## Saturday, 6 February 2016

### On This Day in Math - February 6

 Newton Statue - Trinity Chapel, Cambridge UK
The feeling of it (pure oxygen) to my lungs was not sensibly different from that of common air; but I fancied that my breast felt peculiarly light and easy for some time afterwards. Who can tell but that, in time, this pure air may become a fashionable article in luxury. Hitherto only two mice and myself have had the privilege of breathing it.~Joseph Priestley

The 37th day of the year; 37 is the only prime with a three digit period for the decimal expansion of its reciprocal, 1/37 = .027027....
Big Prime:::

n = integer whose digits are (left to right) 6424 copies of 37, followed by units digit of 3, is prime (n = 3737...373 has 12849 digits) *Republic of Math ‏@republicofmath

An amazing reversal: 37 is the 12th prime & 73 is the 21st prime . This enigma is the only known combination.

If you use multiplication and division operations to combine Fibonacci numbers, (for example, 4 = 2^2, 6 = 2·3, 7 = 21/ 3 ,...) you can make almost any other number. Almost, but you can't make 37.  In fact, there are 12 numbers less than 100 that can not be expressed as "Fibonacci Integers" *Carl Pomerance, et al.

EVENTS
1672 Newton wrote Henry Oldenburg about his optical theories, (including the phrase, "because that Light is a heterogenous mixture of differently refrangible rays." and Oldenburg published them a few days later in the
Philosophical Transactions. The controversy that followed dissuaded Newton from publishing on optics—and also on the calculus—until 1704 *ISIS, 69, p 134 (*VFR)
But it is requisite, that the prism and lens be placed steady, and that the paper, on which the colours are cast be moved to and fro; for, by such motion, you will not only find, at what distance the whiteness is most perfect but also see, how the colours gradually convene, and vanish into whiteness, and afterwards having crossed one another in that place where they compound whiteness, are again dissipated and severed, and in an inverted order retain the same colours, which they had before they entered the composition. You may also see, that, if any of the colours at the lens be intercepted, the whiteness will be changed into the other colours. And therefore, that the composition of whiteness be perfect, care must be taken, that none of the colours fall besides the lens.
Some of his opponents denied the truth of his experiments, refusing to believe in the existence of the spectrum. Others criticized the experiments, saying that the length of the spectrum was never more than three and a half times the breadth, whereas Newton found it to be five times the breadth. It appears that Newton made the mistake of supposing that all prisms would give a spectrum of exactly the same length;

1766 Just a few months before he returns to St. Petersburg, Euler reads his paper (E401) “A New Method for Comparing the Observation of the Moon to Theory” to the Berlin Academy. The paper proposes numerical techniques for approximating a bodys velocity and acceleration. Sandifer suggests that the paper had great influence on LaGrange’s foundational program for the Calculus. *Ed Sandifer, How Euler Did It, MAA

1828 George Biddell Airy appointed Plumian professor of astronomy at Cambridge at a salary of £500 per annum. He was appointed even after he raised a row that the previous salary of £300 was inadequate. For the previous two years he held the Lucasian professorship—the post Newton held—at a salary of £99. *VFR

1930 Kurt G¨odel received his Ph.D. from the University of Vienna for a dissertation, directed by Hans Hahn, that showed the completeness of ﬁrst order logic (every valid ﬁrst-order formula is provable). *VFR

1959 Kilby Files Patent For Integrated Circuit.
Jack Kilby of Texas Instruments files a patent application called "miniaturized electronic circuits" for his work on a multi-transistor device. The patent was only one of 60 that Kilby holds. While Kilby has the earliest patent on the "integrated circuit," it was Robert Noyce, later co-founder of Intel, whose parallel work resulted in a practical device. Kilby's device had several transistors connected by flying wires while Noyce devised the idea of interconnection via a layer of metal conductors. Noyce also adapted Jean Hoerni's planar technique for making transistors to the manufacture of more complex circuits. *CHM
 Two drawings from Kilby's first IC patent *haverford.edu

BIRTHS
1465 Scipione del Ferro (6 February 1465 – 5 November 1526) born in Bologna, Italy. Around 1515 he solved the cubic equations x3+px = q and x3= px + q when p and q are positive. His methods are unknown. This information was passed on to his son-in-law Annibale dalla Nave who was tricked into revealing it to Cardano, who published it in his Ars magna of 1545.*VFR
There are no surviving scripts from del Ferro. This is in large part due to his resistance to communicating his works. Instead of publishing his ideas, he would only show them to a small, select group of friends and students. It is suspected that this is due to the practice of mathematicians at the time of publicly challenging one another. When a mathematician accepted another's challenge, each mathematician needed to solve the other's problems. The loser in a challenge often lost funding or his university position. Del Ferro was fearful of being challenged and likely kept his greatest work secret so that he could use it to defend himself in the event of a challenge.
Despite this secrecy, he had a notebook where he recorded all his important discoveries. After his death in 1526, this notebook was inherited by his son-in-law Hannival Nave, who was married to del Ferro's daughter, Filippa. Nave was also a mathematician and a former student of del Ferro's, and he replaced del Ferro at the University of Bologna after his death. In 1543, Gerolamo Cardano and Ludovico Ferrari (one of Cardano's students) travelled to Bologna to meet Nave and learn about his late father-in-law's notebook, where the solution to the depressed cubic equation appeared.
Del Ferro also made other important contributions to the rationalization of fractions with denominators containing sums of cube roots.
He also investigated geometry problems with a compass set at a fixed angle, but little is known about his work in this area. *Wik (Teachers may need to explain to students how suppression of the squared term allows this to solve general cubics.)

1695 Nikolaus II Bernoulli (February 6, 1695, Basel, Switzerland – July 31, 1726, St. Petersburg, Russia) was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. Nicolaus worked mostly on curves, differential equations, and probability. He was a contemporary of Leonhard Euler. He also contributed to fluid dynamics.*Wik He was the oldest and favorite of three sons of Johann Bernoulli. He made important mathematical contributions to the problem of trajectories while working on the mathematical arguments behind the dispute between Newton and Leibniz.*SAU When the father was asked to come to St. Petersburg to join the Academy, he declined because of his age. He suggested that they take his son Nikolaus, but, so that he not be lonely, they should also take another son Daniel. Unfortunately, Nikolaus II drowned in 1726, only eight months after going to St. Petersburg. His professorship was succeeded in 1727 by Leonhard Euler, whom the Bernoulli brothers had recommended.

