Pat'sBlog

Thursday, 5 March 2020

Islands in the Mist, ----- of Polynomials, and Pretty Geometry

I once read a description of math as like seeing islands in a great ocean covered by a mist. As you learn the subject you work around on an island and clear away some of the mist. Often your education jumps from one island to another at the direction of a teacher and eventually you have mental maps of parts of many separate islands. But at some point, you clear away a fog on part of an island and see it connects off to another island you had partially explored, and now you know something deeper about both islands and the connectedness of math.

I was recently reminded of one of those kinds of connections that ties together several varied topics from the high school education of most good math students. It starts with that over-criticized (and under-appreciated, i) Algebra I technique of factoring.
Almost ever student in introductory algebra is introduced to a "sum and product" rule that relates the factors of a simple quadratic (with quadratic coefficient of one) to the coefficients. The rule says that if the roots are at p and q, then the linear coefficient will be the negative of p+q, and the constant term will be their product, pq. So for example, the simple quadratic with roots at x=2 and x=3 will be x² - 5x + 6.

I know from experience that if you take a cross section of 100 students who enter calculus classes after two+ year of algebra, very few will know that you can extend that idea out to cubics and higher power polynomials. An example for a polynomial with four roots will probably suffice for most to understand. Because the constant terms in linear factors are always the opposite of the roots, {if 3 is a root, (x-3) is a factor} it is easiest to negate all the roots before doing the math involved (at least for me it always was).
So if we wanted to find the simple polynomial with roots at -1, -2, -3, and -4 (chosen so all the multipliers are +) we would find that the fourth degree polynomial will have 10 for the coefficient of x³ because 1+2+3+4 = 10, just as it works in the second term of a quadratic. After that, the method starts to combine sets of them. The next coefficient will be the sum of the products of each pair of factor coefficients. In the example I created we would add 1x2+1x3+1x4+2x3+2x4+3x4 to get 35x². The next term sums all triple products of the numbers, 1x2x3 + 1x2x4 + 1x3x4 + 2x3x4 = 50 for the linear coefficient. And in the constant term, we simply multiply all of them together to get 24.

After you've carried that around for a while and maybe forgotten how to get all the other terms, the easy part may remain; the second term is the sum of the opposite of the roots, and the constant term is their product. Then you get to calculus and you learn how easy it is to take the derivative of a polynomial. Then maybe you are playing around with some simple derivatives and you realize that a function f(x) = xⁿ + Ax^n-1 + ... will have a derivative that is f'(x)=nx^n-1 + A(n-1)x^n-2. You realize that if f(x) has roots that sum to A, then f'(x) has roots that will sum to (n-1)A/n [If your younger and this seems unclear, note that the roots of f(x) are the same as the roots of n*f(x), for example, y= x² - 1 has the same roots (+/-1) as 2x²-2 or 3x²-3 etc].

Much later, you come back across this thought, but because you are at a different place in your understanding of math, you realize that means that the average of the zeros of f(x) is A/n, because there are n of them. So the average of f'(x) must also be A/n because there are n-1 of them... and since f"(x) is related to f'(x) by this same method, A/n must be the average of all the zeros of derivatives of f(x) that do not descend to a constant value.
Because that seems to glib to pass muster with most of my students, an example of these last two paragraphs, to show how interrelated they are. Take the example f(x) = $x^4 + 3x^3 + 7x^2 + 2x + 4 $. We simply inspect to see that the roots have a sum of 3, and since there are four of them, their average is -3/4. Without knowing the derivatives, we know the roots of f' will sum to $ \frac{3(-3)}{4} = \frac{-9}{4} $ and since there are three of them, their average is ...yeah... -3/4. We can find f" and the rest by continuing this, but the big flashing light here is that the average stays the same, so the sum of the roots is just the average root times the highest power of the derivative.

You smugly nest that away in your mind and go on about your business, occasionally refreshing it by relating it to a friend or colleague in the coffee shop or at a conference.

Someday down the line you wonder, or someone you relate it to asks, will that work with numbers that have complex roots, and you quickly convince yourself that it will, and feel pretty smug for knowing all this. Then you stumble across an old copy of Professor Dan Kalman's paper on Marden's Theorem (at least you will if you are as lucky as I was). (Professor Kalman was awarded the 2009 Lester R. Ford Award of the MAA for his 2008 paper on this theorem. Jörg Siebeck discovered this theorem 81 years before Morris Marden wrote about it (1965). However, Prof Kalman writes, "I call this Marden’s Theorem because I first read it in M. Marden’s wonderful book". The theorem says that if you take a trinomial with complex roots (even if the coefficients are complex numbers) there is a really beautiful geometric tie in to the average idea, and more.

I will illustrate with an example that is easy to picture. Suppose we take a trinomial with roots of 2+5i, 2-5i and 6, f(x) =x³ - 10 x² + 53 x - 174. The derivative will be 3x² - 20x + 53, with zeros at the complex conjugates x = 1/3 (10-i sqrt(59)) and x = 1/3 (10+i sqrt(59)). Both of these we can see quickly have averages of 10/3 for the zeros, but these first derivative zeros will play a special geometric role a little later in Marden's theorem.

The second derivative of the original cubic gives us 6x-20, with a zero which agrees with the average of the zeros above.

All those little islands with a common algebraic truth seem somehow connected, but then a little more of the mist clears, and the geometry is revealed.

But if we examine these zeros on a complex plane, the three zeros of the original function can form the vertices of a triangle. And the two zeros of the first derivative fall inside that triangle, with the zero of f" bisecting the segment joining them.

