Pat'sBlog

Wednesday, 23 January 2019

On This Day in Math - January 23

Me Considering Pascal Considering the Roulette

Nature creates curved lines while humans create straight lines.

~Hideki Yukawa

The 23rd day of the year; 23! is 23 digits long (22, 23, and 24 each solve n! is n digits in length (decimal)). 23 is also the answer to the classic Birthday Problem. (How many randomly selected people in a group makes the probability greater than 50% that (at least)two share a common birthdate.)

There are only two values of $ x>0 $ so that $ x^3 + (x+1)^3 +(x+2)^3 $is a perfect square. The largest is 23. $ 23^3+24^3 + 25^3 = 41616 = 204^2 $ *@BenVitale

decimal # 11111111111111111111111 is prime (23 ones) . * ‏@MathYearRound

EVENTS

1656 Blaise Pascal wrote the ﬁrst of his eighteen Provincial Letters.

1640 John Pell wrote Mersenne that Thomas Harriot (1560–1621) had found the law of refraction, now known as Snell’s law.*VFR The first known accurate description of the law was by the scientist Ibn Sahl at Baghdad court, when in 984 he used the law to derive lens shapes that focus light with no geometric aberrations in the manuscript On Burning Mirrors and Lenses.
Snell's law states that the ratio of the sines of the angles of incidence and refraction is equivalent to the ratio of phase velocities in the two media, or equivalent to the reciprocal of the ratio of the indices of refraction:
$ \frac{sin (\theta_1)}{sin(\theta_2)}=\frac{v_1}{v_2}= \frac{n_2}{n_1} $ *Wik

1675 Christiaan Huygens drew a sketch in his notebook of a watch mechanism with a coiled spring regulator and then, “Eureka – I have found it”. He believed he had found a method of regulating a clock that would keep accurate time and not be affected by motion for use in attacking the problem of longitude. (Hooke had presented a watch regulated by a spring in the early 1660’s to the Royal Society for exactly the same goal in response to Huygen’s first pendulum clock. ) *Lisa Jardine, Ingenious Pursuits, pg 148

1764 Harvard Hall, the last of Harvard’s original buildings, burned down in a nor'easter, taking with it almost the entire College library & John Harvard's book collection. The job of replacing the valuable scientific instruments housed in the building fell to John Winthrop, the second Hollis professor of mathematics and natural philosophy at Harvard. He was also friend and advisor of George Washington. *VFR

1812, the second day of powerful earthquakes struck, with an epicenter in the far southeast corner of Missouri. It was a part of a three-month series in the central Mississippi River valley, known as the New Madrid earthquakes. They began on 16 Dec 1811, with the first two major earthquakes, six hours apart, each with an epicenter in northeastern Arkansas. All were felt hundreds of miles away. All were powerful, about magnitude 7-7.5. There were many aftershocks, and an a final major earthquake on 7 Feb 1812. Contemporary accounts tell of houses damaged, chimneys toppled, remarkable geological phenomena and landscapes changed. They remain among the most powerful earthquakes in the United States. The New Madrid fault remains a concern (especially in my neighborhood here in Possum Trot, Ky, only 100 or so miles away). *TIS

1848 Gold discovered in California. Jonas Clark soon was on hand with a wagon load of shovels and so made a wagon load of money. He used it to found Clark University in Worcester, MA, which,in the early 1890’s, had the strongest mathematics department in the country. *VFR

1883 The first session of the Edinburgh Mathematical Society,

*Proceedings of the Edinburgh Mathematical Society, Volumes 1-4

1896, Wilhelm Roentgen first made a public lecture-demonstration of his X-ray device, in Würzburg, Germany. *TIS

1896 Two months after Rontgen discovered X-rays (the x was for unknonwn), Henri Poincare was sent photographs of these X-rays and was so amazed that he passed them on to two doctors and asked if they could duplicate Rontgen's work. On January 23 they would present a paper on their results at the French Academy with Henri Becquerel in the audience. Within months he would discover rays coming from Uranium. *Brody & Brody, The Science Class You Wished You Had

1911, Marie Curie's nomination to the French Academy of Sciences, having already won one Nobel Prize, is nevertheless voted down by the Academy's all-male membership. She went on to win a second Nobel Prize. *TIS

1913 Russian mathematician Andrei Andreyevich Markov addressed the Imperial Academy of Sciences in St. Petersburg, reading a paper titled “An example of statistical investigation of the text Eugene Onegin concerning the connection of samples in chains.” The idea he introduced that day is the mathematical and computational device we now know as a Markov chain.
Markov’s 1913 paper was not his first publication on “samples in chains”; he had written on the same theme as early as 1906, but it was the 1913 paper that was widely noticed, both in Russia and abroad, and that inspired further work in the decades to come. The earlier discussions were abstract and technical, giving no hint of what the new probabilistic method might be good for; in 1913 Markov demonstrated his technique with a novel and intriguing application—analyzing the lexical structure of Alexander Pushkin’s poem Eugene Onegin. Direct extensions of that technique now help to identify genes in DNA and generate gobbledygook text for spammers. *Brian Hayes

1930 Clyde Tombaugh photographed the planet Pluto, the only planet discovered in the twentieth century, after a systematic search instigated by the predictions of other astronomers. Tombaugh was 24 years of age when he made this discovery at Lowell Observatory in Flagstaff, Ariz. *TIS

*Northwestern University

1986 Science reported that a statistical analysis of word frequencies on a newly discovered poem attributed to Shakespeare concluded “There is no convincing evidence for rejecting the hypothesis that Shakespeare wrote it.” Otherwise said, the poem “ﬁts Shakespeare as well as Shakespeare ﬁts Shakespeare.” [Mathematics Magazine 59 (1986), p 183]. *VFR