1802 Sir Charles Wheatstone, (6 Feb 1802, 19 Oct 1875) English physicist who popularized the Wheatstone bridge, a device that accurately measured electrical resistance and became widely used in laboratories. He didn't actually invent the "Wheatstone Bridge". His contemporary, Samuel Hunter Christie, came up with the idea of the bridge circuit, but Wheatstone set the precedent for using it in the way in which it has been most commonly used. Over time, the device became associated with him and took on his name. He did, however, invent the concertina (1829), the stereoscope (1838), and an early form of the telegraph. He also developed a chronoscope (1842) to determine the velocity of projectiles at an English gunnery.*TIS (For students of discrete math, or interested in codes, Wheatstone was also the creator of the Playfair Cipher.) {Wheatstone's work was so diverse that after a lecture at the Science Conference in South Kensington (London) by Prof. W. G. Adams on Wheatstone's acoustical discoveries, William Spottiswoode commented, "It must have struck all those in science... that when they fancied they had found something new, they find it was done by Sir Charles Wheatstone years ago." *Knowledge and Scientific News, Jan 1908, pg 7

1848 Adam Wilhelm Siegmund Günther (6 Feb 1848 in Nuremberg, Germany - 3 Feb 1923 in Munich, Germany) Günther's contributions to mathematics include a treatise on the theory of determinants (1875), hyperbolic functions (1881), and the parabolic logarithm and parabolic trigonometry (1882). He also wrote numerous books and journal articles [which] encompass both pure mathematics and its history and physics physics, geophysics, meteorology, geography, and astronomy. The individual works on the history of science, worth reading even today, bear witness to a thorough study, a remarkable knowledge of the relevant secondary literature, and a superior descriptive ability. *SAU

1916 John Crank (6 February 1916 – 3 October 2006) was a mathematical physicist, best known for his work on the numerical solution of partial differential equations.
He worked on ballistics during the Second World War, and was then a mathematical physicist at Courtaulds Fundamental Research Laboratory from 1945 to 1957. In 1957, he was appointed as the first Head of Department of Mathematics at Brunel College in Acton. He served two terms of office as Vice-Principal of Brunel before his retirement in 1981, when he was granted the title of Professor Emeritus.
Crank's main work was on the numerical solution of partial differential equations and, in particular, the solution of heat-conduction problems. He is best known for his work with Phyllis Nicolson on the heat equation, which resulted in the Crank–Nicolson method.*Wik

DEATHS
1612 Christopher Clavius (March 25, 1538 – February 6, 1612 {some sources give Feb 12 for the date of death}), the Euclid of the sixteenth-century, born in the German town of Bamberg, the see of the prince-bishop of Franconia. He was also the leader of the Gregorian calendar reform. Perhaps his greatest contribution was as an educational reformer. *Renaissance Mathematicus He was a German Jesuit mathematician and astronomer who was the main architect of the modern Gregorian calendar. In his last years he was probably the most respected astronomer in Europe and his textbooks were used for astronomical education for over fifty years in Europe and even in more remote lands (on account of being used by missionaries). As an astronomer Clavius held strictly to the geocentric model of the solar system, in which all the heavens rotate about the Earth. Though he opposed the heliocentric model of Copernicus, he recognized problems with the orthodox model. He was treated with great respect by Galileo, who visited him in 1611 and discussed the new observations being made with the telescope; Clavius had by that time accepted the new discoveries as genuine, though he retained doubts about the reality of the mountains on the Moon. Later, a large crater on the Moon was named in his honour. *Wik

1804 Joseph Priestley (13 Mar 1733, 6 Feb 1804) English chemist, clergyman and political theorist who discovered the element oxygen. His early scientific interest was electricity, but he is remembered for his later work in chemistry, especially gases. He investigated the "fixed air" (carbon dioxide) found in a layer above the liquid in beer brewery fermentation vats. Although known by different names at the time, he also discovered sulphur dioxide, ammonia, nitrogen oxides, carbon monoxide and silicon fluoride. Priestley is remembered for his invention of a way of making soda-water (1772), the pneumatic trough, and recognizing that green plants in light released oxygen. His political opinions and support of the French Revolution, were unpopular. After his home and laboratory were set afire (1791), he sailed for America, arriving at New York on 4 Jun 1794 *TIS He died on the morning of 6 February 1804 and was buried at Riverview Cemetery in Northumberland, Pennsylvania.

Return unto thy rest, O my soul, for the
Lord hath dealt bountifully with thee.
I will lay me down in peace and sleep till
I awake in the morning of the resurrection. *Wik

 *cometography.com
1923 Edward Emerson Barnard (16 Dec 1857; 6 Feb 1923) astronomer who pioneered in celestial photography, specializing in wide-field photography. From the time he began observing in 1881, his skill and keen eyesight combined to make him one of the greatest observers. Barnard came to prominence as an astronomer through the discovery of numerous comets. In the 1880s, a patron of astronomy in Rochester, N.Y. awarded \$200 per new comet was found. Barnard discovered eight - enough to build a "comet house" for his bride. At Lick Observatory (1888-95) he made the first photographic discovery of a comet; photographed the Milky Way; and discovered the fifth moon of Jupiter. Then he joined Yerkes Observatory, making his Photographic Atlas of Selected Regions of the Milky Way.*TIS The faint Barnard's Star is named for Edward Barnard after he discovered in 1916 that it had a very large proper motion, relative to other stars. This is the second nearest star system to the Sun, second only to the Alpha Centauri system. *Wik

1965 Ernst Erich Jacobsthal (16 October 1882, Berlin – 6 February 1965, Überlingen) was a German mathematician, and brother to the archaeologist Paul Jacobsthal.
In 1906, he earned his PhD at the University of Berlin, where he was a student of Georg Frobenius, Hermann Schwarz and Issai Schur; his dissertation, Anwendung einer Formel aus der Theorie der quadratischen Reste (Application of a Formula from the Theory of Quadratic Remainders), provided a proof that prime numbers of the form 4n + 1 are the sum of two square numbers. *Wik The theory was first conjectured by Fermat and proved by Euler.