So the vertices of the triangle are the roots of f(x), the two red points are f' zeros, and they are the foci of the ellipse shown inscribed in the triangle. And f" has a zero at the center of the ellipse. The ellipse passes through the midpoints of the vertices, and it turns out it is the maximal area ellipse you can inscribe in that triangle, called the Steiner inellipse. (A little algebra, a little calculus, a little geometry, a little trig... maybe there are really no islands, just one math land mass. )

I backed it all up one level by integrating f(x) but the four roots did not appear to relate to the three vertices of the trinomial in any pretty way. They do obey the Gauss-Lucas Theorem. The Gauss–Lucas theorem gives a geometrical relation between the roots of a polynomial P and the roots of its derivative P'. The set of roots of a real or complex polynomial is a set of points in the complex plane. The theorem states that the roots of P' all lie within the convex hull of the roots of P, that is the smallest convex polygon containing the roots of P. When P has a single root then this convex hull is a single point and when the roots lie on a line then the convex hull is a segment of this line. The Gauss–Lucas theorem, named after Carl Friedrich Gauss and Félix Lucas is similar in spirit to Rolle's theorem, another high school calculus basic.

And here is a tie-in for the stats students, the line containing the foci and centroid is the least squares regression line for the three vertices.

If you have only three roots to a higher degree polynomial (one with some or all the roots multiple, such as f(x) = (x-a)^J (x-b)^K(x-c)^J then the ellipse will be tangent at points that divide the segments in ratios of J/K, K/L, and L/J. This is due to Linfield who published it in 1920.

And if you have an n-sided polygon which is tangent to an ellipse at all four midpoints, it seems that there is a complex polynomial with those roots whose derivative has zeros at the foci of the ellipse. I managed to create an easy example by using the idea of a rhombus centered at the origin. The polynomial f(x)=x⁴+3x² - 4 has zeros at 2i, -2i, 1 and -1. The derivative, 4x²+6 has zeros at +sqrt(3/2) and -sqrt(3/2). Using this focus and the point (1/2,1) which is the midpoint of one side of the rhombus I get the ellipse 4x² + y²=2 which seems to work.

I don't have a clear easy way to recognize what fourth power polynomials would have that property, so if you want to be next to teach me some math, send me what you know.

I communicated several times in 2007 with Professor Kalman when we shared some information about the history of a problem we were both working on. He went on to include that material, and Marden's Theorem in his wonderful book Uncommon Mathematical Excursions: Polynomia and Related Realms. If you pick it up, check the acknowledgements. There is actually a hat-tip from the professor to yours truly for (a very tiny bit of) assistance with the material for Lill's graphic method of solving for the roots of a polynomial. Still, I'm grateful for any recognition.

On This Day in Math - March 5

Terrestrial globe by Mercator dating from 1541. It is now in the museum collection of the Palazzo Ducale in Urbania, Italy, and is one of about 22 existing Mercator globes.*Wik

But in the present century, thanks in good part to the influence of Hilbert, we have come to see that the unproved postulates with which we start are purely arbitrary. They must be consistent, they had better lead to something interesting.
~Julian Lowell Coolidge

The 65th day of the year; 65 is the smallest hypotenuse of two different primitive Pythagorean triangles (and of two other triangles that are not primitive) with all integral sides. (Don't just sit there, find them!)

And $ 65 = 1^5 + 2^4 + 3^3 + 4^2 + 5^1 $ *jim wilder ‏@wilderlab

OR, $65= 0^2 + 1^4 + 2^5 + 3^3 + 4^1 + 5^0 $ *@Expert_says

65 is the constant of a 5x5 normal magic square.
A magic square with the integers 1 through 25 has a sum of 65 in each row, column, and major diagonal.

Euler found 65 integers, which he called "numeri idonei," that could be used to prove the primality of certain numbers.[idoneal numbers (also called suitable numbers or convenient numbers) are the positive integers D such that any integer expressible in only one way as $x^2 ± Dy^2$ (where x² is relatively prime to Dy²) is a prime, prime power, twice one of these, or a power of 2. In particular, a number that has two distinct representations as a sum of two squares (such as 65) is composite. Every idoneal number generates a set containing infinitely many primes and missing infinitely many other primes.]

65 is also the smallest Fermat semiprime [A Fermat number, $2^{2^n} +1$ that is factorable into two primes. Fermat conjectured they were all prime]

EVENTS

In 1223 BC, the oldest recorded eclipse occurred, according to one plausible interpretation of a date inscribed on a clay tablet retrieved from the ancient city of Ugarit, Syria (as it is now). This date is favoured by recent authors on the subject, although alternatively 3 May 1375 BC has also been proposed as plausible. Certainly by the 8th century BC, the Babylonians were keeping a systematic record of solar eclipses, and possibly by this time they may have been able to apply numerological rules to make fairly accurate predictions of the occurrence of solar eclipses. The first total solar eclipse reliably recorded by the Chinese occurred on 4 Jun 180 BC*TIS

In 1590, Tycho Brahe discovered a comet in the constellation Pisces.*TIS

In 1616, Copernican theory was declared "false and erroneous" in a decree delivered by Cardinal Robert Bellarmine, and issued by the Catholic Church in Rome. Further, no person was to be permitted to hold or teach the theory that the earth revolves around the sun. When Galileo subsequently violated the decree, he was put on trial and held under house arrest for the final eight years of his life. *TIS Copernican theory was declared "false and erroneous" by the 11 theologians, appointed by the Pope to examine it, on 24 February 1616. Bellarmine, who was not one of these 11, was ordered by the Pope to convey this decision to Galileo, which he did verbally on 26 February 1616. The Decree of the Index was issued on 5 March 1616 in which "…the books by Nicolaus Copernicus and Diego Zúñiga be suspended until corrected…" This decree was signed by the Most Illustrious and Reverend Lord Cardinal of St. Cecilia, Bishop of Albano P. (Paolo Sfondrati) and Fra Francisco Magdelenus Capiferreus, O.P., Secretary. *Thony Christie, My thanks to Thony for the correction More detail about this event can be found on the Feb 26 Post about Galileo

1639 Debeaune to Mersenne: “I do not think that one could acquire any solid knowledge of nature in physics without geometry, and the best of geometry consists of analysis, of such kind that without the latter it is quite imperfect.” *VFR

