program) in his entire 45 year career.  He withdrew from championship play during the years 1958–1975, relinquishing the title during that time. (anyone know why?) Tinsley retired from championship play in 1991. In August 1992, he defeated the Chinook computer program 4–2 (with 33 draws) in a match. Chinook had placed second at the U.S. Nationals in 1990, which usually qualifies one to compete for a national title. However, the American Checkers Federation and the English Draughts Association refused to allow a computer to play for the title. Unable to appeal their decision, Tinsley resigned his title as World Champion and immediately indicated his desire to play against Chinook. The unofficial yet highly publicized match was quickly organized, and was won by Tinsley.
In one game, Chinook, playing with white pieces, made a mistake on the tenth move. Tinsley remarked, "You're going to regret that." Chinook resigned after move 36, fully 26 moves later. The ACF and the EDA were placed in the awkward position of naming a new world champion, a title which would be worthless as long as Tinsley was alive. They granted Tinsley the title of World Champion Emeritus as a solution.
In August 1994, a second match with Chinook was organized, but Tinsley withdrew after only six games (all draws) for health reasons. Don Lafferty, rated the number two player in the world at the time, replaced Tinsley and fought Chinook to a draw. Tinsley was diagnosed with pancreatic cancer a week later. Seven months later, he died. *Wik

2008 Buzz Lightyear lifts off the Earth for real. A Buzz Lightyear toy was launched aboard the space shuttle Discovery with mission STS-124 and returned on Discovery 15 months later with STS-128, the 12-inch action figure is the longest-serving toy in space. Disney Parks partnered with NASA to send Buzz Lightyear to the International Space Station and create interactive games, educational worksheets and special messages encouraging students to pursue careers in science, technology, engineering and mathematics (STEM). The action figure will go on display in the museum’s "Moving Beyond Earth" gallery in the summer. in 2012 the Toy Story character became part of the National Air and Space Museum’s popular culture collection. * [I still have a Buzz Lightyear toy on my book case given to me by some students because I used to use his trademark quote in (my very questionable) Latin, "ad infinitum, et ultra." ]
2013  At a Harvard seminar on May 13, 2013, the first chink was  produced in solving the twin primes conjecture.  A lecturer from the University of New Hampshire, Yitang Zhang, had proved that there are infinitely many pairs of primes that differ by no ,ore than 70,000,000.  It was a long way from differing by two, but it was an even greater distance from infinity.  He had submitted his paper to the Annals of Mathematics in April, and they had rushed to get reviews, which turned out to be enthusiastically positive. By May 21, 2013, the paper was accepted for publication on the 1st of May 2014.
By the 31st of May 2013, a group led by Scott Morrison and Terry Tao had lowered the gap to 42,342,946; game on!

This work led to a 2013 Ostrowski Prize, a 2014 Cole Prize, a 2014 Rolf Schock Prize, and a 2014 MacArthur Fellowship. Zhang became a professor of mathematics at the University of California, Santa Barbara in fall 2015.


1683 Jean-Pierre Christin (May 31, 1683 – January 19, 1755) was a French physicist, mathematician, astronomer and musician. His proposal to reverse the Celsius thermometer scale (from water boiling at 0 degrees and ice melting at 100 degrees, to water boiling at 100 degrees and ice melting at 0 degrees) was widely accepted and is still in use today.
Christin was born in Lyon. He was a founding member of the Académie des sciences, belles-lettres et arts de Lyon and served as its Permanent Secretary from 1713 until 1755. His thermometer was known in France before the Revolution as the thermometer of Lyon. *Wik

1872 Charles Greeley Abbot (31 May 1872; 17 Dec 1973 at age 101) was an American astrophysicist who is thought to have been the first scientist to suspect that the radiation of the Sun might vary over time. In 1906, Abbot became director of the Smithsonian Astrophysical Observatory and, in 1928, fifth Secretary of the Smithsonian. To study the Sun, SAO established a network of solar radiation observatories around the world-- usually at remote and desolate spots chosen primarily for their high percentage of sunny days. Beginning in May 1905 and continuing over decades, his studies of solar radiation led him to discover, in 1953, a connection between solar variations and weather on Earth, allowing general weather patterns to be predicted up to 50 years ahead. *TIS

1912 Martin Schwarzschild (31 May 1912; 10 Apr 1997 at age 84) German-born American astronomer who in 1957 introduced the use of high-altitude hot-air balloons to carry scientific instruments and photographic equipment into the stratosphere for solar research.*TIS