1973 Ira Sprague Bowen (21 Dec 1898; 6 Feb 1973) was an American astrophysicist. His investigation of the ultraviolet spectra of highly ionized atoms led to his explanation of the unidentified strong green spectral lines of gaseous nebulae (clouds of rarefied gas) as forbidden lines of ionized oxygen and nitrogen. This emission, appearing to match no known element, had formerly been suggested to be due to a hypothetical element, "nebulium." Bowen was able to show, that in reality, the emission lines exactly matched those calculated to be the "forbidden lines" of ionized oxygen and nitrogen under extremely low pressure. This made a major advance in the knowledge of celestial composition. He was director of the Mt. Wilson and Palomar Observatories from 1948-64.*TIS

1992 Caius Jacob (29 March 1912 , Arad - 6 February 1992 , Bucharest ) was a Romanian mathematician and member of the Romanian Academy. He made ​​contributions in the fields of fluid mechanics and mathematical analysis , in particular vigilance in plane movements of incompressible fluids, speeds of movement at subsonic and supersonic , approximate solutions in gas dynamics and the old problem of potential theory. His most important publishing was Mathematical introduction to the mechanics of fluids. *Wik

Credits :
*CHM=Computer History Museum
*FFF=Kane, Famous First Facts
*NSEC= NASA Solar Eclipse Calendar
*RMAT= The Renaissance Mathematicus, Thony Christie
*SAU=St Andrews Univ. Math History
*TIA = Today in Astronomy
*TIS= Today in Science History
*VFR = V Frederick Rickey, USMA
*Wik = Wikipedia
*WM = Women of Mathematics, Grinstein & Campbell

## Friday, 5 February 2016

### On This Day in Math - February 5

 See Events:1897

It is a mathematical fact that the casting of this pebble from my hand alters the center of gravity of the universe.
~Thomas Carlyle

The 36th day of the year; 36 is the smallest non trivial number which is both triangular and square. What's the next?
36 is the sum of the first three cubes, $1 ^3 + 2^3 + 3^3 = 36$  The sums of the first n cubes is always a square number. $\sum_{k=1}^n k^3 = (\frac{(n)(n+1)}{2}) ^2$ Note that this sequence and its formula were known to (and possibly discovered by) Nicomachus, 100 CE)

The sum of the first 36 integers, $\sum_{k=1}^{36} k = 666$ the so called "number of the beast."

And Mario Livio pointed out in a tweet that this is 5/2 in European style dating, and 52 is the maximum number of moves needed to solve the "15" sliding puzzle from any solvable position.

A special historical tribute to 36: The thirty-six officers problem is a mathematical puzzle proposed by Leonhard Euler in 1782. He asked if it were possible to place officers of six ranks from each of six regiments in a 6x6 square so that no row or column would have an officer of the same rank, or the same regiment. Euler suspected that it could not be done. Euler knew how to construct such squares for nxn when n was odd, or a multiple of four, and he believed that all such squares with n = 4m+2 (6, 10, 14...) were impossible ( Euler didn't say it couldn't be done. He just said that his method does not work for numbers of that form.) Proof that he was right for n=6 took a while. French mathematician (and obviously a very patient man) Gaston Tarry proved it in 1901 by the method of exhaustion. He wrote out each of the 9408 6x6 squares and found that none of them worked. Then in 1959, R.C. Bose and S. S. Shrikhande proved that all the others could be constructed. So the thirty-six square is the only one that can't be done.

EVENTS
1575 Jan De Groot entered the University of Leiden, in the Netherlands, on its opening day. With Simon Stevin he later performed an experiment proving that bodies of different weights fall the same distance in the same time (published 1586 by Stevin). This anti-Aristotelian experiment anticipated Galileo’s famous, but apocryphal, experiment at the Leaning Tower of Pisa. His son Hugo De Groot was a famous jurist. *VFR Thony Christie pointed out that "The anti-Aristotle tower and ball experiment was first carried out by Johannes Philiponus in 6th century CE". Philiponus proposed a kinetic theory for motion in place of Aristotle's impetus.

1673 Robert Hooke writes in his journal that he had, "Told the Society of Arithmetick engine.‏*@HookesLondon It is said that Newton had this, and other Hooke items, including Hooke's portrait, removed from the Royal Society after Hooke's death but this does not seem to be supported by most math historians

5 Feb 1675 (OS) 15 Feb 1676(NS) Newton wrote Hooke: "What DesCartes did was a good step....If I have seen further it is by standing on ye sholders of Giants." *VFR
The letter is at the Historical Society of Pennsylvania.

1689 The Convention Parliament, with Cambridge U. MP Isaac Newton voting in the majority, declared the throne of England vacant after James II escaped to France with the permission of his Son-in-Law and daughter, William and Mary, who were offered the crown jointly. The only record of a comment by Newton during the Parliament except to ask for a servant to close a drafty window. *Thomas Levenson, Newton and The Counterfeiter.

1772 Laplace presented his ﬁrst probability memoir to the Acad´emie des Sciences. *VFR

1796 Schiller (1759–1815) wrote to Goethe (1749–1832): “Wo es die Sache leidet, halte ich es immer f¨ur besser, nicht mit dem Anfang anzufangen, der immer das Schwerste ist.” (I always think it better, whenever possible, not to begin at the beginning, as it is always the most difficult part). Although this is advice from one poet to another, it seems to apply to mathematics, especially the foundations of mathematics. Quoted from Numbers (1990) by H.-D. Ebinghaus et al., p. 6. *VFR

1835 A ceremony to honor "The Genius and Discoveries of Sir Isaac Newton" was organized by the citizens of the Lincolnshire, his area of birth, a few years after the centennial of his death. By unanimous choice, the committee selected as the speaker, the 19 yr old George Boole. "All present were struck by the youthful age of the speaker and not a little amazed by both his knowledge of the subject and his confident lecturing style."
*Desmond MacHale, The Life and Work of George Boole

1840 The American Statistical Association held its ﬁrst annual meeting, in Boston. "On November 27, 1839, five men held a meeting in the rooms of the American Education Society at No. 15 Cornhill in Boston, Massachusetts, to organize a statistical society. Its purpose, as stated in the society's first constitution, was to "collect, preserve, and diffuse statistical information in the different departments of human knowledge." Originally called the American Statistical Society, the organization's name was changed to the American Statistical Association (ASA) at its first annual meeting, held in Boston on February 5, 1840. " *Robert L. Mason, ASA: The First 160 Years