1959 Robert Noyce Conceives the Idea for a Practical Integrated Circuit:
Robert Noyce, as a co-founder and research director of Fairchild Semiconductor, was responsible for the initial development of silicon mesa and planar transistors, which led to a commercially applicable integrated circuit. In 1968 Noyce went on to found Intel Corp. with Gordon Moore and Andy Grove.*CHM

2003 And Then it was gone.... The final, very weak signal from Pioneer 10 was received on January 23, 2003 when it was 12 billion kilometers (80 AU) from Earth. Launched on Mar 10,1972, Pioneer 10 crossed the orbit of Saturn in 1976 and the orbit of Uranus in 1979. On June 13, 1983, Pioneer 10 crossed the orbit of Neptune, the outermost planet at the time, and so became the first man-made object to leave the proximity of the major planets of the solar system.
The last successful reception of telemetry was received from Pioneer 10 on April 27, 2002; subsequent signals were barely strong enough to detect, and provided no usable data. *Wik
2013 The Institute for Applied Computational Science of the Harvard School of Engineering and Applied Sciences celebrated the centenary of Markov’s 1913 paper which promoted the wide study and use of Markov Chains. (see 1913 above) **Brian Hayes

2013 On January 23rd Dr. Curtis Cooper of Central Missouri University discovered the 48th known Mersenne prime, 2^57,885,161-1, a 17,425,170 digit number. The GIMP site records this as the 25th of January, so I shall use both dates until I figure out why two different dates reported.

BIRTHS

1693 Georg Bernhard Bilfinger (23 Jan 1693; 18 Feb 1750) German philosopher, mathematician, statesman, and author of treatises in astronomy, physics, botany, and theology. He is best known for his Leibniz-Wolffian philosophy, a term he coined to refer to his own position midway between those of the philosophers Gottfried Wilhelm Leibniz and Christian Wolff.*TIS

1719 John Landen (23 Jan 1719; died 15 Jan 1790) British mathematician who made important contributions on elliptic integrals. As a trained surveyor and land agent (1762-88), Landen's interest in mathematics was for leisure. He sent his results on making the differential calculus into a purely algebraic theory to the Royal Society, and also wrote on dynamics, and summation of series. Landen devised an important transformation, known by his name, giving a relation between elliptic functions which expresses a hyperbolic arc in terms of two elliptic ones. He also solved the problem of the spinning top and explained Newton's error in calculating the precession. Landen was elected a Fellow of the Royal Society in 1766. He corrected Stewart's result on the Sun-Earth distance (1771).*TIS

1785 Matthew Stewart (15 Jan 1717 in Rothesay, Isle of Bute, Scotland - 23 Jan 1785 in Catrine, Ayrshire, Scotland)was a Scottish geometer who wrote on geometry and planetary motion. Stewart's fame is based on General theorems of considerable use in the higher parts of mathematics (1746), described by Playfair as, "... among the most beautiful, as well as most general, propositions known in the whole compass of geometry." *SAU

1806 Ernst Ferdinand Adolf Minding (23 Jan 1806 in Kalisz,Russian Empire (now Poland) - 3 May 1885 in Dorpat, Russia (now Tartu, Estonia))His work, which continued Gauss's study of 1828 on the differential geometry of surfaces, greatly influenced Peterson. In 1830 Minding published on the problem of the shortest closed curve on a given surface enclosing a given area. He introduced the geodesic curvature although he did not use the term which was due to Bonnet who discovered it independently in 1848. In fact Gauss had proved these results, before either Minding of Bonnet, in 1825 but he had not published them.
Minding also studied the bending of surfaces proving what is today called Minding's theorem in 1839. The following year he published in Crelle's Journal a paper giving results about trigonometric formulae on surfaces of constant curvature. Lobachevsky had published, also in Crelle's Journal, related results three years earlier and these results by Lobachevsky and Minding formed the basis of Beltrami's interpretation of hyperbolic geometry in 1868.
Minding also worked on differential equations, algebraic functions, continued fractions and analytic mechanics. In differential equations he used integrating factor methods. This work won Minding the Demidov prize of the St Petersburg Academy in 1861. It was further developed by A N Korkin. Darboux and Émile Picard pushed these results still further in 1878. *SAU

1840 Ernst Abbe (23 Jan 1840, 14 Jan 1905) German physicist who made theoretical and technical innovations in optical theory. He improved microscope design, such as the use of a condenser lens to provide strong, even illumination (1870). His optical formula, now called the Abbe sine condition, applies to a lens to form a sharp, distortion-free image He invented the Abbe refractometer for determining the refractive index of substances. In 1866, he joined Carl Zeiss' optical works, later became his partner in the company, and in 1888 became the owner of the company upon Zeiss' death. Concurrently, he was appointed professor at the Univ. of Jena in 1870 and director of its astronomical and meteorological observatories in 1878.*TIS

1853 Kazimierz Żorawski (June 22, 1866 – January 23, 1953) was a Polish mathematician. His work earned him an honored place in mathematics alongside such Polish mathematicians as Wojciech Brudzewski, Jan Brożek (Broscius), Nicolas Copernicus, Samuel Dickstein, and Stefan Banach,.
Żorawski's main interests were invariants of differential forms, integral invariants of Lie groups, differential geometry and fluid mechanics. His work in these disciplines was to prove important in other fields of mathematics and science, such as differential equations, geometry and physics (especially astrophysics and cosmology).*Wik