1912 Chien-Shiung Wu (simplified Chinese: 吴健雄; traditional Chinese: 吳健雄; pinyin: Wú Jiànxióng, May 31, 1912 – February 16, 1997) was a Chinese American experimental physicist who made significant contributions in the field of nuclear physics. Wu worked on the Manhattan Project, where she helped develop the process for separating uranium metal into uranium-235 and uranium-238 isotopes by gaseous diffusion. She is best known for conducting the Wu experiment, which contradicted the hypothetical law of conservation of parity. This discovery resulted in her colleagues Tsung-Dao Lee and Chen-Ning Yang winning the 1957 Nobel Prize in physics, and also earned Wu the inaugural Wolf Prize in Physics in 1978. Her expertise in experimental physics evoked comparisons to Marie Curie. Her nicknames include "the First Lady of Physics", "the Chinese Madame Curie", and the "Queen of Nuclear Research".*Wik

1926 John Kemeney  (May 31, 1926 – December 26, 1992) born in Budapest, Hungary. He worked on logic with Alonzo Church at Princeton, was Einstein’s assistant at the IAS, developed the computer language BASIC, and served as President of Dartmouth College. To learn more about him, see the interview in Mathematical People. Profiles and Interviews (1985), edited by Donald J. Albers and G. L. Alexanderson. *VFR
  In his 66-year life, Kemeny had a significant impact on the history of computers, particularly during his years at Dartmouth College, where he worked with Thomas Kurtz to create BASIC, an easy-to-use programming language for his computer students. Kemeny earlier had worked with John von Neumann in Los Alamos, N.M., during the Manhattan Project years of World War II. *CHM

1930 Ronald Valentine Toomer (31 May 1930; 26 Sep 2011 at age 81) was an American engineer who was a legendary creator of steel roller coasters. His early career, was in the aerospace industry, where he helped design the heat shield for Apollo spacecraft and was also involved with NASA's first satellite launches. In 1965, he joined the Arrow Development Company to apply tubular steel technology to the design the Runaway Mine Ride, the world's first all-steel roller coaster. It opened the following year at Six Flags over Texas. By 1975, he designed the Roaring 20's Corkscrew for Knott's Berry Farm, introducing first 360° looping rolls, in fact two of them. Later, his design included seven inversions in the Shockwave roller coaster for Six Flags Great America. He produced over 80 roller coasters before 1998. *TIS

1931 John Robert Schrieffer (31 May 1931; Oak Park, Illinois,USA- )John Robert Schrieffer is an American physicist who shared (with John Bardeen and Leon N. Cooper) the 1972 Nobel Prize for Physics for developing the BCS theory (for their initials), the first successful microscopic theory of superconductivity. Although first described by Kamerlingh Onnes (1911), no theoretical explanation had been accepted. It explains how certain metals and alloys lose all resistance to electrical current at extremely low temperatures. The insight of the BCS theory is that at very low temperatures, under certain conditions, electrons can form bound pairs (Cooper pairs). This pair of electrons acts as a single particle in superconductivity. Schrieffer continued to focus his research on particle physics, metal impurities, spin fluctuations, and chemisorption. *TIS


1832 Evariste Galois (25 October 1811 – 31 May 1832) died of peritonitis from a gunshot wound of the previous day. He died in the Cochin Hospital – this is now at 27 Rue du Faubourg St. Jacques,in the 14th district of Paris. He was buried in a common grave at Montparnasse Cemetery, but no trace of the grave remains.

1841 George Green (14 July 1793 – 31 May 1841) British mathematical physicist who wrote An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism (Green, 1828). The essay introduced several important concepts, among them a theorem similar to the modern Green's theorem, the idea of potential functions as currently used in physics, and the concept of what are now called Green's functions. George Green was the first person to create a mathematical theory of electricity and magnetism and his theory formed the foundation for the work of other scientists such as James Clerk Maxwell, William Thomson, and others. His work ran parallel to that of the great mathematician Gauss (potential theory).

Green's life story is remarkable in that he was almost entirely self-taught. He was born and lived for most of his life in the English town of Sneinton, Nottinghamshire, nowadays part of the city of Nottingham. His father (also named George) was a baker who had built and owned a brick windmill used to grind grain. The younger Green only had about one year of formal schooling as a child, between the ages of 8 and 9.
Self taught at a reading library while working full time as the manager of the family mill, He wrote a pivotal paper in applied calculus. George Green is buried in the family grave in the north east corner of St Stephens churchyard, just across the road from Green's Mill and car park. After his death the plaque below was placed in Westminster Abbey near the memorial to Newton. There are also memorials to Faraday, and Lord Kelvin. The Green Family mill has been completely restored and is now a Science center.