1843 The great comet of 1843, A night-time view showing an eyewitness account of the Great Comet of 1843, painted by the astronomer Charles Piazzi Smyth. The earliest observation occurred on the evening of 5 of February, 1843 and Smyth recorded its appearance at the Royal Observatory, Cape of Good Hope, South Africa between 3 and 6 of March. When at its greatest brilliance, it was visible only from southern latitudes. The view in the painting is probably taken from the Observatory. It shows Table Bay with Table Mountain visible in the background on the left. A large sailing ship sits in the foreground on the right, with other shipping in the distance. One of the great British astronomers, Smyth was 42 years Astronomer Royal for Scotland. *Royal Museums Greenwich

1850 D. D. Parmalee issued a patent (US Patent # 7074) for the ﬁrst key-driven adding machine. *VFR
While this was the first US patent, an earlier key-driven machine had been patented "as early as 1844 by Jean-Baptiste Schwilgue´ (1776– 1856), together with his son Charles. Jean-Baptiste Schwilgue´ was the architect of Strasbourg’s third astronomical clock during the years 1838–1843. He was trained as a clockmaker,but also became professor of mathematics,weights and measures controller, and an industry man, whose particular focus was on improving scales." *Denis Roegel, An Early (1844) Key-Driven Adding Machine, IEEE Annals of the History of Computing, Volume 30, Number 1, January-March 2008, pp. 59-65

In 1897, the Indiana State House legislature presented Bill No.246 which in effect gave 3.2 exactly as the value of pi. It stated, in part, "the ratio of the diameter and circumference [pi] is as five-fourths to four." That is (4 divided by 5/4) = 16/5 = 3.2 exactly. It was introduced by Representative Taylor I. Record, a farmer and lumber merchant, on behalf of a mathematical hobbyist, Dr. Edwin J. Goodwin, M.D. Neither they, nor the House politicians, understood it was mathematically incorrect. That was shortly recognized by Clarence A. Waldo, mathematics professor at Purdue University, who advised the Indiana Senators. They indefinitely postponed the bill on 12 Feb 1897. Pi is, in fact, an irrational number, approx. 3.141592.*TIS (more detail here)

1924 The Royal Greenwich Observatory begins broadcasting the time "pips" on BBC, a series of six short tones broadcast at one-second intervals by many BBC Radio stations. The pips were introduced in 1924 and have been generated by the BBC since 1990. The pips were the idea of the Astronomer Royal, Sir Frank Watson Dyson, and the head of the BBC, John Reith.*Wik
This eight-day wall-mounted astronomical regulator by Edward John Dent & Co was originally made for use in observing the Transit of Venus in 1874. In 1923 it was adapted as the primary standard for the new six-pip time signal. The clock sent electrical impulses down a telephone wire to the BBC for conversion into audio pips for radio broadcasts. It has a zinc tube temperature-compensated pendulum and was corrected from 1929 by the Shortt master clock number 16. The three sets of contacts for closing the six-pip circuit every quarter of an hour can be seen in two of the holes within the seconds dial, and halfway down the pendulum, operated by a roller. This clock was in service for the BBC signal at the Observatory from 1924 to 1949, when it was superseded by a quartz clock. *Royal Observatory Greenwich

1958 Kilby Files a Patent for the Integrated Circuit. Jack Kilby of Texas Instruments files a patent application called miniaturized electronic circuits for his work on a multi-transistor device. The patent was only one of 60 that Kilby holds. While Kilby has the earliest patent on the integrated circuit, it was Robert Noyce, later co-founder of Intel, whose parallel work resulted in a practical device. Kilby's device had several transistors connected by flying wires while Noyce devised the idea of interconnection via a layer of metal conductors. Noyce also adapted Jean Hoerni's planar technique for making transistors to the manufacture of more complex circuits. *CHM

 *Wik
In 1962, the Sun, the Moon, and the five naked-eye visible planets - Mercury, Venus, Mars, Jupiter, and Saturn - were in conjunction. Though not in a straight line along their orbital paths, as viewed in the sky, they were within 16 degrees of each other (meaning all appeared within a circle just 16 º across). This conjunction coincided with a total solar eclipse, which made viewing Mercury, Venus, Mars, Jupiter, and Saturn possible for a brief period of time from a small stretch of Earth where the eclipse's shadow hit. The five naked-eye visible planets cluster together in the sky within a circle 25 degrees or less in diameter once every 57 years, on average. The next time in the 21st century that this will happen is 8 Sep 2040. *TIS (image...In May of 2011 a planetary conjunction of Mercury, Venus, Mars and Jupiter appeared very close to each other in the sky.) And for St. Valentines day this year (2012) I have ordered up a conjunction with Mercury and Neptune less than 1.5 o apart for my beautiful Jeannie, but the rest of you may enjoy it as well.

2040 The near-Earth asteroid 2011 AG5 currently has an impact probability of 1 in 625 for Feb. 5, 2040, according to Donald Yeomans, head of the Near-Earth Object Observations Program at NASA’s Jet Propulsion Laboratory in Pasadena, California.

BIRTHS
1608 Caspar Schott SJ, and Gaspar Schott or Kaspar Schott (February 5 1608 in Königshofen, May 22 1666 in Würzburg) was a scientific author and educator.
Schott attended the Würzburg Jesuit High School and entered the Order in 1627. During his studies in Würzburg one of his teachers was Athanasius Kircher. When the Jesuits fled before the approaching Swedish army in 1631,Schott went to Palermo to complete his studies. He stayed in Sicily 20 years as a teacher of mathematics, philosophy, moral theology at the Jesuit school in Palermo. In 1652 was sent to Rome as support in the scientific work of Kircher. He decided to publish Kircher's work. In 1655, he returned as Professor in the Würzburg school, where he taught mathematics and physics. He was Hofmathematker and confessor of the Elector Johann Philipp von Schönborn who had just purchased the vacuum pump invented by Otto von Guericke and used at Magdeburg.
He corresponded with leading scientists including Otto von Guericke, Christiaan Huygens, and Robert Boyle. The term "technology" was probably invented by Schott in his "Technica curiosa" which inspired Boyle and Hooke's vacuum experiments.
In the posthumously published work Organum mathematicum he describes his Cistula invented by him, a computing device with which you can multiply and divide. *Wik

1797 Jean-Marie-Constant Duhamel (5 Feb 1797; 29 Apr 1872) French mathematician and physicist who proposed a theory dealing with the transmission of heat in crystal structures based on the work of the French mathematicians Jean-Baptiste-Joseph Fourier and Siméon-Denis Poisson. *TIS