1857 Andrija Mohorovicic (23 Jan 1857; 18 Dec 1936) Croatian meteorologist and geophysicist who discovered the boundary between the Earth's crust and mantle, a boundary now named the Mohorovicic discontinuity. In 1901 he was appointed head of the complete meteorological service of Croatia and Slavonia, he gradually extended the activities of the observatory to other fields of geophysics: seismology, geomagnetism and gravitation. After the Pokuplje (Kupa Valley) earthquake of 8 Oct 1909, he analyzed the spreading of seismic waves with shallow depths through the Earth. From these, he was the first to establish, on the basis of seismic waves, a surface of velocity discontinuity separating the crust of the Earth from the mantle, now known as the Mohorovicic discontinuity.
*TIS

1862 David Hilbert (23 Jan 1862; 14 Feb 1943) German mathematician who reduced geometry to a series of axioms and contributed substantially to the establishment of the formalistic foundations of mathematics. In his book, Foundations of Geometry, he presented the first complete set of axioms since Euclid. His work in 1909 on integral equations led to 20th-century research in functional analysis (in which functions are studied as groups.) Today Hilbert's name is often best remembered through the concept of Hilbert space in quantum physics, a space of infinite dimensions.*TIS
He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He also formulated the theory of Hilbert spaces, one of the foundations of functional analysis.
Hilbert adopted and warmly defended Georg Cantor's set theory and transfinite numbers. A famous example of his leadership in mathematics is his 1900 presentation of a collection of problems that set the course for much of the mathematical research of the 20th century.
Hilbert and his students contributed significantly to establishing rigor and developed important tools used in modern mathematical physics. Hilbert is known as one of the founders of proof theory and mathematical logic, as well as for being among the first to distinguish between mathematics and metamathematics.*Wik
In Constance Reid's "Hilbert", she describes his mother thus: "She was an unusual woman-in the German way of expression, 'an original' - interested in philosophy and astronomy and fascinated by prime numbers."

1872 Paul Langevin (23 Jan 1872; 19 Dec 1946) French physicist who was the first to explain (1905) the effects of paramagnetism and diamagnetism (the weak attraction or repulsion of substances in a magnetic field) using statistical mechanics. He further theorized how the effects could be explained by how electron charges behaved within the atom. He popularized Einstein's theories for the French public. During WW I, he began developing a source for high intensity ultrasonic waves, which made sonar detection of submarines possible. He created the ultrasound from piezoelectric crystals vibrated by high-frequency radio circuits. In WW II, he spoke out against the Nazis, for which he was arrested and imprisoned, though he managed to escaped and fled to Switzerland.*TIS

1876 Alfredo Niceforo (23 Jan 1876;2 Mar 1960) Italian sociologist, criminologist, and statistician who posited the theory that every person has a "deep ego" of antisocial, subconscious impulses that represent a throwback to precivilized existence. Accompanying this ego, and attempting to keep its latent delinquency in check, according to his concept, is a "superior ego" formed by man's social interaction. This theory, which he published in 1902, bears some resemblance to the discoveries of psychoanalysis that were being made about the same time. *TIS

1878 Edwin Plimpton Adams (23 Jan 1878 in Prague - 31 Dec 1956 in Princeton, USA) studied at Harvard, Göttingen and Cambridge and became Physics Professor at Princeton. He is best known for his translations of some of Einstein's lectures. *SAU

1907 Hideki Yukawa (23 Jan 1907; 8 Sep 1981) Japanese physician and physicist who shared the 1949 Nobel Prize for Physics for “his prediction of the existence of mesons on the basis of theoretical work on nuclear forces.” In his 1935 paper, On the Interaction of Elementary Particles*, he proposed a new field theory of nuclear forces that predicted the existence of the previously unknown meson. Mesons are particles heavier than electrons but lighter than protons. One type of meson was subsequently discovered in cosmic rays in 1937 by American physicists, encouraging him to further develop meson theory. From 1947, he worked mainly on the general theory of elementary particles in connection with the concept of the “non-local” field. He was the first Japanese Nobel Prize winner. *TIS (Yukawa donated a bronze crane that works as a wind chime when pushed against a traditional peace bell from which it is suspended at the Children's Peace Museum in Hiroshima. On the bell in his handwriting is the wish, "A Thousand Paper Cranes. Peace on Earth and in the Heavens."(Is this crane still there? I understood it had been moved."

1924 Sir Michael James Lighthill (23 Jan 1924, 17 Jul 1998) was a British mathematician who contributed to supersonic aerofoil theory and, aeroacoustics which became relevant in the design of the Concorde supersonic jet, and reduction of jet engine noise. Lighthill's eighth power law which states that the acoustic power radiated by a jet is proportional to the eighth power of the jet speed. His work in nonlinear acoutics found application in the lithotripsy machine used to break up kidney stones, the study of flood waves in rivers and road traffic flow. Lighthill also introduced the field of mathematical biofluiddynamics. Lighthill followed Paul Dirac as Lucasian professor of Mathematics (1969) and was succeeded by Stephen Hawking (1989) *TIS

1947 Peter Jephson Cameron (23 January 1947;Toowoomba, Queensland, Australia -) is an Australian mathematician who works in group theory, combinatorics, coding theory, and model theory. He is currently half-time Professor of Mathematics at the University of St Andrews, and Emeritus Professor at Queen Mary University of London.
Cameron received a B.Sc. from the University of Queensland and a D.Phil. in 1971 from University of Oxford, with Peter M. Neumann as his supervisor. Subsequently he was a Junior Research Fellow and later a Tutorial Fellow at Merton College, Oxford, and also lecturer at Bedford College, London. He was awarded the London Mathematical Society's Whitehead Prize in 1979 and is joint winner of the 2003 Euler Medal.
Cameron specialises in algebra and combinatorics; he has written books about combinatorics, algebra, permutation groups, and logic, and has produced over 250 academic papers.[3] He has an Erdős number of 1, with whom he posed the Cameron–Erdős conjecture. *Wik