1931 Eugène Maurice Pierre Cosserat (4 March 1866 in Amiens, France - 31 May 1931 in Toulouse, France) Cosserat studied the deformation of surfaces which led him to a theory of elasticity. *SAU

1986 (Leo) James Rainwater (9 Dec 1917, 31 May 1986 at age 68)was an American physicist who won a share of the Nobel Prize for Physics in 1975 for his part in determining the asymmetrical shapes of certain atomic nuclei. During WW II, Rainwater worked on the Manhattan Project to develop the atomic bomb. In 1949 he began formulating a theory that not all atomic nuclei are spherical, as was then generally believed. The theory was tested experimentally and confirmed by Danish physicists Aage N. Bohr(4th son of Niels Bohr) and Ben R. Mottelson. For their work the three scientists were awarded jointly the 1975 Nobel Prize for Physics. He also conducted valuable research on X rays and took part in Atomic Energy Commission and naval research projects. *TIS

1998 Michio Suzuki (October 2, 1926 – May 31, 1998) was a Japanese mathematician who studied group theory.
A Professor at the University of Illinois at Urbana-Champaign from 1953 until his death. Suzuki received his Ph.D in 1952 from the University of Tokyo, despite having moved to the United States the previous year. He was the first to attack the Burnside conjecture, that every finite non-abelian simple group has even order.
A notable achievement was his discovery in 1960 of the Suzuki groups, an infinite family of the only non-abelian simple groups whose order is not divisible by 3. The smallest, of order 29120, was the first simple group of order less than 1 million to be discovered since Dickson's list of 1900.
There is also a sporadic simple group called the Suzuki group, which he announced in 1968. The Tits ovoid is also referred to as the Suzuki ovoid. *Wik

2000 Erich Kähler (16 January 1906, Leipzig – 31 May 2000, Wedel) was a German
Kähler was born in Leipzig, and studied there.
As a mathematician he is known for a number of contributions: the Cartan–Kähler theorem on singular solutions of non-linear analytic differential systems; the idea of a Kähler metric on complex manifolds; and the Kähler differentials, which provide a purely algebraic theory and have generally been adopted in algebraic geometry. In all of these the theory of differential forms plays a part, and Kähler counts as a major developer of the theory from its formal genesis with Élie Cartan.
His earlier work was on celestial mechanics; and he was one of the forerunners of scheme theory, though his ideas on that were never widely adopted. *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

Solving Geometric Problem with Pure Logic

I recently came across one of those geometry puzzles with multiple choice answers people like to post.

It struck me that a clever student could figure out the answer without calculations.  Before I go on I'll post the puzzle and let you play at it, but no calculating.  

You see these things all over the place if you are on the internet much.  This one at least was more straightforward than the "95% of people can't do this arithmetic problem" and the do something with order of operations using an obelus  (the spit symbol, ÷, used only in English speaking countries for division, strangely first used in a German book, Teutsche Algebra by Johann Rahn.)  

Anyway, that diversion was to take up a little space between the problem and the answer.

Since I knew the answer I read some of the different approaches in the comments. My favorite was a guy whose solution was pure logic.  

Answers a) and c) rule each other out,  since sides b and  a are the legs of the right triangle, if you switched their names, you get a different radius for congruent right triangles.  Answer b)  can't be right either.  The incircle can not be longer than the hypotenuse, because it is inside the triangle, but in any triangle, the two shorter sides are more than the longest side, which means that the b) answer woud be greater than the hypotenuse.  

So if any of them are true, d) must be the one.  

And it is true, but I'm too lazy to write it out, here is a  solution I plucked from  Quora.  

That's the kind of clever thinking that makes teachers smile.

Tuesday 30 May 2023

On This Day in Math - May 30

The best review of arithmetic 
consists in the study of algebra.
~Florian Cajori

The 150th day of the year; 150 is the largest gap between consecutive twin prime pairs less than a thousand. It occurs between {659, 661} and {809, 811}. *Prime Curios

150 is a palindrome in base 4(2112), and in base 7(303) .

A Poly divisible number is an n-digit number so that for the first digit is divisible by one, the first two digits are divisible by two, the first three digits are divisible by three, etc up to n. There are 150 three-digit poly divisible numbers. Hat tip to Derek Orr .

In February of 1657 Fermat proposed a new problem to Frenicle: Find a number x which will make (ax2 + 1) a square, where a is a (nonsquare) integer.  Frenicle found solutions of the problem. In the second part of the Solutio (pp. 18–30) he cited his table of solutions for all values of a up to 150 and explained his method of solution. 

150 is the sum of eight consecutive primes starting with 7.

150 is a Harshad(joy-giver) number, divisible by the sum of its digits.

150 year celebration is called sesquicentennial of the event.
And... 150 is the number of degrees in the quincunx astrological aspect explored by Johannes Kepler.

Rubix Cube gotten too easy for you? Try the Professor's Cube, 150 movable facets.


1667 After much debate about the presence of a woman at a Royal Society meeting, the Duchess of Newcastle was allowed to observe a demonstration of a "experiments of colours", the "weighing of air in an exhausted receiver", and "the dissolving of flesh with a certain liquor of Mr. Boyle's suggesting." This was probably the first visit by a woman to the Royal Society. The Duchess, Margaret Cavendish, was a competent scientist in her own right. Her prolific writings in the nature of science earned her the nickname “Mad Madge”. I have a note from VFR that she was elected to FRS, but can not confirm, the note says "No other woman was elected FRS until 1945" .