1673 Hooke presents Arithmetic Engine to Royal Society. After a presentation of a calculating machine by Leibniz on January 22, (after which Leibniz complained to Oldgenburg that Hooke's examination of the machine had shown "almost indecent interest") Hooke became interested in creating a better machine and announced such intention to the Royal Society. Working with Richard Shortgrave, Harry Hunt and John Pell he produced a machine which would multiply to twenty places over the next six weeks. His diary entry seemed to indicate the demonstration went well, but within a few days he seemed to have dismissed such machines entirely. *Stephen Inwood, Forgotten Genius

1684 Halley's father mysteriously went missing and five weeks later was found murdered on the banks of the Medway. *Kate Morant, halleyslog.wordpress.com

On March 5, 1750, Euler read his own Recherches sur la Précession at the Berlin Academy. Two days later he wrote d'Alembert giving an extended account of his struggle to derive the precession and giving d'Alembert credit for re-inspiring his efforts to solve it. * Curtis Wilson, Historia Mathematica, Volume 35, Issue 4, November 2008, Pages 329–332

1831 Birth of "The Average Man". Adolphe Quetelet read a memoir to the Brussels Academy Royal. The newborn l'homme moyen would not be officially named by quetelet until July. *Statistics on the Table: The History of Statistical Concepts and Methods
By Stephen M. Stigler

1876 Sylvester, at age 61, appointed professor of mathematics at Johns Hopkins University. This was the real beginning of graduate mathematics education in the United States. *VFR

1960 Gao–Guenie (H5 ordinary chondrite) meteorites fell in Burkina Faso on March 5, 1960 at 17:00 (local time). After three separate detonations, several thousands of stones rained down over an area of about 70 square kilometres (27 sq mi). The sound of the fall was heard as far as Ouagadougou, which is 100 kilometers (62 mi) away. Eyewitnesses said that some trees were broken and henhouses destroyed. The largest stones recovered weigh up to 10 kilograms (22 lb)*Wik

1963 On this day in 1963, the Hula-Hoop, a hip-swiveling toy that became a huge fad across America when it was first marketed by Wham-O in 1958, is patented by the company’s co-founder, Arthur “Spud” Melin. An estimated 25 million Hula-Hoops were sold in its first four months of production alone. *http://www.history.com

1993 Talking Laptop Helps Blind Student Earn B.S.:
In an early demonstration of the impact computers could have on people's lives, the Los Angeles Times reports that a blind student was taking advantage of a talking laptop computer to help him complete courses necessary to graduate from UCLA. After 15 years of going to college on and off, the computer provided Robert Antunez the independence and aid he needed to complete a bachelor's degree in political science. *CHM

BIRTHS

1512 Gerardus Mercator (5 Mar 1512- 2 Dec 1594) Flemish cartographer whose most important innovation was a map, embodying what was later known as the Mercator projection, on which parallels and meridians are rendered as straight lines spaced so as to produce at any point an accurate ratio of latitude to longitude. He also introduced the term atlas for a collection of maps. *TIS A nice blog about the Mercator projection, which he suggests should be called the Mercator Wright projection is at the Renaissance Mathematicus blogsite.
For those interested in a quick look at the math involved in the Merator-Wright projection, this Endeavour blog by John D. Cook may help.

1575 William Oughtred (5 Mar 1575; 30 Jun 1660 at age 85) English mathematician and Episcopal minister who invented the earliest form of the slide rule, two identical linear or circular logarithmic scales held together and adjusted by hand. Improvements involving the familiar inner rule with tongue-in-groove linear construction came later. He also introduced the familiar multiplication sign x in a 1631 textbook, along with the first use of the abbreviations sin, cos and tan.*Tis There is an Oughtred Society dedicated to the history and preservation of slide rules.

1624/25 John Collins (5 March 1624 in Wood Eaton (4km north of Oxford), England - 10 Nov 1683 in London, England) was an accountant and publisher who corresponded extensively with the mathematicians of his day. Collins's importance is, as Barrow said, being "the English Mersenne" . He corresponded with Barrow, David Gregory, James Gregory, Newton, Wallis, Borelli, Huygens, Leibniz, Tschirnhaus and Sluze.
Collins published books by Barrow and Wallis and left a collection of 2000 books and an uncounted number of manuscripts.
He did publish works of his own, however. For instance he published works on sundials, trigonometry for navigation and the use of the quadrant. He had a paper on cartography published and also wrote on accounting, compound interest and annuities. His major works were An introduction to merchant's accounts (1652), The sector on a quadrant (1658), Geometrical dialling (1659), The mariner's plain scale new plained (1659) and, in 1664, he published Doctrine of Decimal Arithmetick. *SAU

1779 Benjamin Gompertz (March 5, 1779 – July 14, 1865), was a self educated mathematician, denied admission to university because he was Jewish.[citation needed] Nevertheless he was made Fellow of the Royal Society in 1819. Gompertz is today mostly known for his Gompertz law (of mortality), a demographic model published in 1825. The model can be written in this way:

N(t) = N(0) e^{-c (e^{at}-1)},

where N(t) represents the number of individuals at time t, and c and a are constants.

This model is a refinement of the demographic model of Malthus. It was used by insurance companies to calculate the cost of life insurance. The equation, known as a Gompertz curve, is now used in many areas to model a time series where growth is slowest at the start and end of a period. The model has been extended to the Gompertz–Makeham law of mortality.