DEATHS

1805 Claude Chappe (25 Dec 1763, 23 Jan 1805)French engineer who invented the semaphore visual telegraph. He began experimenting in 1790, trying various types of telegraph. An early trial used telescopes, synchronised pendulum clocks and a large white board, painted black on the back, with which he succeeded in sending a message a few sentences long across a 16km (10mi) distance. To simplify construction, yet still easily visible to read from far away, he changed to using his semaphore telegraph in 1793. Smaller indicators were pivoted at each end of large horizontal member. The two indicators could each be rotated to stand in any of eight equally spaced positions. By setting them at different orientations, a set of corresponding codes was used to send a message.*TIS

1810 Johann Wilhelm Ritter (16 Dec 1776, 23 Jan 1810) German physicist who discovered the ultraviolet region of the spectrum (1801) and thus helped broaden man's view beyond the narrow region of visible light to encompass the entire electromagnetic spectrum from the shortest gamma rays to the longest radio waves. After studying Herschel's discovery of infrared radiation, he observed the effects of solar radiation on silver salts and deduced the existence of radiation outside the visible spectrum. He also made contributions to spectroscopy and the study of electricity. *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

Tuesday, 22 January 2019

How Pi was Almost Equal to Three in Indiana

So what about the state that passed a law that set Pi = 3? Well, it was in the paper, and on the internet, but it never happened, although it did get close once. A hoax article was printed and widely circulated (on April 1, 1998) that said that NASA engineers in Huntsville, Alabama were upset about the discovery that the Alabama legislature had just passed a law making Pi=3. When the perpetrators of the hoax realized that the article was being paraphrased (without all the hints that it was a joke, such as the authors name, April Holiday) and circulating as truth, they tried to circulate a notice of the hoax, but found the truth spread much more slowly than the sensational story. (A portent of the nature of fake news well before the term was common)

But it was January 19, 1897 that an actual bill was introduced into the state House of Indiana to pass a law which would have, in effect, make pi equal to ... well several numbers it seems. And I should point out that the proposed bill was the idea idea of a fellow who had already proved many of the impossible constructions of geometry, such as squaring the circle. Here is a another description of the bizarre incident by Cecil Adams from his web column, "Straight Dope":
----------------
It happened in Indiana. Although the attempt to legislate pi was ultimately unsuccessful, it did come pretty close. In 1897 Representative T.I. Record of Posen county introduced House Bill #246 in the Indiana House of Representatives. The bill, based on the work of a physician and amateur mathematician named Edward J. Goodwin (Edwin in some accounts), suggests not one but three numbers for pi, among them 3.2, as we shall see. The punishment for unbelievers I have not been able to learn, but I place no credence in the rumor that you had to spend the rest of your natural life in Indiana. [ Although it is often called the Pi Bill, the main result claimed by the bill is a method to square the circle, rather than to establish a certain value for the mathematical constant π (pi). In fact pi is not mentioned in the text of the bill.]

The text of the bill consists of a series of mathematical claims followed by a recitation of Goodwin's previous accomplishments:

"... his solutions of the trisection of the angle, doubling the cube and quadrature of the circle having been already accepted as contributions to science by the American Mathematical Monthly ... And be it remembered that these noted problems had been long since given up by scientific bodies as unsolvable mysteries and above man's ability to comprehend."

Goodwin's "solutions" were indeed published in the AMM, though with a disclaimer of 'published by request of the author'.

If you feel up to the task, the bill can be found here.

Just as people today have a hard time accepting the idea that the speed of light is the speed limit of the universe, Goodwin and Record apparently couldn't handle the fact that pi was not a rational number. "Since the rule in present use [presumably pi equals 3.14159...] fails to work ..., it should be discarded as wholly wanting and misleading in the practical applications," the bill declared. Instead, mathematically inclined Hoosiers could take their pick among the following formulae:
(1) The ratio of the diameter of a circle to its circumference is 5/4 to 4. In other words, pi equals 16/5 or 3.2
(2) The area of a circle equals the area of a square whose side is 1/4 the circumference of the circle. Working this out algebraically, we see that pi must be equal to 4.
(3) The ratio of the length of a 90 degree arc to the length of a segment connecting the arc's two endpoints is 8 to 7. This gives us pi equal to the square root of 2 x 16/7, or about 3.23.

There may have been other values for pi as well; the bill was so confusingly written that it's impossible to tell exactly what Goodwin was getting at. Mathematician David Singmaster says he found six different values in the bill, plus three more in Goodwin's other writings and comments, for a total of nine.

Lord knows how all this was supposedly to clarify pi or anything else, but as we shall see, they do things a little differently in Indiana. Bill #246 was initially sent to the Committee on Swamp Lands. The committee deliberated gravely on the question, decided it was not the appropriate body to consider such a measure and turned it over to the Committee on Education. The latter committee gave the bill a "pass" recommendation and sent it on to the full House, which approved it unanimously, 67 to 0.

In the state Senate, the bill was referred to the Committee on Temperance. (One begins to suspect it was silly season in the Indiana legislature at the time.) It passed first reading, but that's as far as it got. According to The Penguin Dictionary of Curious and Interesting Numbers, the bill "was held up before a second reading due to the intervention of C.A. Waldo, a professor of mathematics [at Purdue] who happened to be passing through." Waldo, describing the experience later, wrote, "A member [of the legislature] then showed the writer [i.e., Waldo] a copy of the bill just passed and asked him if he would like an introduction to the learned doctor, its author. He declined the courtesy with thanks, remarking that he was acquainted with as many crazy people as he cared to know."