1836 Alexander Stewart Herschel (5 February 1836 – 18 June 1907) was a British astronomer, born in Feldhausen, South Africa.
He was the son of John Herschel and the grandson of William Herschel. Although much less well known than either of them, he did pioneering work in meteor spectroscopy. He also worked on identifying comets as the source of meteor showers. The Herschel graph, the smallest non-Hamiltonian polyhedral graph, is named after Herschel due to his pioneering work on Hamilton's Icosian game. *Wik
The image of the graph at right is from Christian Perfect at the Aperiodical Blog.  You can’t draw a path on it that visits each vertex exactly once, but you can make a polyhedron whose vertices and edges correspond with the graph exactly. It’s also bipartite – you can color the vertices using two colors so that edges only connect vertices of different colors.
I think the polyhedron is the only enneahedron (9 faces children) that has all quadrilateral faces. You can see the solid here.

1907 Wilhelm Magnus (February 5, 1907, Berlin, Germany – October 15, 1990, New York City) made important contributions in combinatorial group theory, Lie algebras, mathematical physics, elliptic functions, and the study of tessellations.*Wik

1915 Robert Hofstadter (5 Feb 1915, 17 Nov 1990) American scientist who was a joint recipient of the Nobel Prize for Physics in 1961 for his investigations in which he measured the sizes of the neutron and proton in the nuclei of atoms. He revealed the hitherto unknown structure of these particles and helped create an identifying order for subatomic particles. He also correctly predicted the existence of hte omega-meson and rho-meson. He also studied controlled nuclear fission. Hofstadter was one of the driving forces behind the creation of the Stanford Linear Accelerator. He also made substantial contributions to gamma ray spectroscopy, leading to the use of radioactive tracers to locate tumors and other disorders.*TIS

1930 Urbanik Kazimierz (born 5 February 1930 in Krzemieniec - 29 May 2005 in Wrocław ) - Polish mathematician, rector of the University of Wroclaw ( 1975 - 1981 ), Doctor Honoris Causa of the University of Lodz and the Technical University of Wroclaw. He dealt with problems from different fields of mathematics, but his research interests were focused on the theory of probability . He obtained several important results in the theory of stochastic processes , information theory , theoretical physics , universal algebra , topology and measure theory . He published about 180 scientific papers. *Wik

DEATHS
1881 Thomas Carlyle (4 Dec 1795 in Ecclefechan, Dumfriesshire, Scotland - 5 Feb 1881 in Chelsea, London, England) was a Scottish writer who was also interested in mathematics. He translated Legendre's work.*SAU

1939 Gheorghe Ţiţeica ((October 4, 1873–February 5, 1939) publishing as George or Georges Tzitzeica) was a Romanian mathematician with important contributions in geometry. He is recognized as the founder of the Romanian school of differential geometry.*Wik

1977 Oskar Benjamin Klein (September 15, 1894 – February 5, 1977) was a Swedish theoretical physicist. Klein retired as professor emeritus in 1962. He was awarded the Max Planck medal in 1959. He is credited for inventing the idea, part of Kaluza–Klein theory, that extra dimensions may be physically real but curled up and very small, an idea essential to string theory / M-theory. *Wik

1980 Nachman Aronszajn (26 July 1907, Warsaw, Poland – 5 February 1980 Corvallis, Oregon, U.S) was a Polish American mathematician of Ashkenazi Jewish descent. Aronszajn's main field of study and expertise was mathematical analysis. He also contributed to mathematical logic.
He received his Ph.D. from the University of Warsaw, in 1930, in Poland. Stefan Mazurkiewicz was his thesis advisor. He also received a Ph.D. from Paris University, in 1935; this time Maurice Fréchet was his thesis advisor. He joined the Oklahoma A&M faculty, but moved to the University of Kansas in 1951 with his colleague Ainsley Diamond after Diamond, a quaker, was fired for refusing to sign a newly-instituted loyalty oath. Aronszajn retired in 1977. He was a Summerfield Distinguished Scholar from 1964 to his death.
He introduced, together with Prom Panitchpakdi, the injective metric spaces under the name of "hyperconvex metric spaces". Together with Kennan T. Smith, Aronszajn offered proof of the Aronszajn–Smith theorem. Also, the existence of Aronszajn trees was proven by Aronszajn; Aronszajn lines, also named after him, are the lexicographic orderings of Aronszajn trees.
He also has a fundamental contribution to the theory of reproducing kernel Hilbert space, the Moore–Aronszajn theorem is named after him. *Wik

1988 Dorothy Lewis Bernstein (April 11, 1914 – February 5, 1988) was an American mathematician known for her work in applied mathematics, statistics, computer programming, and her research on the Laplace transform.
Dorothy Bernstein was born in Chicago, the daughter of Russian immigrants to the US. She was a member of the American Mathematical Society and the first woman elected president of the Mathematical Association of America. Due in great part to Bernstein's ability to get grants from the National Science Foundation, Goucher College (where she taught for decades) was the first women's university to use computers in mathematics instruction in the 1960s.*Wik

1997 Frederick Justin Almgren,(3 July 1933 in Birmingham, Alabama, USA - 5 Feb 1997 in Princeton, USA) Almost certainly Almgren's most impressive and important result was only published in 2000, three years after his death. Why was this? The paper was just too long to be accepted by any journal. Brian Cabell White explains the background in a review of the book published in 2000 containing the result:
By the early 1970s, geometric analysts had made spectacular discoveries about the regularity of mass-minimizing hypersurfaces. (Mass is area counting multiplicity, so that if k sheets of a surface overlap, the overlap region is counted k times.) In particular, the singular set of an m-dimensional mass-minimizing hypersurface was known to have dimension at most m - 7. By contrast, for an m-dimensional mass-minimizing surface of codimension greater than one, the singular set was not even known to have m-measure 0. Around 1974, Almgren started on what would become his most massive project, culminating ten years later in a three-volume, 1700-page preprint containing a proof that the singular set not only has m-dimensional measure 0, but in fact has dimension at most (m - 2). This dimension is optimal, since by an earlier result of H Federer there are examples for which the dimension of the singular set is exactly (m - 2). ...
Now, thanks to the efforts of editors Jean Taylor and Vladimir Scheffer, Almgren's three-volume, 1700-page typed preprint has been published as a single, attractively typeset volume of less than 1000 pages.*SAU