1794 Jacques Babinet (5 March 1794 – 21 October 1872) was a French physicist, mathematician, and astronomer who is best known for his contributions to optics. A graduate of the École Polytechnique, which he left in 1812 for the Military School at Metz, he was later a professor at the Sorbonne and at the Collège de France. In 1840, he was elected as a member of the Académie Royale des Sciences. He was also an astronomer of the Bureau des Longitudes.
Among Babinet's accomplishments are the 1827 standardization of the Ångström unit for measuring light using the red Cadmium line's wavelength, and the principle (Babinet's principle) that similar diffraction patterns are produced by two complementary screens. He was the first to suggest using wavelengths of light to standardize measurements. His idea was first used between 1960 and 1983, when a meter was defined as a wavelength of light from krypton gas.
In addition to his brilliant lectures on meteorology and optics research, Babinet was also a great promoter of science, an amusing and clever lecturer, and a brilliant, entertaining and prolific author of popular scientific articles. Unlike the majority of his contemporaries, Babinet was beloved by many for his kindly and charitable nature. He is known for the invention of polariscope and an optical goniometer. *Wik

1815 Angelo Genocchi (5 March 1817 – 7 March 1889) was an Italian mathematician who specialized in number theory. He worked with Giuseppe Peano. The Genocchi numbers are named after him. G(t)= 2t/(e^t+1)for integer values of t. The first few are 1, −1, 0, 1, 0, −3, 0, 17...(A001469 in OEIS)
Genocchi was President of the Academy of Sciences of Turin.*Wik

1880 Sergei Natanovich Bernstein (March 5, 1880 – October 26, 1968) was a Russian and Soviet mathematician. His doctoral dissertation, submitted in 1904 to the Sorbonne, solved Hilbert's nineteenth problem on the analytic solution of elliptic differential equations. Later, he published numerous works on Probability theory, Constructive function theory, and mathematical foundations of genetics. From 1906 until 1933, Bernstein was a member of the Kharkov Mathematical Society. *Wik

1885 Pauline Sperry born in Peabody, Massachusetts. After graduating Phi Beta Kappa from Smith College in 1906 she taught several years before doing graduate work at the University of Chicago under the projective differential geometer Ernest Julius Wilczynski (1876–1932). Her doctoral thesis, "Properties of a certain projectively defined two-parameter family of curves on a general surface", drew on his work as the founder of the American school of projective differential geometry. After receiving her Ph.D. in 1916 she taught at the University of California at Berkeley, becomming the ﬁrst woman to be promoted to assistant professor in mathematics (in 1923). In 1950 she was ﬁred for refusing to sign a loyalty oath.

1915 Laurent-Moïse Schwartz (5 March 1915 in Paris – 4 July 2002 in Paris) was a French mathematician. He pioneered the theory of distributions, which gives a well-defined meaning to objects such as the Dirac delta function. He was awarded the Fields medal in 1950 for his work (developing the theory of distributions, a new notion of generalized functions motivated by the Dirac delta-function of theoretical physics). He was the first French mathematician to receive the Fields medal. For a long time he taught at the École polytechnique. *Wik

1931 Vera S. Pless (born 1931) is an American mathematician specializing in combinatorics and coding theory. She is professor emeritus at the University of Illinois at Chicago. She has co-authored several articles with John H. Conway, giving her an Erdős number of 2.*Wik

DEATHS

1827 Pierre Simon, Marquis de Laplace (23 Mar 1749, 5 Mar 1827 at age 78) was a French mathematician, physicist, statistician and astronomer known for his mathematical analysis of the stability of the solar system (1773), alleviating Isaac Newton's concerns about perturbations between planets. He took an exact approach to science. He developed an explanation of surface tension of a liquid in terms of inter-molecular attractions, investigated capillary action and the speed of sound. He assisted Antoine Lavoisier (1783) investigating specific heat and heats of combustion, initiating the science of thermochemistry. He believed the solar system formed from a collapsing nebula. He contributed to the mathematics of probability and calculus, in which a differential equation is known by his name, and was involved in establishing the metric system.*TIS His last words were, “What we know is very slight; what we don’t know is immense.” *Eves, Revisited, 319◦

1827 Count Alessandro Giuseppe Antonio Anastasio Volta (18 Feb 1745; 5 Mar 1827 at age 82) Italian physicist who invented the electric battery (1800), which for the first time enabled the reliable, sustained supply of current. His voltaic pile used plates of two dissimilar metals and an electrolyte, a number of alternated zinc and silver disks, each separated with porous brine-soaked cardboard. Previously, only discharge of static electricity had been available, so his device opened a new door to new uses of electricity. Shortly thereafter, William Nicholson decomposed water by electrolysis. That same process later enabled Humphry Davy to isolate potassium and other metals. Volta also invented the electrophorus, the condenser and the electroscope. He made important contributions to meteorology. His study of gases included the discovery of methane. The volt, a unit of electrical measurement, is named after him.*TIS

1875 Claude-Louis Mathieu (25 Nov 1783; 5 Mar 1875) French astronomer and mathematician who worked particularly on the determination of the distances of the stars. He began his career as an engineer, but soon became a mathematician at the Bureau des Longitudes in 1817 and later professor of astronomy in Paris. For many years Claude Mathieu edited the work on population statistics L'Annuaire du Bureau des Longitudes produced by the Bureau des Longitudes. His work in astronomy focussed on determining the distances to stars. He published L'Histoire de l'astronomie au XVIII siècle in 1827. *TIS

1905 Karol Borsuk (May 8, 1905, Warsaw – January 24, 1982, Warsaw) Polish mathematician. His main interest was topology.
Borsuk introduced the theory of absolute retracts (ARs) and absolute neighborhood retracts (ANRs), and the cohomotopy groups, later called Borsuk-Spanier cohomotopy groups. He also founded the so called Shape theory. He has constructed various beautiful examples of topological spaces, e.g. an acyclic, 3-dimensional continuum which admits a fixed point free homeomorphism onto itself; also 2-dimensional, contractible polyhedra which have no free edge. His topological and geometric conjectures and themes stimulated research for more than half a century. *Wikipedia

1925 Johan Ludwig William Valdemar Jensen (8 May 1859 in Nakskov, Denmark - 5 March 1925 in Copenhagen, Denmark)contributed to the Riemann Hypothesis, proving a theorem which he sent to Mittag-Leffler who published it in 1899. The theorem is important, but does not lead to a solution of the Riemann Hypothesis as Jensen had hoped. It expresses, "... the mean value of the logarithm of the absolute value of a holomorphic function on a circle by means of the distances of the zeros from the center and the value at the center. "
He also studied infinite series, the gamma function and inequalities for convex functions.*SAU