1765 "Ms. Catherine Price, Daughter of the late Dr. Halley " was paid a sum of 100 Pounds for "causing to be delivered to the Commissioners of the Longitude, several of the said Dr. Halley's manuscript papers, which... may lead to discoveries useful to navigation." *Derek Howse, Britain's Board of Longitude, the Finances

1832 Galois mortally wounded by a gunshot wound to the abdomen in a duel of honor. He was left for dead after the duel but a peasant took him to a hospital. *VFR

The infamous duel with Pescheux d'Herbinville took place near the Glassier pond in the southern suburb of Gentilly. The duel was over Galois's involvement with Stéphanie-Félicie Poterine du Motel, who was d'Herbinville's fiancée, but it has been claimed that the affair was a political frame-up by government agents in order to eliminate Galois He died in the Cochin Hospital – this is now at 27 Rue du Faubourg St. Jacques, 14e, but I don't know how long it has been there. He was buried in a common grave at Montparnasse Cemetery, but no trace of the grave remains.
The Galois memorial in the cemetery of Bourg-la-Reine.
Évariste Galois was buried in a common grave and
 the exact location is still unknown. *Wik

1896   Widely considered to be the real first accident, this occurred on May 30, 1896, during a “horseless wagon race” in New York City. Henry Wells lost control of his vehicle and crashed into a bicyclist named Ebeling Thomas. The bicyclist broke his leg, and the driver was arrested. If only there was a New York Defensive Driving course back then, a lot of chaos could have been avoided. However, as there were several other bicyclists arrested that day for the 1896 equivalent of speeding, perhaps a certain amount of chaos was just par for the course at that time.*Improv

1903 Minor planet (511) Davida Discovered 1903 May 30 by R. S. Dugan at Heidelberg. Named by the discoverer in honor of David P. Todd (1855-1939), professor of astronomy and director of the Amherst College Observatory (1881-1920). David Todd was the husband of Mabel Todd, who wrote books about solar eclipses. David has also a drawing of a painting of a solar eclipse in one of his books. *NSEC

2000 almost 200 years after the (now called) tangrams exploded across Europe, the nation of Finland issued a stamp panel designed as a tangram square.  Only four of the seven shapes were postage stamps.  Each tangram shape featured an idea of education and science.  One triangle showed a Sierpinski triangle.


1423 Georg von Peurbach (or Peuerbach) (May 30, 1423 in Peuerbach near Linz – April 8, 1461 in Vienna)  He worked on trigonometry astronomy, and was the teacher of Regiomontanus. *VFR
He promoted the use of Arabic numerals (introduced 250 years earlier in place of Roman numerals), especially in a table of sines he calculated with unprecedented accuracy. He died before this project was finished, and his pupil, Regiomontanus continued it until his own death. Peurbach was a follower of Ptolomy's astronomy. He insisted on the solid reality of the crystal spheres of the planets, going somewhat further than in Ptolomy's writings. He calculated tables of eclipses in Tabulae Ecclipsium,observed Halley's comet in Jun 1456 and the lunar eclipse of 3 Sep 1457 from a site near Vienna. Peurbach wrote on astronomy, his observations and devised astronomical instruments. *TIS  The Renaissance Mathematicus has a nice piece about Peurbach and his life... the kind of detail that comes from a passion for his subject.  Check it out. 
Georg von Peuerbach:
Theoricarum novarum
planetarum testus, Paris 1515

1800 Karl Wilhelm Feuerbach (30 May 1800 – 12 March 1834) born in Jena, Germany. His mathematical fame rests entirely on three papers. Most important was this contribution to Euclidean geometry: The circle which passes through the feet of the altitudes of a triangle touches all four of the circles which are tangent to the three sides; it is internally tangent to the inscribed circle and externally tangent to each of the circles which touches the sides of the triangle externally. *VFR

The circle is also commonly called the Nine-point circle. It passes through the feet of the altitudes, the midpoints of the three sides, and the point half way between the orthocenter and the vertices.

1814 Eugene Charles Catalan (30 May 1814 – 14 February 1894) was a Belgian mathematician who defined the numbers called after him, while considering the solution of the problem of dissecting a polygon into triangles by means of non-intersecting diagonals. *SAU The Catalan numbers have a multitude of uses in combinatorics.  There are many counting problems in combinatorics whose solution is given by the Catalan numbers. The book Enumerative Combinatorics: Volume 2 by combinatorialist Richard P. Stanley contains a set of exercises which describe 66 different interpretations of the Catalan numbers. 
One of the ways is Cn is the number of different ways n + 1 factors can be completely parenthesized (or the number of ways of associating n applications of a binary operator, as in the matrix chain multiplication problem). For n = 3, for example, we have the following five different parenthesizations of four factors:
((ab)c)d     (a(bc))d     (ab)(cd)     a((bc)d)     a(b(cd))