Credits :
*CHM=Computer History Museum
*FFF=Kane, Famous First Facts
*NSEC= NASA Solar Eclipse Calendar
*RMAT= The Renaissance Mathematicus, Thony Christie
*SAU=St Andrews Univ. Math History
*TIA = Today in Astronomy
*TIS= Today in Science History
*VFR = V Frederick Rickey, USMA
*Wik = Wikipedia
*WM = Women of Mathematics, Grinstein & Campbell

## Thursday, 4 February 2016

### On This Day in Math - February 4

 Archimedes *http://www.w-volk.de/museum

Technical skill is mastery of complexity while creativity is mastery of simplicity.
~ Sir Erik Christopher Zeeman

The 35th day of the year; There are 35 hexominos, the polyominoes made from 6 squares. *Number Gossip
(I only recently learned that, Although a complete set of 35 hexominoes has a total of 210 squares, which offers several possible rectangular configurations, it is not possible to pack the hexominoies into a rectangle.)

The longest open uncrossed (doesn't cross it's own path) knight's path on an 8x8 chessboard is 35 moves.   (longest cycle(end where you start) is only 32 moves)

In Base 35 (A=10, B=11, etc) NERD is Prime, $23*35^3+14*35^2+27*35+13 = 1,004,233$

EVENTS
1600 Johannes Kepler arrives at Benatek Castle near Prague, where Tycho Brahe had moved his observatory, and retinue, after his benefactor King Frederick drank himself to death. *Timothy Ferris, Coming of Age in the Milky Way

1703 46 of the 47 Ronin commit seppuku (ritual suicide) as recompense for avenging their master's death. . This I mention here because one of the 47, was the greatest Asian mathematician of his age, Shigekiyo Matsumura, who among other things, approximated the value of pi as 3.141592648, which is accurate to eight significant figures..."
More of the story here.

1751 Franklin electrocutes a turkey, opines culinary improvement:
My Respects to Mr. Watson. He desir’d you to enquire what Success we had in our Attempts to kill a Turkey by the Electrical Strokes. Please to acquaint him, that we made several Experiments on Fowls this Winter; That we found two large thin glass Jars, gilt (holding each about 6 Gallons, and taking 2000 Turns of a Globe of 9 Inches Diameter to charge them full, when the Globe works very well, and will charge a common half pint Vial with 50 Turns) were sufficient to kill common Hens outright; but the Turkies, tho’ thrown into violent Convulsions, and then lying as dead for some Minutes, would recover in less than a quarter of an Hour. However, having added Mr. Kinnersley’s Jarrs and mine together, in all 5, tho’ not fully charg’d, we kill’d a Turky with them of about 10 lb.wt. and suppose they would have kill’d a much larger. I conceit that the Birds kill’d in this Manner eat uncommonly tender.
*Franklinpapers

1753 Writing about his Experiments and Observations on Electricity made at Philadelphia in America, a work Diderot called the best example of the experimental art with which he was acquainted, Benjamin Franklin (in a letter to John Perkins) boasted that he had “not, with some of our learned moderns, disguised [his] nonsense in Greek, clothed it in algebra, or adorned it with ﬂuxions.” *Thomas L. Hankins, Jean d’Alembert, p 4 (via VFR) (This is in contrast to the quote by Lelande that d'Alembert "had never held a prism in his hand.")

1841 First recorded reference to "Groundhog Day" in America:
When German settlers arrived in the 1700s, they brought a tradition known as Candlemas Day, which has an early origin in the pagan celebration of Imbolc. It came at the mid-point between the Winter Solstice and the Spring Equinox. Superstition held that if the weather was fair, the second half of Winter would be stormy and cold. For the early Christians in Europe, it was the custom on Candlemas Day for clergy to bless candles and distribute them to the people in the dark of Winter. A lighted candle was placed in each window of the home. The day's weather continued to be important. If the sun came out February 2, halfway between Winter and Spring, it meant six more weeks of wintry weather.

The earliest American reference to Groundhog Day can be found at the Pennsylvania Dutch Folklore Center at Franklin and Marshall College:

February 4, 1841 - from Morgantown, Berks County (Pennsylvania) storekeeper James Morris' diary..."Last Tuesday, the 2nd, was Candlemas day, the day on which, according to the Germans, the Groundhog peeps out of his winter quarters and if he sees his shadow he pops back for another six weeks nap, but if the day be cloudy he remains out, as the weather is to be moderate."

According to the old English saying:
If Candlemas be fair and bright,
Winter has another flight.
If Candlemas brings clouds and rain,
Winter will not come again.
*Stormfax.com

In 1868, Charles Darwin began writing his book The Descent of Man and Selection in Relation to Sex. He was now 69 years old, working in his home in Downe, England. *TIS

1995 The Connect Four game was mathematically solved first by James D. Allen (Oct 1, 1988), and independently by Victor Allis (Oct 16, 1988). First player can force a win. Strongly solved by John Tromp's 8-ply database (Feb 4, 1995). Weakly solved for all boardsizes where width+height is at most 15 (Feb 18, 2006). *Wik

BIRTHS

1896 Friedrich (Hermann) Hund (4 Feb 1896 - 31 Mar 1997) was a German physicist known for his work on the electronic structure of atoms and molecules. He introduced a method of using molecular orbitals to determine the electronic structure of molecules and chemical bond formation. His empirical Hund's Rules (1925) for atomic spectra determine the lowest energy level for two electrons having the same n and l quantum numbers in a many-electron atom. The lowest energy state has the maximum multiplicity consistent with the Pauli exclusion principle. The lowest energy state has the maximum total electron orbital angular momentum quantum number, consistent with rule. They are explained by the quantum theory of atoms by calculations involving the repulsion between two electrons. *TIS