1840 Franz Carl Joseph Mertens (20 March 1840 in Schroda, Posen, Prussia (now Środa Wielkopolska, Poland) - 5 March 1927 in Vienna, Austria) Mertens worked on a number of different topics including potential theory, geometrical applications to determinants, algebra and analytic number theory, publishing 126 papers. Bruce C Berndt writes, "Mertens is perhaps best known for his determination of the sign of Gauss sums, his work on the irreducibility of the cyclotomic equation, and the hypothesis which bears his name. "
Many people are aware of Mertens contributions since his elementary proof of the Dirichlet theorem appears in most modern textbooks. However he made many deep contributions including Mertens' theorems, three results in number theory related to the density of the primes. He proved these results using Chebyshev's theorem, a weak version of the prime number theorem. *SAU
In his youth, Mertens moved to Berlin where he became a student at Berlin
University, and where he studied under Kronecker and Kummer. Mertens first worked in Krakow, and then moved to Austria. Ernst Fischer and Schrodinger, for instance, were students of Mertens at the University of Vienna. *Julio Gonzalez Cabillon, Historia Matematica Discussions

1885 John Radford Young (1799– March 5,1885; Peckam, England) was a mathematician, professor and author, who was almost entirely self-educated. At an early age he became acquainted with Olinthus Gilbert Gregory, who perceived his mathematical ability and assisted him in his studies.
In 1833, he was appointed Professor of Mathematics at Belfast College. When Queen's College, Belfast, opened in 1849, the Presbyterian party in control there prevented Young's reappointment as Professor in the new establishment. From that time he devoted himself more completely to the study of mathematical analysis, and made several original discoveries. He appears to have been the first to use the term "circular function" when he used it in 1831 in the an edition of Elements of the Differential Calculus "Thus, ax, a log x, sin x, &c., are transcendental functions: the first is an exponential function, the second a logarithmic function, and the third a circular function"
In 1847, he published in the Transactions of the Cambridge Philosophical Society a paper "On the Principle of Continuity in reference to certain Results of Analysis", and, in 1848, in the Transactions of the Royal Irish Academy a paper "On an Extension of a Theorem of Euler". As early as 1844, he had discovered and published a proof of Newton's rule for determining the number of imaginary roots in an equation. In 1866, he completed his proof, publishing in The Philosophical Magazine a demonstration of a principle which in his earlier paper he had assumed as axiomatic. In 1868, he contributed to the Proceedings of the Royal Irish Academy a memoir "On the Imaginary Roots of Numerical Equations".
*Wik

1930 Christine Ladd-Franklin (1 Dec 1847; 5 Mar 1930) American scientist and logician known for contributions to the theory of colour vision accounting for the development of man's color sense which countered the established views of Helmholtz, Young, and Hering. Her position was that color-sense developed in stages. Ladd- Franklin's conclusions were particularly useful in accounting for color-blindness in some individuals. In logic, she published an original method for reducing all syllogisms to a single formula *TIS Ladd-Franklin was the first woman to have a published paper in the Analyst. She was also the first woman to receive a Ph.D. in mathematics and logic. The majority of her publications were based on visual processes and logic. Her views on logic influenced Charles S. Peirce’s logic and she was highly praised by Prior. *Wik

1954 Julian Lowell Coolidge (28 Sep 1873, 5 Mar 1954 at age 80) American mathematician and educator who published numerous works on theoretical mathematics along the lines of the Study-Segre school. Coolidge received a B.A. at Harvard (1895), then in England he graduated (1897) with a B.Sc. from Balliol College Oxford. (It is interesting that this degree from Oxford was in natural science and it was the first natural science degree ever awarded by Oxford.) He taught at Groton School, Conn. (1897-9) where one of his pupils was Franklin D Roosevelt, the future U.S. president. From 1899 he taught at Harvard University. Between 1902 and 1904, he went to Turin to study under Corrado Segre and then to Bonn where he studied under Eduard Study. His Mathematics of the Great Amateurs is perhaps his best-known work. *TIS . This geometer wrote several noteworthy books on the history of geometry.*VFR

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

Wednesday, 4 March 2020

On This Day in Math - March 4

This granite memorial, to William Willett, is in a clearing in Petts Wood in south east London. On the south face of the memorial is a sundial that is "set" to British Summer Time (BST) *http://www.waymarking.com/

There is no philosophy which is not founded upon knowledge of the phenomena, but to get any profit from this knowledge it is absolutely necessary to be a mathematician.
~Daniel Bernoulli

The 64th day of the year; 64 is the smallest power of two with no prime neighbor. (What is next value of 2ⁿ with no prime neighbor?)

64 is also the first whole number that is both a perfect square and a perfect cube.

There were 64 disks in Eduard Lucas' myth about the Towers of Hanoi.

64 is also the number of hexagrams in the I Ching, and the number of sexual positions in the Kama Sutra. (I draw no conclusions about that information)

There are 64 ordered permutations of nonempty subsets of {1,..., 4}: Eighteenth- and nineteenth-century combinatorialists call this the number of (nonnull) "variations" of 4 distinct objects.

EVENTS

1675 date of Charles II’s Royal Warrant that ordered the Board of Ordnance to pay for “the support and Maintenance” of John Flamsteed, appointed “our astronomical observator” and charged:

“to apply himself with the most exact care and diligence to the rectifying the tables of the motions of the heavens, and the places of the fixed stars, so as to find our the so much-desired longitude of places for the perfecting the art of navigation.”