The first Catalan numbers for n = 0, 1, 2, 3, ... are

1, 1, 2, 5, 14, 42, 132, 429
A convex polygon with n + 2 sides can be cut into triangles by connecting vertices with non-crossing line segments (a form of polygon triangulation). The number of triangles formed is n and the number of different ways that this can be achieved is Cn. The following hexagons illustrate the case n = 4:


1889 Paul Ernest Klopsteg (May 30, 1889 – April 28, 1991) was an American physicist. The asteroid 3520 Klopsteg was named after him and the yearly Klopsteg Memorial Award was founded in his memory.
He performed ballistics research during World War I at the US Army's Aberdeen Proving Grounds in Maryland. He applied his knowledge of ballistics to the study of archery.
He was director of research at Northwestern University Technical Institution. From 1951 through 1958 he was an associate director of the National Science Foundation and was president of the American Association for the Advancement of Science from 1958 through 1959.*Wik

1908 Hannes Olof Gösta Alfvén (30 May 1908 in Norrköping, Sweden; 2 April 1995 in Djursholm, Sweden) Alfvén developed the theory of magnetohydrodynamics (MHD), the branch of physics that helps astrophysicists understand sunspot formation and the magnetic field-plasma interactions (now called Alfvén waves in his honor) taking place in the outer regions of the Sun and other stars. For this pioneering work and its applications to many areas of plasma physics, he shared the 1970 Nobel Prize in physics. *DEBORAH TODD AND JOSEPH A. ANGELO, JR., A TO Z OF SCIENTISTS IN SPACE AND ASTRONOMY

1778 (François Marie Arouet) Voltaire (21 November 1694 – 30 May 1778) was a French author who popularized Isaac Newton's work in France by arranging a translation of Principia Mathematica to which he added his own commentary (1737). The work of the translation was done by the marquise de Châtelet who was one of his mistresses, but Voltaire's commentary bridged the gap between non-scientists and Newton's ideas at a time in France when the pre-Newtonian views of Descartes were still prevalent. Although a philosopher, Voltaire advocated rational analysis. He died on the eve of the French Revolution.*TIS

1912 Wilbur Wright  (April 16, 1867 – May 30, 1912), American aviation pioneer, who with his brother Orville, invented the first powered airplane, Flyer, capable of sustained, controlled flight (17 Dec 1903). Orville made the first flight, airborn for 12-sec. Wilbur took the second flight, covering 853-ft (260-m) in 59 seconds. By 1905, they had improved the design, built and and made several long flights in Flyer III, which was the first fully practical airplane (1905), able to fly up to 38-min and travel 24 miles (39-km). Their Model A was produced in 1908, capable of flight for over two hours of flight. They sold considerable numbers, but European designers became strong competitors. After Wilbur died of typhoid in 1912, Orville sold his interest in the Wright Company in 1915 *TIS

1926 Vladimir Andreevich Steklov (9 January 1864 – 30 May 1926)  made many important contributions to applied mathematics. In addition to the work for his master's thesis and his doctoral thesis referred to above, he reduced problems to boundary value problems of Dirichlet type where Laplace's equation must be solved on a surface. He wrote General Theory of Fundamental Functions in which he examined expansions of functions as series in an infinite system of orthogonal eigenfunctions. In fact the term "Fundamental Functions", which is due to Poincaré, means eigenfunctions in today's terminology.
Steklov was not the first to examine series expansions in terms of infinite sets of orthogonal eigenfunctions, of course Fourier had examined a special case of this situation many years before. Steklov, however, produced many papers on this topic which led him to a general theory to replace the special cases examined by others. He studied a generalisation of Parseval's equality for Fourier series to his general setting showing this to be a fundamental property. In all his list of publications contains 154 items. *SAU

1943 Anderson McKendrick (September 8, 1876 - May 30, 1943) trained as a medical doctor in Glasgow and came to Edinburgh as Superintendent of the College of Physicians Laboratory. He made some significant mathematical contributions to biology. *SAU

1964 Leo Szilard (11 Feb 1898; 30 May 1964 at age 66) Hungarian-American physicist who, with Enrico Fermi, designed the first nuclear reactor that sustained nuclear chain reaction (2 Dec 1942). In 1933, Szilard had left Nazi Germany for England. The same year he conceived the neutron chain reaction. Moving to N.Y. City in 1938, he conducted fission experiments at Columbia University. Aware of the danger of nuclear fission in the hands of the German government, he persuaded Albert Einstein to write to President Roosevelt, urging him to commission American development of atomic weapons. In 1943, Major General Leslie Groves, leader of the Manhattan Project designing the atomic bomb, forced Szilard to sell his atomic energy patent rights to the U.S. government. *TIS Frederik Pohl , talks about Szilard's epiphany about chain reactions in Chasing Science (pg 25),
".. we know the exact spot where Leo Szilard got the idea that led to the atomic bomb. There isn't even a plaque to mark it, but it happened in 1938, while he was waiting for a traffic light to change on London's Southampton Row. Szilard had been remembering H. G. Well's old science-fiction novel about atomic power, The World Set Free and had been reading about the nuclear-fission experiment of Otto Hahn and Lise Meitner, and the lightbulb went on over his head."