The bill was postponed indefinitely and died a quiet death. According to a local newspaper, however, "Although the bill was not acted on favorably no one who spoke against it intimated that there was anything wrong with the theories it advances. All of the Senators who spoke on the bill admitted that they were ignorant of the merits of the proposition. It was simply regarded as not being a subject for legislation."

Tennessee is also frequently mentioned as a state that "passed a pi=3 bill" but that seems to come from a small reference by Robert Heinlein in Stranger in a Strange Land. "In the Tennessee legislature a bill was again introduced to make the ratio pi exactly equal to three"."

* Much of the text was taken from http://www.straightdope.com. and Wikipedia

On This Day in Math - January 22

Niels & Harald Bohr talking football w/ children (see Deaths 1951)

Prudens interrogatio quasi dimidium sapientiae.

A prudent question is, as it were, one half of wisdom.

~Sir Francis Bacon

The 22nd day of the year; 22 is the smallest Hoax number (the sum of its digits is equal to the sum of the digits of its distinct prime factors). Can you find the next? [these sums that Hoax numbers add up to are an interesting study also]

Arrange the whole numbers from 1 to 22 into pairs so that the sum of the numbers in each pair is a perfect square. (Turns out that you can't, and 22 is the largest even number for which this is true) * Henri Picciotto@hpicciotto

Extra bonus: 22! has exactly 22 digits. *Mario Livio @Mario_Livio

22 is the smallest number which can be expressed as the sum of two primes in three ways.

EVENTS
1673 Leibniz presents a calculation machine at the Royal Society. Leibniz would complain to Oldenburg that Hooke took an "almost obscene" interest in the machine. Sure enough, by Feb 2 Hooke was actively working on an "arithmetic engine" that he would complete and show to the Royal Society within the month. By the following month his interest waned and he decided that no mechanical device could compare to paper and pencil or "Lord Napier's metal or parchment rods" (Napiers bones)*Stephen Inwood, The Forgotten Genius: The Biography Of Robert Hooke 1635-1703

1779 The parish register of Madron (the parish church) records ‘Humphry Davy, son of Robert Davy, baptized at Penzance, January 22nd, 1779. Davy was born in Penzance in Cornwall, United Kingdom, on 17 December 1778.

1833 In his notebook, Gauss introduces the linking number of two knots. "Gauss' note presents the first deep incursion into knot theory. *History of Topology edited by Ian Mackenzie James
From the Classic Carl Friedrich Gauss: Titan of Science By Guy Waldo Dunnington, Jeremy Gray, Fritz-Egbert Dohse, a partial translation comments on the developing theory of knots.

1876 The Johns Hopkins University Founded commonly referred to as Johns Hopkins, JHU, or simply Hopkins, is a private research university based in Baltimore, Maryland, United States. Johns Hopkins maintains campuses in Maryland, Washington, D.C., Italy, China, and Singapore.
The university was founded on January 22, 1876 and named for its benefactor, the philanthropist Johns Hopkins. Daniel Coit Gilman was inaugurated as first president on February 22, 1876. On his death in 1873, Johns Hopkins, a Quaker entrepreneur and childless bachelor, bequeathed $7 million to fund a hospital and university in Baltimore, Maryland. At that time this fortune, generated primarily from the Baltimore and Ohio Railroad, was the largest philanthropic gift in the history of the United States.*JHU Web page

1879, the English were embroiled in a series of running conflicts in South Africa known as the Zulu War. On Jan. 22, 1879, a numerically superior Zulu force overwhelmed a smaller but technologically more advanced British contingent, in what became known as the Battle of Isandlwana. By coincidence, an annular solar eclipse (where the Moon is visually too small to cover the Sun) occurred around 2:30 p.m. at the tail end of the skirmish. The event would have been a deep partial from the battlefield, and the name “Isandlwana” in Zulu means “the day of the dead moon.” *listosaur.com web page

1889 Oskar Bolza gave his ﬁrst lecture to a non-German audience. At Johns Hopkins University he gave twenty lectures “on the theory of substitution groups and its application to algebraic equations.” This was the ﬁrst course on Galois theory in this country. It was published in 1891 in the American Journal of Mathematics.*VFR

1919 Richard Courant married Nina Runge in G¨ottingen. She was the daughter of the mathematician Carl Runge and granddaughter of the physiologist and philosopher of science Emil DuBois-Reymond. This provides another example of mathematical talent being passed from father to son-in-law. [Constance Reid, Courant in G¨ottingen and New York. The Story of an Improbable Mathematician (Springer 1976), p. 75–76] *VFR

In 1980, Soviet dissident physicist Dr. Andrei Sakharov was arrested, stripped of his honors and exiled to Gorky from Moscow. *TIS

1984 Apple Computer Launches the Macintosh, the first successful mouse-driven computer with a graphic user interface, with a single $1.5 million commercial during the Super Bowl. Apple's commercial played on the theme of George Orwell's 1984 and featured the destruction of Big Brother -- a veiled reference to IBM -- with the power of personal computing found in a Macintosh.*CHM Surprise (to most) bit of feminism at the end, wait for it.