*Rebekah Higgitt, Teleskopos (although Ms. Higgitt is not fond of historical anniversaries)

The Royal Observatory web page contains a little more information about the events that precipitated the founding of the observatory:

If you'd stood here on the hill in Greenwich Park on 10 August 1675 you would have seen an important event. At 3.14pm the first Astronomer Royal John Flamsteed laid the foundation stone of the new Royal Observatory, Britain’s first state-funded scientific research institution. Events had moved quickly after the initial visit by the French astronomer, Sieur de St. Pierre in December 1674. Thanks to Charles II’s French mistress, Louise de Kéroualle, rumours started to circulate at court that St. Pierre had devised a means of determining longitude at sea by using observations of the Moon’s position in relation to the background stars. Improving navigation at sea was a major challenge for 17th century merchants and their sailors who undertook long voyages across the globe to bring back precious cargoes of tea, spices, timber, porcelain and textiles. While the French astronomer’s claims were rejected by a committee of English scholars in February 1675, the emergence of this idea highlighted the need for something to be done to address this challenge which offered many lucrative financial and political benefits. On 4 March 1675, the King signed a Royal Warrant appointing Flamsteed as 'astronomical observator..[..]..so as to find out the so much-desired longitude of places for the perfecting the art of navigation'.

1801 Thomas Jefferson became the third president of the United States. During his two terms in office he repeatedly sponsored bills providing governmental support of science for the common good. *VFR

1837 Adolphe Quetelet Predicts a meteor shower for the night of August 10th. First published prediction that Persid meteors were annual event.
The 1833 Leonid storm had galvanized interest in meteors, and the time was ripe. Adolphe Quetelet, a Belgian statistician and founder and director of the Brussels Observatory, had mentioned mid-August meteors very tentatively six months earlier. His attention had been called to meteors by François Arago of France, who dominated European science at the time with his skill in discerning important scientific problems and suggesting experiments to solve them. What, asked Arago in the wake of the 1833 display, constituted a shower of meteors, and what was the rate of the ordinary, everynight drizzle?
The problem was ideal for Quetelet, whose passion was statistics. In a speech to the Royal Academy of Sciences and Arts of Brussels on December 3, 1836, Quetelet gave his answer: averaged over the night and year, a single observer should expect to see eight sporadic (nonshower) meteors per hour. That figure is still good today. After his speech Quetelet made a brief mention of unusual August meteors, and in his 1836 annual report of the Brussels Observatory he presented the idea timidly and almost in passing: "I thought I also noticed a greater frequency of these meteors in the month of August (from the 8th to the 15th)."
By the following year, Quetelet had accidentally found records in his observatory of exceptional meteor displays on August 10th of 1834 and 1835 to accompany the increase he had seen in 1836. He called for scientists at the March 4, 1837, session of the Royal Academy of Brussel to watch the sky on August 10, 1837. *Sky and Telescope

1891 David Hilbert submits article on his space filling curve, Über die stetige Abbildung einer Linie auf ein Flächenstück to the journal Mathematische Annalen. *Wik

1929 When Herbert Hoover was sworn in as President of the United States, his wife, Lou Henry Hoover, became the first “First Lady” with a degree in a scientific field. Like her husband, she had graduated from Stanford with a degree in geology. *FFF pg 313

1949 The first time the carbon-14 radioactive dating technique was used. To test the theory the method was used to determine the age of Egyptian artifacts where their age was already known. Willard Frank Libby dated a piece of wood from the Third Dynasty Pharaoh Djoser's tomb that was about 4,700 years old. This age was nearly the same as the half-life of carbon-14, they expected the concentration of carbon-14 would be half that found today. This test was successful. *about.com

1956 An Wang Sells Core Memory Patent to IBM:
An Wang sells his patent for ferrite core memory to IBM for $$500,000$. One of the most important inventions in computer history, ferrite core memory was widely used in digital computers from the mid-1950s until the mid-1970s. The U.S. Patent Office awarded Wang the patent for what he called a pulse transfer controlling device in 1949. Jay Forrester at MIT is considered the inventor of core memory. *CHM

In 1977, the first Freon-cooled Cray-1 supercomputer, costing $$19,000,000$ , was shipped to Los Alamos Laboratories, NM, and was used to help the defense industry create sophisticated weapons systems. This system had a peak performance of 133 megaflops and used the newest technology, integrated circuits and vector register technology. The Cray-1 looked like no other computer before or since. It was a cylindrical machine 7 feet tall and 9 feet in diameter, weighed 30 tons and required its own electrical substation to provide it with power (an electric bill around $$35,000/month$). The inventor, Seymour Cray, died 5 Oct 1996 in an auto accident. His innovations included vector register technology, cooling technologies, and magnetic amplifiers. *TIS

1979 Voyager I photo reveals rings of Jupiter. *VFR

*NASA

2012 Today's date can be written (yr/mo/day) as 12/3/4 (I missed this until it was pointed out to me by Don McDonald)

BIRTHS

1822 Jules Antoine Lissajous (4 March 1822 in Versailles, France - 24 June 1880 in Plombières, France) was a French mathematician best known for the Lissajous figures produced from a pair of sine waves. *SAU The curves are also called Bowditch curves for the early American mathematician, Nathanial Bowditch, who worked with them earlier. In general, a parametric curve with equations x= A sin(k t ); y= B sin(m t), the curves can describe things as simple as a circle or ellipse to more complex open and closed curves. If the ratio of k/m is rational, the curve will eventually close.(EEB)

Lissajous was interested in waves and developed an optical method for studying vibrations. He wanted to be able to see the waves that were created by vibrations, usually expressed in the form of sound. At first he studied waves produced by a tuning fork in contact with water, studying the ripples that were caused. Working on these ideas, he published Sur la position des noeuds dans les lames qui vibrent transversalement (1850). In 1855 he described a way of studying acoustic vibrations by reflecting a light beam from a mirror attached to a vibrating object onto a screen. *SAU

1833 John Monroe Van Vleck (March 4, 1833–November 4, 1912) was an American mathematician and astronomer. He taught astronomy and mathematics at Wesleyan University in Middletown, Connecticut for more than 50 years (1853-1912), and served as acting university president twice.[1][2] The Van Vleck Observatory (at Wesleyan University)[3] and the crater Van Vleck on the Moon are named after him. *Wik