1992 Antoni Zygmund (December 25, 1900 – May 30, 1992) Polish-born mathematician who created a major analysis research centre at Chicago, and recognized in 1986 for this with the National Medal for Science. In 1940, he escaped with his wife and son from German controlled Poland to the USA. He did much work in harmonic analysis, a statistical method for determining the amplitude and period of certain harmonic or wave components in a set of data with the aid of Fourier series. Such technique can be applied in various fields of science and technology, including natural phenomena such as sea tides. He also did major work in Fourier analysis and its application to partial differential equations. Zygmund's book Trigonometric Series (1935) is a classic, definitive work on the subject.

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

The Cubic Attractiveness of 153

From 2012 post, with additional detail and a guest followup:

Just in time for June 1st, which is the 153rd day of this leap year.  I recently discovered an interesting quality about the number 153.

Ok, I was amazed. I was lead from a comment (below) that this is all shown in the online integer sequence, but it is still great fun.

Pick any old number you want, and multiply by three (or just pick a number that is a multiple of three).
Now take all the digits and cube them and add the cubes together.
For example, if you picked 231, you would add 23 + 3 3 + 1 3 to get 36. 
Yeah, So what you're probably thinking... but take that new number and do the same thing... cube the digits and add them up... Nothing?  Keep going... eventually you get to 153, and then when you  do it again, you get 153 forever.
A digression about notation and terminology
In slightly more formal language, it seems that 153 is the fixed point attractor of any multiple of three under the process of summing the cubes of the digits. This process of an n-digit number, k,  which has a sun of the nth power of its digits equal to k, is variously called a narcissistic number, an Armstrong number, or a perfect digital invariant(often written PDI and PPDI).  So 153 is a Narcissistic number.  The other three digit numbers that are narcisscistic are 370, 371, and 407.  

In his famous A Mathematicians Apology, G, H. Hardy dismisses these numbers with the comment that, "There is nothing in these odd facts which appeals to the mathematician."  

But recreational mathematicians move on undisturbed by Hardy's judgement.  The first use of Narcissistic numbers that I have found, although defined a little more broadly than is now done, was in  Joseph S. Madachy's Mathematics on Vacation in 1966. He described them as a number that is equal to a function of its digits, and included numbers like 145 = 1! + 4! + 5! 

The earliest record of Armstrong Numbers was in a 1971 book,  Computer Science Laboratory Exercises  by F. D. Federighi, ‎Edwin D. Reilly . I suspect much of the popularity of the numbers sprang from being a good programming exercise.

END Digression

The 231 that I used above goes to 36, which goes to 243, then 99, 1458, 702 and 351, which give 153. 

If you started with something like 72, it gets there pretty quickly,  72 →   351 →  153.  

Other numbers take a little longer.  717, for example, takes 13 iterations of the cubing process to get there...but it does.  
And 153 is special... ONLY the multiples of three go there.  Pick another number that's not a multiple of three and run the process and several things might happen, but what won't happen, is going to 153. 

When I began to explore why this happened I could figure out a few things easy enough.  The first realization was that the iterations couldn't just get bigger and bigger and diverge to infinity.  93 is 729 and so any four digit number has to iterate to a value less than 4 * 729 or 2916, and anything with more digits just keeps getting pushed down until it gets under that limit.
So all the numbers in the world have to end up doing something other than just keep getting bigger.
That only leaves a few options.  They might just keep jumping around in some orbit that cycles through several numbers.  This happens with 46 for example.  
After a few iterations you fall into a three cycle. 

46 → 280 → 520 →  133 → 55  → 250 →133

Other numbers also do this, and there are a few different three cycles and  two cycles, but that is sort of an oddity, at least for numbers less than 1000 (still have work to do here).
Most numbers go to a fixed aatractor.  For multiples of three, that seems to always be 153.   For numbers that are not multiples of three, the general fixed points are either 370, or 371.  And they have a modulo three relationship as well.  Numbers that are congruent to 2 Mod3 (they have a remainder of 2 when you divide by three) generally go to 371.  The exceptions are a couple of numbers; 47 , 74, 77,  89 & 98,   which go to 407 (of course 707, 908, 980, etc would also, I counted 30 numbers less than 1000 that go to 407 as a fixed point).  

All the cycles that I have found are numbers that are equivalent to 1 Mod3.  The cycles seem pretty common with less than 1/2 the smaller numbers going to a fixed value of 370 and the occasional few like 1, 10, 100, that have a fixed point at one. Similar to the Happy numbers under the squared sum.  118 is in that group as well, for example.  

The reason for the separation into modulus classes of three is easy enough to explain.  When you cube a number, it's modulus in base three isn't changed.  For example, 4 Mod3 is 1, and 43 = 64 is also equivalent to 1 Mod3.  So the Modulus of a number doesn't change under this process, grouping the results together.  