In 1997, American Lottie Williams was reportedly the first human to be struck by a remnant of a space vehicle after re-entering the earth's atmosphere. At 3 a.m., while walking in a park in Tulsa, Oklahoma, she saw a light pass over her head. “It looked like a meteor,” she said. Minutes later, she was hit on the shoulder by a six-inch piece of blackened metallic material. The debris that struck Ms. Williams has not been examined to confirm its origin, but a used Delta II rocket, launched nine months earlier, had crashed into the Earth's atmosphere half an hour earlier. NASA scientists believe that Williams was hit by a part of it, making her the only person in the world known to have been hit by man-made space debris. *TIS

BIRTHS
1561 Sir Francis Bacon (22 Jan 1561; 9 Apr 1626) English philosopher remembered for his influence promoting a scientific method. He held that the aim of scientific investigation is practical application of the understanding of nature to improve man's condition. He wrote that scientists should concentrate on certain important kinds of experimentally reproducible situations, (which he called "prerogative instances"). After tabulating such phenomena, the investigator should also aim to make a gradual ascent to more and more comprehensive laws, and will acquire greater and greater certainty as he or she moves up the pyramid of laws. At the same time each law that is reached should lead him to new kinds of experiment, that is, to kinds of experiment over and above those that led to the discovery of the law. *TIS He died a month after performing his ﬁrst scientific experiment. He stuﬀed a chicken with snow to see if this would cause it to spoil less rapidly. The chill he caught during this experiment led to his death. [A. Hellemans and B. Bunch. The Timetables of Science, p . 32]. *VFR

1592 Pierre Gassendi (22 Jan 1592; 24 Oct 1655) French scientist, mathematician, and philosopher who revived Epicureanism as a substitute for Aristotelianism, attempting in the process to reconcile Atomism's mechanistic explanation of nature with Christian belief in immortality, free will, an infinite God, and creation. Johannes Kepler had predicted a transit of Mercury would occur in 1631. Gassendi used a Galilean telescope to observed the transit, by projecting the sun's image on a screen of paper. He wrote on astronomy, his own astronomical observations and on falling bodies.*TIS

1775 André-Marie Ampère (22 Jan 1775; 10 Jun 1836) French mathematician, physicist and chemist who founded and named the science of electrodynamics, now known as electromagnetism. His interests included mathematics, metaphysics, physics and chemistry. In mathematics he worked on partial differential equations. Ampère made significant contributions to chemistry. In 1811 he suggested that an anhydrous acid prepared two years earlier was a compound of hydrogen with an unknown element, analogous to chlorine, for which he suggested the name fluorine. He produced a classification of elements in 1816. Ampère also worked on the wave theory of light. By the early 1820's, Ampère was working on a combined theory of electricity and magnetism, after hearing about Oersted's experiments. *TIS (It is said that Ampere was capable of intense concentration leading to absent-mindedness. Once walking in Paris he had an insight and pulled a piece of chalk out of his pocket and finding the back of a cab he began to cover the back of the cab with equations, and was then shocked to see his solution begin to pull away and disappear down the street.)

1865 Louis Carl Heinrich Friedrich Paschen (22 Jan 1865; 25 Feb 1947) was a German physicist who was an outstanding experimental spectroscopist. In 1895, in a detailed study of the spectral series of helium, an element then newly discovered on earth, he showed the identical match with the spectral lines of helium as originally found in the solar spectrum by Janssen and Lockyer nearly 40 years earlier. He is remembered for the Paschen Series of spectral lines of hydrogen which he elucidated in 1908. *TIS

1866 Gustav de Vries (22 Jan 1866 in Amsterdam, The Netherlands
- 16 Dec 1934 in Haarlem, The Netherlands) was a Dutch mathematician who introduced the famous Korteweg-de Vries equation which characterizes traveling waves. *SAU

1874 Leonard Eugene Dickson (22 Jan 1874,Independence, Iowa, 17 Jan 1954, Harlingen, Texas)American mathematician who made important contributions to the theory of numbers and the theory of groups. He published 18 books including Linear groups with an exposition of the Galois field theory. The 3-volume History of the Theory of Numbers (1919-23) is another famous work still much consulted today. *TIS

1880 Frigyes Riesz (22 Jan 1880; 28 Feb 1956) Hungarian mathematician and pioneer of functional analysis, which has found important applications to mathematical physics. His theorem, now called the Riesz-Fischer theorem, which he proved in 1907, is fundamental in the Fourier analysis of Hilbert space. It was the mathematical basis for proving that matrix mechanics and wave mechanics were equivalent. This is of fundamental importance in early quantum theory. His book Leçon's d'analyse fonctionnelle (written jointly with his student B Szökefalvi-Nagy) is one of the most readable accounts of functional analysis ever written. Beyond any mere abstraction for the sake of a structure theory, he was always turning back to the applications in some concrete and substantial situation. *TIS

1908 Lev Davidovich Landau (22 Jan 1908; 1 Apr 1968) Soviet physicist who worked in such fields as low-temperature physics, atomic and nuclear physics, and solid-state, stellar-energy, and plasma physics. Several physics terms bear his name. He was awarded the 1962 Nobel Prize for Physics for his theory to explain the peculiar superfluid behaviour of liquid helium at very low temperature (2.18 K). Landau's further contributions are partly reflected in such terms as Landau diamagnetism and Landau levels in solid-state physics, Landau damping in plasma physics, the Landau energy spectrum in low-temperature physics, or Landau cuts in high-energy physics. *TIS

1929 Walter Volodymyr Petryshyn (Vladimir Petryshin) (22 January 1929, Liashky Murovani, Lviv - ) is a famous Ukrainian mathematician. He had commenced his studies in Lviv during World War II, but he became a displaced person at the end of the war and continued his schooling in Germany. In 1950 he emigrated from Germany to the United States and completed his education there, living in Paterson, New Jersey. He studied at Columbia University and was awarded a B.A. in 1953, an M.S. in 1954, and a Ph.D. in 1961. Petryshyn's main achievements are in functional analysis. His major results include the development of the theory of iterative and projective methods for the constructive solution of linear and nonlinear abstract and differential equations.*Wik