1906 Clyde William Tombaugh (4 Feb 1906 on Ranch near Streator, Illinois - 17 Jan 1997) was an American astronomer who discovered what was then recognized as the planet Pluto, which he photographed on 23 Jan 1930, 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. He also discovered several clusters of stars and galaxies, studied the apparent distribution of extragalactic nebulae, and made observations of the surfaces of Mars, Venus, Jupiter, Saturn, and the Moon.Born of poor farmers, his first telescope was made of parts from worn-out farming equipment. *TIS
From my personal blog after a visit to Mars Hill, Flagstaff, Az. (much material from Wikipedia)
In the late 19th and early 20th century, observers of Mars drew long straight lines that appeared on the surface between 60 degrees north and south of the martian equator. Italian astronomer Giovanni Schiaparelli called these lines canali, which became canals in English. Lowell extended this observation to a theory that Mars had polar ice caps that would melt in the martian spring and fill the canals. He even extended the theory to include intelligent life on Mars that had designed the canals.
Eventually it became clear that there were no martian canals, but Mars hill went on to be the sight where a self educated Kansas schoolboy found his dream of working in astronomy in 1929, when the observatory director, V M Slipher, "handed the job of locating Planet X to Clyde Tombaugh, a 23-year-old Kansas man who had just arrived at the Lowell Observatory after Slipher had been impressed by a sample of his astronomical drawings."
On the nights of Jan 23 and 30th of January, 1830, he found a planet in the images that he thought was the Planet X. "The discovery made front page news around the world. The Lowell Observatory, who had the right to name the new object, received over 1000 suggestions, from "Atlas" to "Zymal". Tombaugh urged Slipher to suggest a name for the new object quickly before someone else did. Name suggestions poured in from all over the world. Constance Lowell proposed Zeus, then Lowell, and finally her own first name. These suggestions were disregarded.
The name "Pluto" was proposed by Venetia Burney (later Venetia Phair), an eleven-year-old schoolgirl in Oxford, England. Venetia was interested in classical mythology as well as astronomy, and considered the name, one of the alternate names of Hades, the Greek god of the Underworld, appropriate for such a presumably dark and cold world. She suggested it in a conversation with her grandfather Falconer Madan, a former librarian of Oxford University's Bodleian Library. Madan passed the name to Professor Herbert Hall Turner, who then cabled it to colleagues in America. The object was officially named on March 24, 1930."
Among the many awards Tombaugh received was a scholarship to the Univ of Kansas, where he would eventually earn a Bachelors and Masters Degree. It is said that the Astronomy Dept head refused to allow him to take the introductory astronomy class because it would be undignified for the discoverer of a planet.
Tombaugh died on January 17, 1997, when he was in Las Cruces, New Mexico, at the age of 90. A small portion of his ashes were placed aboard the New Horizons spacecraft. The container includes the inscription: "Interned (sic) herein are remains of American Clyde W. Tombaugh, discoverer of Pluto and the solar system's 'third zone'. Adelle and Muron's boy, Patricia's husband, Annette and Alden's father, astronomer, teacher, punster, and friend: Clyde W. Tombaugh

1925 Sir Erik Christopher Zeeman FRS (born 4 February 1925), is a Japanese-born British mathematician known for his work in geometric topology and singularity theory. His main contributions to mathematics were in topology, particularly in knot theory, the piecewise linear category, and dynamical systems.
Zeeman is known among the wider scientific public for his contribution to, and spreading awareness of catastrophe theory, which was due initially to another topologist, René Thom, and for his Christmas lectures about mathematics on television in 1978. He was especially active encouraging the application of mathematics, and catastrophe theory in particular, to biology and behavioral sciences.*Wik

1926 Jaroslav Hájek (4 Feb 1926 in Podebrady, Bohemia (now Czech Republic) - 10 June 1974 in Prague, Czechoslovakia) He was among the pioneers of unequal probability sampling. The name "Hájek predictor" now labels his contributions to the use of auxiliary data in estimating population means. In 1967 Hájek published (jointly with Z Sidak) Theory of rank tests but it was a work which had in fact been written four years before in 1963. Their methods use three lemmas of Le Cam in order to treat rank statistics under local alternatives and they established the efficiency of rank tests. *SAU

1927 Rolf William Landauer (4 Feb 1927; 27 Apr 1999) German-born American physicist known for his formulation of Landauer's principle concerning the energy used during a computer's operation. Whenever the machine is resetting for another computation, bits are flushed from the computer's memory, and in that electronic operation, a certain amount of energy is lost. Thus, when information is erased, there is an inevitable "thermodynamic cost of forgetting," which governs the development of more energy-efficient computers. While engineers dealt with practical limitations of compacting ever more circuitry onto tiny chips, Landauer considered the theoretical limit, that if technology improved indefinitely, how soon will it run into the insuperable barriers set by nature?*TIS

1948 Ken Thompson Is Born.  Thompson, who with Dennis Ritchie, developed UNIX at AT&T Bell Laboratories, is born. The UNIX operating system combined many of the timesharing and file management features offered by Multics, from which it took its name. (Multics, a projects of the mid - 1960s, represented the first effort at creating a multi-user, multi-tasking operating system.) The UNIX operating system quickly secured a wide following, particularly among engineers and scientists. *CHM

DEATHS

1615 Giambattista della Porta (1 November 1535 Vico Equense (near Naples), Italy
- 4 February 1615 Naples, Italy) was an Italian scholar who worked on cryptography and also on optics. He claimed to be the inventor of the telescope although he does not appear to have constructed one before Galileo.
In 1563, della Porta published De Furtivis Literarum Notis, a work about cryptography. In it he described the first known digraphic substitution cipher. Charles J. Mendelsohn commented, "He was, in my opinion, the outstanding cryptographer of the Renaissance. Some unknown who worked in a hidden room behind closed doors may possibly have surpassed him in general grasp of the subject, but among those whose work can be studied he towers like a giant."
Della Porta invented a method which allowed him to write secret messages on the inside of eggs. During the Spanish Inquisition, some of his friends were imprisoned. At the gate of the prison, everything was checked except for eggs. Della Porta wrote messages on the egg shell using a mixture made of plant pigments and alum. The ink penetrated the egg shell which is semi-porous. When the egg shell was dry, he boiled the egg in hot water and the ink on the outside of the egg was washed away. When the recipient in prison peeled off the shell, the message was revealed once again on the egg white.