Now when you add in the fact that there are not really that many of these smaller (less than 3000, say) numbers that can be made with the sum of cubes unless you allow for weird numbers like 11,111 or something to get five.  

I haven't explored much beyond 1000, so I'm not sure if I will come across other cyclic orbits. Or why only the 3n+1 type numbers produce cycles. And I'm not sure what else I may find, but I'm thinking that what I've seen so far makes 153 a very special number. 

After I wrote this back in 2012, a young friend working on his degree at Pitt sent me notes on his reaults when he  took off on the fourth power,  He surmised that all the numbers two through nine went to a fixed value of 13139 and then repeated the cycle (13139 --> 6725 --> 4338 --> 4514 --> 1138 --> 4179 --> 9219 --> 13139...)  I checked this for a couple and it took a few iterations to get there, but the ones I checked did.  That means numbers like 11, 20, 30, .. 101, 111, 200,  and similar numbers if digits sum to 2-9.

He also found that 12 had a fixed point attractor at 8208, ( 12 --> 17 --> 2402 --> 288 --> 8208 --> 8208 --> 8208 ) which means that 17, 21, 71, 102, 107, 170, 201, 210, 224, 242, 288, 422, 701, 710,828, 882 will all go to the same attractor (and of course many more).  

I have now convinced myself that 13, 14, 15, 16, 18, 19,   go to the same cycle as 2 through 9 so now 2 through 9, 11, 13, 14, 15, 16, 18, 19, 20, 22, 23, 24, 25, 26,27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 50, 51, 52, 56, 62, 65, 72, 81, 91 101, 111, 200...
All end in the cycle (13139 --> 6725 --> 4338 --> 4514 --> 1138 --> 4179 --> 9219 --> 13139...

If my calculating does not contain any big errors, so far I have found end behavior for the following numbers under 50:

Two  have fixed attractors at 1 (1,10)

forty (listed above) end at the cycle (13139 --> 6725 --> 4338 --> 4514 --> 1138 --> 4179 --> 9219 --> 13139...

Three end in the fixed point 8208 (12, 17, 21 )  

5 of the numbers are not yet checked. In time I will reduce these. 

A little over a year after I did this, Derek had spent more time working on this and I asked him to guest post his work in progress. 

So here are his notes (I'll add notes of what I have found over the last decade in italics along the way):

Digits to the Fourth Power

By Derek J. Orr, University of Pittsburgh

I was perusing through Pat’s blog here and I noticed two pages that involved summing the squares of digits and summing the cubes of digits. I, then, proposed taking the fourth power of these digits. When I did so, I found some pretty interesting results. Unfortunately, it is time consuming to do each of these numbers manually but I plan to get up to 1,000 and maybe even 10,000 (if motivation and free time permit).

Some numbers that I found went to 1 and are called “happy numbers”. Unfortunately, the only ones I’ve found were 1, 10 and 100 (I went up to 225 so far). However, I am assuming the next smallest happy number is 1,000 just because it’s hard to find one. I wanted to find one and, since I couldn’t, I forced myself to.

I know that a happy number is a number that will eventually reduce to the number 1 and repeat forever. So, the guaranteed happy numbers are 1, 10, 100, 1000, 10000, etc. Thus, I figured out what numbers can get me to these guaranteed happy numbers and, though they aren’t small, I found a few. First, I’ll write out the fourth powers of the single digit numbers: 

Using these values, I found different combinations that could get me to the guaranteed happy numbers. Also, since we have 1 as a possibility, it’s impossible to skip any number I choose. The table below (above) lists a few combinations that I found.

The numbers inside the table represent how many of the digits we need. So, for the first line of the table, the number 1,111,111,111 is a happy number because added together 10 times gets us 10. From this table, the smallest happy number that is not a multiple of 10 (and that is not 1) is 11,123. However, in just two steps, this number can reach 1. What if there are longer iterations that still get us to 1? What if there is a number that can get us 11,123? So, I experimented:

(the other equation I found has 18 digits, so I won’t write it out)

So, we see that there is a new happy number, 22,233,489. Once again, what numbers get to this number? I could do this all day but I won’t, mainly because 22,233,489 can be divided by over 3,388 times. So, this means that the next number has more than 3,388 digits, which is far too many. I’ve assumed that up to now, the numbers will only get bigger. However, what if we tried a different number instead of 11,123? Since the digits are added, we can always change the order. So, 12113, 13112, 21113, 31112, 12131, 13121, 12311, 13211, etc. are also happy numbers. Again, I could experiment on these but I won’t just to save time. I believe it is safe to say that the smallest happy number without a zero (and that is not 1) is 11,123. Since the happy numbers are so hard to find, I looked at different numbers.

One loop that I found was with the number 2,178. I saw it for the first time when I tried the number 127. Here is the iteration for 127.