DEATHS
1779 Jeremiah Fenwicke Dixon (27 July 1733 – 22 January 1779) was an English surveyor and astronomer who is best known for his work with Charles Mason, from 1763 to 1767, in determining what was later called the Mason-Dixon line.
Dixon was born in Cockfield, near Bishop Auckland, County Durham, the fifth of seven children, to Sir George Fenwick Dixon 5th Bt. and Lady Mary Hunter. His father was a wealthy Quaker coal mine owner and aristocrat. His mother came from Newcastle, and was said to have been "the cleverest woman" to ever marry into the Dixon family. Dixon became interested in astronomy and mathematics during his education at Barnard Castle. Early in life he made acquaintances with the eminent intellectuals of Southern Durham: mathematician William Emerson, and astronomers John Bird and Thomas Wright. In all probability it was John Bird, who was an active Fellow of the Royal Society, who recommended Dixon as a suitable companion to accompany Mason.
Jeremiah Dixon served as assistant to Charles Mason in 1761 when the Royal Society selected Mason to observe the transit of Venus from Sumatra. However, their passage to Sumatra was delayed, and they landed instead at the Cape of Good Hope where the transit was observed on June 6, 1761. Dixon returned to the Cape once again with Nevil Maskelyne's clock to work on experiments with gravity.
Dixon and Mason signed an agreement in 1763 with the proprietors of Pennsylvania and Maryland, Thomas Penn and Frederick Calvert, sixth Baron Baltimore, to assist with resolving a boundary dispute between the two provinces. They arrived in Philadelphia in November 1763 and began work towards the end of the year. The survey was not complete until late 1766, following which they stayed on to measure a degree of Earth's meridian on the Delmarva Peninsula in Maryland, on behalf of the Royal Society. They also made a number of gravity measurements with the same instrument that Dixon had used with Maskelyne in 1761. Before returning to England in 1768, they were both admitted to the American Society for Promoting Useful Knowledge, in Philadelphia.
Dixon sailed to Norway in 1769 with William Bayly to observe another transit of Venus. The two split up, with Dixon at Hammerfest Island and Bayly at North Cape, in order to minimize the possibility of inclement weather obstructing their measurements. Following their return to England in July, Dixon resumed his work as a surveyor in Durham. He died unmarried in Cockfield on 22 January 1779, and was buried in an unmarked grave in the Quaker cemetery in Staindrop.
Although he was recognized as a Quaker, he was not a very good one, dressing in a long red coat and occasionally drinking to excess. *Wik

1904 The Reverend George Salmon (25 September 1819 - 22 January 1904) was, firstly, a mathematician whose publications in algebraic geometry were widely read in the second half of the 19th century. He was also an Anglican theologian who devoted himself mostly to theology for the last forty years of his life. His publications in theology were widely read, too. He spent his entire career at Trinity College Dublin. In 1848 Salmon had published an undergraduate textbook entitled A Treatise on Conic Sections. This text remained in print for over fifty years, going though five updated editions in English, and was translated into German, French and Italian. In the late 1840s and the 1850s Salmon was in regular and frequent communication with Arthur Cayley and J.J. Sylvester. The three of them together with a small number of other mathematicians (including Charles Hermite) were developing a system for dealing with n-dimensional algebra and geometry. During this period Salmon published about 36 papers in journals. In these papers for the most part he solved narrowly defined, concrete problems in algebraic geometry, as opposed to more broadly systematic or foundational questions. But he was an early adopter of the foundational innovations of Cayley and the others. In 1859 he published the book Lessons Introductory to the Modern Higher Algebra (where the word "higher" means n-dimensional). This was for a while simultaneously the state-of-the-art and the standard presentation of the subject, and went through updated and expanded editions in 1866, 1876 and 1885, and was translated into German and French. *Wik

1951 Harald August Bohr (22 Apr 1887, 22 Jan 1951) Danish mathematician who devised a theory that concerned generalizations of functions with periodic properties, the theory of almost periodic functions. His brother was noted physicist Niels Bohr.*TIS Harald was an excellent football(soccer) player in his youth and played for the National team. Niels played also, but not at the same high level. An interesting anecdote about Niels Bohr as an athlete is here.

1921 Marie Georges Humbert (7 Jan 1859 in Paris, France - 22 Jan 1921 in Paris, France) His doctorate extended Clebsch's work on curves. He then studied Abel's work which he developed and put into a geometric setting. It was as a direct consequence of his work on using abelian functions in geometry which won for him the 1892 Académie des Sciences prize for work on Kummer surfaces. As Costabel writes, "He thus enriched analysis and gave the complete solution of the two great questions of the transformation of hyperelliptic functions and of their complex multiplication. "
He also extended work of Hermite considering applications to number theory throughout his life.
Humbert would be better known today if the area of mathematics in which he worked had remained in favor. Since it has now become merely something of an historical curiosity rather than mainstream mathematics, his contribution is less well known. It does, however, indicate the quality of his mathematics that, despite this, his name and results are known today. To some extent this is a consequence of the fact that although he worked in a specialized area he had a remarkably broad knowledge of mathematics and his results form links between areas. *SAU

1922 Camille Jordan (5 Jan 1838, 22 Jan 1922) French mathematician and engineer who prepared a foundation for group theory and built on the prior work of Évariste Galois. As a mathematician, Jordan's interests were diverse, covering topics throughout the aspects of mathematics being studied in his era. The topics in his published works include finite groups, linear and multilinear algebra, the theory of numbers, topology of polyhedra, differential equations, and mechanics.*TIS (His date of death is listed as 22 Jan by *SAU & *Wik but 20 Jan by *TIS)