Della Porta was the founder of a scientific society called the Academia Secretorum Naturae (Accademia dei Segreti). This group was more commonly known as the Otiosi, (Men of Leisure). Founded sometime before 1580, the Otiosi were one of the first scientific societies in Europe and their aim was to study the "secrets of nature." Any person applying for membership had to demonstrate they had made a new discovery in the natural sciences.
His private museum was visited by travelers and was one of the earliest examples of natural history museums. It inspired the Jesuit Athanasius Kircher to begin a similar, even more renowned, collection in Rome.
*SAU *Wik

1774 Charles-Marie de La Condamine (27 Jan 1701, 4 Feb 1774) French naturalist and mathematician who became particularly interested in geodesy (earth measurement). He was put in charge by the King of France of an expedition to Equador to measure a meridional arc at the equator (1735-43). It was wished to determine whether the Earth was either flattened or elongated at its poles. He then accomplished the first scientific exploration of the Amazon River (1743) on a raft, studying the region, and brought the drug curare to Europe. He also worked on establishment of a universal unit of length, and is credited with developing the idea of vaccination against smallpox, later perfected by Edward Jenner. However, he was almost constantly ill and died in 1773, deaf and completely paralyzed.*TIS

1895 Thomas Penyngton Kirkman FRS (31 March 1806 – 3 February 1895) was a British mathematician. Despite being primarily a churchman, he maintained an active interest in research-level mathematics, and was listed by Alexander Macfarlane as one of ten leading 19th-century British mathematicians. Kirkman's schoolgirl problem, an existence theorem for Steiner triple systems that founded the field of combinatorial design theory, is named after him.
Kirkman's first mathematical publication was in the Cambridge and Dublin Mathematical Journal in 1846, on a problem involving Steiner triple systems that had been published two years earlier in the Lady's and Gentleman's Diary by Wesley S. B. Woolhouse. Despite Kirkman's and Woolhouse's contributions to the problem, Steiner triple systems were named after Jakob Steiner who wrote a later paper in 1853. Kirkman's second research paper paper, in 1848, concerned hypercomplex numbers.
In 1850, Kirkman observed that his 1846 solution to Woolhouse's problem had an additional property, which he set out as a puzzle in the Lady's and Gentleman's Diary:
Fifteen young ladies in a school walk out three abreast for seven days in succession: it is required to arrange them daily, so that no two shall walk twice abreast.
This problem became known as Kirkman's schoolgirl problem, subsequently to become Kirkman's most famous result. He published several additional works on combinatorial design theory in later years. Kirkman also studied the Pascal lines determined by the intersection points of opposite sides of a hexagon inscribed within a conic section. Any six points on a conic may be joined into a hexagon in 60 different ways, forming 60 different Pascal lines. Extending previous work of Steiner, Kirkman showed that these lines intersect in triples to form 60 points (now known as the Kirkman points), so that each line contains three of the points and each point lies on three of the lines. *Wik

1928 Hendrik Antoon Lorentz (18 Jul 1853 - 4 Feb 1928) Dutch physicist who shared (with Pieter Zeeman) the Nobel Prize for Physics in 1902 for his theory of the influence of magnetism upon electromagnetic radiation phenomena. The theory was confirmed by findings of Zeeman and gave rise to Albert Einstein's special theory of relativity. From the start, Lorentz made it his task to extend James Clerk Maxwell's theory of electricity and of light. Already in his doctor's thesis, he treated the reflection and refraction phenomena of light from this new standpoint. His fundamental work in the fields of optics and electricity revolutionized conceptions of the nature of matter. In 1878, he published an essay relating the velocity of light in a medium, to its density and composition. *TIS

1974 Satyendra Nath Bose (1 Jan 1894; 4 Feb 1974) Indian physicist and mathematician who collaborated with Albert Einstein to develop a theory of statistical quantum mechanics, now called Bose-Einstein statistics. In his early work in quantum theory (1924), Bose wrote about the Planck black-body radiation law using a quantum statistics of photons, Plank's Law and the Light Quantum Hypothesis. Bose sent his ideas to Einstein, who extended this technique to integral spin particles. Dirac coined the name boson for particles obeying these statistics. Among other things, Bose-Einstein statistics explain how an electric current can flow in superconductors forever, with no loss. Bose also worked on X-ray diffraction, electrical properties of the ionosphere and thermoluminescence. *TIS

2003 Jean Brossel ( 15 August 1918 in Périgueux , France - 4 February 2003 in France)developed with Alfred Kastler the technique of optical pumping at origin of lasers. *Arjen Dijksman ‏@materion

2004 Valentina Mikhailovna Borok (9 July 1931, Kharkiv, Ukraine, USSR–4 February 2004, Haifa, Israel) was a Soviet Ukrainian mathematician. She is mainly known for her work on partial differential equations.*Wik

Credits :
*CHM=Computer History Museum
*FFF=Kane, Famous First Facts
*NSEC= NASA Solar Eclipse Calendar
*RMAT= The Renaissance Mathematicus, Thony Christie
*SAU=St Andrews Univ. Math History
*TIA = Today in Astronomy
*TIS= Today in Science History
*VFR = V Frederick Rickey, USMA
*Wik = Wikipedia
*WM = Women of Mathematics, Grinstein & Campbell

## Wednesday, 3 February 2016

### The Very Mathematical US Flag Starfield

Regular visitors here might recall that the good people at Firefly Books sent me a review copy of Ivan Moscovitch's beautiful puzzle book at top just before Christmas. If you missed my first recommendation, let me urge you to check it out.

I mention it again because one of the puzzles came up in a nice illustration of one of the mathematical concepts in the US flag's star field.
So first, the problem.

So there is the set up, a tightly packed square with 100 circles in a 10 by 10 array. And the challenge: Is it possible to squeeze one more circle into the same space? If you need to take some time, just stop reading and grab a pencil or whatever and work on it.

W
A
I
T
I
N
G

OK, and now for the upgraded challenge, how many more can you squeeze into the same space.

Could you see a way to squeeze a total of 105 into the same space? (actualy I think it is very slightly less space.)
Here is a way:

Ok, and now the tie-in question, what does this puzzle have to do with the US flag star field.

If you've been around since the sixties, you probably know that the 48 star flag, the one we had from 1912 to 1959 was a rectangle of 6 rows of eight stars. Within a little less than two years, we had added a 49th, and then a 50th star. If you look at the two star fields, you may notice a similarity.

Although we've had some unusual shaped flags, usually the star field is in a rectangle with the stars displaying some kind of (generally rectangular) similarity. Some have strayed greatly from the rectangle form however. This one with 38 stars from 1877 until 1890 is an example.
But the preference for a rectangular field over a square field was clearly demonstrated in the flags with 25,  which seems to be one of the least symmetric flags ever, the 36 star flag, and the briefly flown 49 star flag with 7 rows of 7, but not in a square. .

So what do you think the 51 star flag might look like?