1854 Sir (William) Napier Shaw (4 Mar 1854; 23 Mar 1945 at age 90) was an English meteorologist who applied his training in mathematics. He studied the upper atmosphere, using instruments carried by kites and high-altitude balloons. He measured (1906) the movement of air in two anti-cyclones, finding descent rates of 350 and 450 metres per day. He calculated the reduction in pressure due to a certain depression to correspond to the removal of two million million tons of air. He introduced the millibar unit for measurement of air pressure (1000 millibar = 1 bar = 1 standard atmosphere) and the tephigram to illustrate the temperature of a vertical profile of the atmosphere. He also co-authored an early work on atmospheric polluiton, The Smoke Problem of Great Cities (1925).*TIS

1862 Robert Emden (4 Mar 1862, 8 Oct 1940) Swiss astrophysicist and mathematician who wrote Gaskugeln (Gas Spheres, 1907), giving a mathematical model of stellar structure as the expansion and compression of gas spheres, wherein the forces of gravity and gas pressure are in equilibrium. He expanded on earlier work by J. H. Lane (1869) and A. Ritter (1878-83) who first derived equations describing stars as gaseous chemical, spherical bodies held together by their own gravity and obeying the known gas laws of thermodynamics. For four decades, the Lane-Emden equation was the foundation of theoretical work on the structure of stars: their central temperatures and pressures, masses, and equilibria. Emden also devised a hypothesis, no longer taken seriously, to explain sunspots. *TIS

1866 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

1881 Richard C(hace) Tolman (4 Mar 1881, 5 Sep 1948) was an American physicist and chemist who demonstrated that electrons are the charge-carrying entities in the flow of electricity, and also made a measurement of its mass. During the Manhattan Project of WW II, he was the chief scientific adviser to Brig. General Leslie Groves, the head of military affairs overseeing the development of the atomic bomb. After the war he was adviser to the U.S. representative to the United Nations Atomic Energy Commission. *TIS

1904 George Gamow (4 Mar 1904,19 Aug 1968) Russian-born American nuclear physicist, cosmologist and writer who was one of the foremost advocates of the big-bang theory, which desribes the origin of the universe as a colossal explosion that took place billions of years ago. In 1954, he expanded his interests into biochemistry and his work on deoxyribonucleic acid (DNA) made a basic contribution to modern genetic theory. *TIS

1914 Robert Rathbun Wilson (4 Mar 1914, 16 Jan 2000) was an American physicist who was the first director of Fermilab. From 1967, he led the design and construction of Fermilab (the Fermi National Accelerator Laboratory) near Chicago, Illinois. He also improved the environment by restoring prairie at the site. It began operating in 1972 with the world's most powerful particle accelerator. With later improvements, it retained that status for well over three decades until it was superceded by the LHC (Large Hadron Collider) at the CERN laboratory in Geneva, Switzerland. Wilson is remembered for his justification of the needed financing at a Senate hearing in 1969, where he said “It has nothing to do with defending our country, except to make it worth defending.” He resigned in 1978 because he did not believe the government was giving it sufficient funding for its research mission.*TIS The stately 16-story Robert Rathbun Wilson Hall rises above the surrounding Illinois countryside. Inspired by a Gothic cathedral in Beauvais, France, its twin towers are joined by crossovers beginning at the seventh floor. Spent a wonderful week there one summer in pursuit of knowledge in non-linear dynamics.

1923 Patrick (Alfred Caldwell) Moore, (4 Mar 1923, )English amateur astronomer, writer and broadcaster. He was educated at home due to childhood illness, from which time he acquired his interest in observational astronomy. Moore is best known as the enthusiastic and knowledgeable presenter of the BBC TV program The Sky at Night, which he began in 1957. With a half-century of broadcasts, this is the world's longest-running television series, and it remains so with the original presenter. Moore has written over 60 books, including The Amateur Astronomer (1970), The A-Z of Astronomy (1986), and Mission to the Planets (1990). As an accomplished xylophone player, his interest in astronomy also shows in the title of one of his musical compositions: Perseus and Andromeda (1975)*TIS

DEATHS

1816 Josef (also José or Joseph) de Mendoza y Ríos (29 January 1761; Sevilla, Spain - 4 March 1816 Brighton, England) was a Spanish astronomer and mathematician of the 18th century, famous for his work on navigation. The first work of Mendoza y Ríos was published in 1787: his treatise, Tratado de Navegación, about the science and technique of navigation in two tomes. He also published several tables for facilitating the calculations of nautical astronomy and useful in navigation to calculate the latitude of a ship at sea from two altitudes of the sun, and the longitude from the distances of the moon from a celestial body.
In the field of the nautical instruments, he improved the reflecting circle.
In 1816, he was elected a foreign member of the Royal Swedish Academy of Sciences. @Wik

1910 Knut Johan Angstrom (12 Jan 1857; 4 Mar 1910) Swedish physicist, son of Anders Angstrom, who invented an electric compensation pyrheliometer and other devices for infra-red photography. With these, he studied the sun's heat radiation*TIS

1915 William Willett (10 Aug 1856, 4 Mar 1915 at age 58)English builder who invented Daylight Saving Time. He claimed he had the idea while taking an early summer morning ride in Petts Wood near to his home in Chislehurst, London. He observed that many blinds were still down, although there was already good daylight, yet many made no use of it. He used his wealth as a prominent home builder to campaign for a scheme of adjusting clocks with the season and published a pamphlet in 1907. His original idea was to make four weekly changes of 20-mins each, for a total of 80-mins. The first Daylight Saving Bill, proposing a single one hour at the change of season failed in 1908. After his death, the idea was adopted during WW I for wartime fuel savings. A memorial was erected in Petts Wood.*TIS A sun dial Memorial was erected in the Petts Wood in his honor.*TIS

1976 Walter Schottky (23 Jul 1886, 4 Mar 1976 at age 89)Swiss-born German physicist whose research in solid-state physics led to development of a number of electronic devices. He discovered the Schottky effect, an irregularity in the emission of thermions in a vacuum tube and invented the screen-grid tetrode tube (1915). The Schottky diode is a high speed diode with very little junction capacitance (also known as a "hot-carrier diode" or a "surface-barrier diode.") It uses a metal-semiconductor junction as a Schottky barrier, rather than the semiconductor-semiconductor junction of a conventional diode. *TIS