1936 V Ramaswami Aiyar (1871 in Coimbatore district, India - 22 Jan 1936 in Chittoor, India) was an enthusiastic amateur mathematician who worked as a civil servant in India. He was a founder of the Indian Mathematical Society. *SAU

1981 Rudolf Oskar Robert Williams Geiger (24 Aug 1894, 22 Jan 1981) German meteorologist, one of the founders of microclimatology, the study of the climatic conditions within a few metres of the ground surface. His observations, made above grassy fields or areas of crops and below forest canopies, elucidated the complex and subtle interactions between vegetation and the heat, radiation, and water balances of the air and soil.*TIS

1987 Patrick du Val (March 26, 1903–January 22, 1987) was a British mathematician, known for his work on algebraic geometry, differential geometry, and general relativity. The concept of Du Val singularity of an algebraic surface is named after him. Du Val's early work before becoming a research student was on relativity, including a paper on the De Sitter model of the universe and Grassmann's tensor calculus. His doctorate was on algebraic geometry and in his thesis he generalised a result of Schoute. He worked on algebraic surfaces and later in his career became interested in elliptic functions.*Wik

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

1990 Bill Ferrar graduated from Oxford after an undergraduate career interrupted by World War I. He lectured at Bangor and Edinburgh before moving back to Oxford. He worked in college administration and eventually became Principal of Hertford College. He worked on the convergence of series. *SAU

Monday, 21 January 2019

Splitters, Cleavers Equalizers and... fractionalizers?

A week, or so, ago, James Tanton posted the image above with the question,
Given a square of area A, perimeter P. For which values of r between zero and one is it possible to draw a line across the square so that it cuts off a section of area equal to rA and a piece of perimeter equal to rP?

OKAY, go ahead and try it, I'll wait a bit.....

dah

de

dah...
...............................................................................................

As you probably discovered, there is only one, r=1/2. But then there are an infinite number of lines that make that so. One of the things I like about this problem is that it calls up so many possible generalizations. Is it possible to slice a Cube with a plane that divides volume and surface area into equal halves?

And what about triangles, Equilateral or in general. As early as 1959 Dov Aveshalom was writing about Cleavers. Unfortunately for me, his first paper was in Hebrew, but in 1963 he wrote "The Perimetric Bisection of Triangles." and used (created?) the term for perimeter bisectors which passed through a midpoint of one side of a triangle.

As far as I am aware, the first use of Splitters was in Ross Honsberger's Episodes in Nineteenth and Twentieth Century Euclidean Geometry in 1995. Shortly after that, in 1997, George Berzsenyi used (created?) the term equalizers in an article in Quantum Magazine. He gave no proofs, but posed three interesting conjectures:
1) Prove that every triangle has at least one equalizer
2) Prove that No triangle has more than three equlizers
3) Prove that there can (or cannot) be a triangle with exactly two equalizers.

The three questions resisted cracking until in 2010, Dimitrios Kodokostas published "Triangle Equalizers" in Mathematics Magazine. He not only proved that every triangle had at lest one equalizer, he described the conditions under which a triangle could have one, two or three equalizers, and that none had more. The breakthrough idea was his discovery that:
any equalizer goes through the incenter and any line through the incenter is an area splitter, if, and only if, it is a perimeter splitter.

By rotating a line through the incenter, a triangle would be cut off between two sides of the triangle and the cutting leg. When the area of that triangle changed for greater than 1/2 to less than, or vice-versa, there must exist an equalizer. He also worked out why there had been no discovery of a triangle with only two equalizers. The only conditions under which that can occur is a triangle with a smallest angle of less than about 49 degrees, and the second angle fell into a very narrow range of values dependent upon the measure of angle A.

So Squares have an infinite number of equalizing lines, and triangles have between one and three, but what about Mr Tanton's challenge to see if some other r was possible, ie, some sort of fractionalizer in general.

I set my sights on an equilateral triangle first, figuring it would be my easiest to work with. Experimenting after a few seconds I realized that a regular unit triangle that had a line cutting smaller equilateral triangle with all sides of 2/3, would be exactly the r=4/9 that was impossible in the square. The perimeter cut off would be 2/3+2/3 = 4/3.

After a moment of playing with the numbers, I came up with two equations that established the conditions under which lengths of a and b cut off from a vertex of the unit triangle would produce a "fractionalizer" $ \frac{a+b}{3} = r$ and $ ab = r $ (since both the original triangle and the cut off triangle could be computed using the formula A= 1/2(side*side)sin(60deg) we could ignore the actual area, and deal with the product of the adjacent sides.

Restructuring the equations we arrive at the condition that a+b = 3ab. So I quickly set up a geogebra sketch to test the range of possible r's that could occur.

At a=b=2/3 I knew that 4/9=r, and as I adjusted the line on the x-axis to more than 2/3, the ratio increased toward 1/2 as the bottom (a side) approached a length of one, and the b side approached 1/2. When I shortened side a, the same thing happened, as the b side became larger. When a was 3/5 for example, the b side worked out to be 3/4, and r was 9/20.

I switched to a 3-4-5 right triangle to make sure that what I found was not limited to regular triangles, and indeed it was not. For example cutting off a right triangle with sides of 2 and 2, popped up right away. a+b= 4 which is 1/3 of P=12, and ab/2 = 2, or 1/3 of 6. In this case since (a+b)/12=r and (ab/2)/6 = r, it worked out to a+b=ab. Cutting off sides of 5/2 and 5/3 for example, gave r=25/72 or abt .347. As the length of side a on the 4 side of the 3-4-5 reached 4, r=4/9, the smallest possible r in the equilateral triangle.

I suspect that there are an infinite range of fractionalizers for every triangle, and that any number in the range $ 0 < r < 1/2 $ is achievable..... but alas the hour has grown late.