127 -- 2418 -- 4369 -- 8194 --10914 -- 6819 -- 11954 -- 7444 -- 3169 -- 7939 -- 15604 -- 2178 -- 6514 -- 2178 -- 6514 --…

So, we can see it goes through this 2,178----6,514 loop. Now, I’ve only seen this work for 127, 172, and 217 (so far) but the four- and five-digit numbers above will also bring about this loop.

Also, I found that there seems to be a fixed point where some numbers end up at, similar to what Pat found when cubing the digits. This number was 8,208, and satisfies the condition (ie 8^4 + 2^4 + 0^4 + 8^4 = 8208) . What numbers gave me 8,208? I computed a bit more than the first 200 digits (225 to be exact) and found the numbers 12, 17, 21, 46, 64, 71, 102, 107, 120, 137, 145, 154, 170, 173, 201, 210, and 224 (and of course 288) will get to 8,208 and stay there. Now, past these, it’s obvious that 317, 415, 514, 710, 701, 713, 422, etc. also work. When you cubed the digits, Pat found that you reach the number 153 and it repeats forever. However, Pat found that if you have any multiple of three, you will reach 153. There is a pattern there. I am sadly not able to find any pattern with these; they seem to be random, like the happy numbers when you square the digits (1, 7, 10, 13, 19, 23, 28…etc.).

Other four digit 

Another loop number these could go to is 13,139. With 13,139, there is a loop involved (13139 -- 6725 -- 4338 -- 4514 -- 1138 -- 4179 -- 9219 -- 13139…). This has happened with every number I haven’t mentioned (around 90% of the numbers I’ve tried).(lots of numbers take a very long time to drop into their cycle)  

Going back to 8208, I keep wondering if there is another fixed point. When cubing the digits, there are five numbers that equal the sum of their digits cubed: 1, 153, 370, 371, and 407. When squaring the digits, there is only one number that equals the sum of its digits squared: 1. But, when taking the fourth power, I have only found two numbers that work: 1 and 8,208.  I do wonder if there are more or not; perhaps a good problem for a computer programmer because doing these manually, though possible, takes up a lot of time. (Derek didn't find the other two fixed point attractors, 1634 and 9474)

It would seem that the only sum of four fourth powers that sum to 1634 are the actual digits of 1,6,3 and 4. My reasoning is that there can not be any digit greater than 6, since the fourth powers of 7,8, and 9 all exceed 1634.  No four digits picked from 0, 1, 2, 3, and 4 can have a sum large enough since 1634 exceeds 4 times 4^4 .  So there must be at least one five or one six, but not one of each , nor two sixes.  So we need only to check using one six and three less than 5, or one or two fives with the rest less than five. trying each of these combinations fails.

9474 seems to suffer a similar fate of attracting only the numbers made up of the same four digits.


I will mention one oddity I came across over the years playing with these, the palindrome 11^4 =14641  is the smallest fourth power whose digits sum to a fourth power, 16.

I haven't pursued the behavior of the fifth power of these, but have been told that the fixed point attractors are 1, 4150, 4151, 54748, 92727, 93084, 194979

If you are one of those number curious folks who also happens to be a really good programmer, I would love to receive a list of other fixed points and cycles in third, fourth, and fifth powers of digits. (Or maybe you can send your own guest blog.)

Monday 29 May 2023

The Distracted Goalie


The great Physicist, Niels Bohr, was brother to an outstanding mathematician, Harald who founded the field of almost periodic functions. In their youth both were very good athletes, with Harald clearly the more dedicated sportsman. Harald had been a member of the Danish National Football team while still a student and had earned a silver medal as such in the 1908 Olympic games; the first time the Olympic games had football. Harald scored two goals in the opening game defeating the French nine-zero. Denmark lost to the UK in the final game. He was such an accomplished football player that it is said when he defended his PhD thesis there were more football fans in the audience than mathematicians.
Brother Niels was also a good athlete, but often seemed to have his focus somewhere other than sports. Both brothers played several games for the Copenhagen-based Akademisk Boldklub, with Niels in goal. The story is told that during one game when almost all the action was happening in the attacking half for his club, a long clearing kick from the other end of the field began to roll toward his goal. Niels stood near the goalpost and seemed unaware of the ball rolling toward his goal with players rushing in from many yards behind it. Alerted by the screaming crown behind him, Niels made the save and cleared away the threat.
After the game his explanation was that he had been distracted by a math problem and was carrying out calculations on the edge of the goal post.
Apparently he kept his love for the game.  The photo at the top shows Niels Bohr with  a group that is unidentified, but he looks to me to be the one handling the ball.  A goal-keeper to the end, it seems.

An Anon. comment suggested, "I think there's also Gamow (upright), Pauli (back to camera) and possibly Heisenberg opposite Bohr. Must have been during a conference."  If anyone can spot brother Harald, and if someone recognizes young version of later great science/math wizards, share, please