1997 Robert Henry Dicke (6 May 1916 St. Louis, Missouri, USA - 4 Mar 1997 at age 80) American physicist who worked in such wide-ranging fields as microwave physics, cosmology, and relativity. As an inspired theorist and a successful experimentalist, his unifying theme was the application of powerful and scrupulously controlled experimental methods to issues that really matter. He also made a number of significant contributions to radar technology and to the field of atomic physics. His visualization of an oscillating universe stimulated the discovery of the cosmic microwave background, the most direct evidence that our universe really did expand from a dense state. A key instrument in measurements of this fossil of the Big Bang is the microwave radiometer he invented. His patents ranged from clothes dryers to lasers. *TIS

2000 Hermann Alexander Brück (15 August 1905 in Berlin, Germany – 4 March 2000 in Edinburgh, Scotland) was a German-born astronomer who spent the great portion of his career in the United Kingdom.
Upon graduation from Munich, Brück followed his friend Albrecht Unsöld to the Potsdam Astrophysical Observatory; Unsöld had earned his doctorate the year before, also under Sommerfeld. While there, he participated in the physics colloquium at the Humboldt University of Berlin with the physicists Max von Laue and Albert Einstein and the astronomer Walter Grotrian. With growing difficulties under National Socialism, Brück left Germany in 1936 to take a temporary research assistantship at the Vatican Observatory. In 1937 he moved to the University of Cambridge to join the circle of the modern astrophysicists around Arthur Eddington. In time, Brück became Assistant Director of the Observatories and John Couch Adams, specializing in solar spectroscopy. He taught a course in classical astronomy and started the student astronomical society, which fostered the careers of many astronomers.
In 1947, at the invitation of Éamon de Valera, Brück moved to Dublin to direct the Dunsink Observatory, which was part of the Dublin Institute for Advanced Studies, where he associated with Erwin Schrödinger. In 1950, the Observatory, along with the Royal Irish Academy, hosted the first meeting of the Royal Astronomical Society. In 1955, the International Astronomical Union held their triennial Assembly in Dublin. At this gathering, the Observatory demonstrated photoelectric equipment for photometry, which had been developed by M. J. Smyth, who had been Brück’s student in Cambridge. Also displayed was the UV solar spectroscopy which extended the Utrecht Atlas and formed part of the revised Rowland tables of the Solar spectrum; Brück’s wife, Dr. Mary Brück (née Conway), was a leading figure in this work.
In 1957, Brück moved to the University of Edinburgh. With his vision and drive, he transformed the Royal Observatory into an internationally-ranked center of research. He put together a team of astronomers and engineers headed initially by P. B. Fellgett and later by V. C. Reddish *Wik

2011 Simon van der Meer (24 Nov 1925, 4 March, 2011)Dutch engineer and physicist who along with Italian physicist Carlo Rubbia, discovered the W particle and the Z particle by colliding protons and antiprotons, for which both men shared the Nobel Prize for Physics. These subatomic particles (units of matter smaller than an atom) transmit the weak nuclear force, one of four fundamental forces in nature. The discovery supported the unified electroweak theory put forward in the 1970's. Working at CERN in Switzerland, Van der Meer improved the design of particle accelerators used produce collisions between beams of subatomic particles. He invented a device that would monitor and adjust the particle beam with correcting magnetic fields by a system of 'kickers' placed around the accelerator ring.*TIS

Tuesday, 3 March 2020

One more Preview for the Etymological and Historical Math Dictionary - Graph

Graph has come to have multiple meanings in mathematics, but for most students it relates to the graph of functions on the coordinate axes. The origin is from the Greek graphon, to write, perhaps with earlier references to carving or scratching. Jeff Miller's web site suggests that the use of graph as a verb may have first been introduced as late as 1898.
In a post to a history newsgroup, Karen Dee Michalowicz commented on the history of graphing:

It is interesting to note that the coordinate geometry that Descartes introduced in the 1600's did not appear in textbooks in the context of graphing equations until much later. In fact, I find it appearing in the mid 1800's in my old college texts in analytic geometry. It isn't until the first decade of the 20th century that graphing appears in standard high school algebra texts. [This matches rise of graph paper in the same periods]. Graphing is most often found in books by Wentworth. Even so, the texts written in the 20th century, perhaps until the 1960's, did not all have graphing. Taking Algebra I in the middle 1950's, I did not learn to graph until I took Algebra II

See my Notes on the History of Graph Paper here

Math historian Bea Lumpkin has written about the early use of graphs by the Egyptians in what was an early use of what painters call the grid method:

In my article ... I suggest, "It is possible that the concept of coordinates grew out of the Egyptian use of square grids to copy or enlarge artwork, square by square. It needs just one short, important step from the use of square grids to the location of points by coordinates.

In the same posting she comments on the finding of graphs in Egyptian finds dating back to 2700 BC:

"An architect's diagram of great importance has lately been found by the Department of Antiquities at Saqqara. It is a limestone flake, apparently complete, measuring about 5 x 7 x 2 inches, inscribed on one face in red ink, and probably belongs to the IIIrd dynasty" Here is the reason that Clark and Engelbach attached great importance to the diagram. It shows a curve with vertical line segments labeled with coordinates that give the height of points on the curve that are equally spaced horizontally. The vertical coordinates are given in cubits, palms and fingers. The horizontal spacing, the authors write "... most probably that is to be understood as one cubit, an implied unit elsewhere." To clinch their analysis, Clarke and Engelbach observe: "This ostrakon was found near the remains of a solid saddle-backed construction, the top of which, as far as could be ascertained from its half-destroyed condition, closely approximated tot he curve obtained from the data on the ostrakon.

This certainly lays claim to the oldest line graph I have ever heard.