## Sunday, 24 January 2021

### Looking for Perfectly Amicable Boxes

It started with a Twitter post by Mario Livio, who is head of public outreach at the Space Telescope Science Institute (the folks who operate the Hubble Telescope) but most of us know him better as a writer of popular math and science. I especially liked his book on the so-called Golden Ratio.

Mario's tweet was, "For buffs:118=14+50+54=18+30+70=15+40+63=21+25+72 and product of each triple is 37,800."

I got to wondering what was the smallest number that would have more than one partition into three parts so that the product of the three was the same. Turns out after some pencil scribbles, that 13 appears to be the smallest: 13= 1+6+6 = 2+2+9 and both have a product of 36.
Fourteen had two pairs. One with a product of 40 and the other with a product of 72 (I leave the actual values to the interested reader to find.)

Twenty-one seemed to be the smallest with three different pairs. Twenty-two and twenty three were interesting because they had the same product: 5+8+9 = 22, 4+9+10 = 23, but both have a product of 360. Twenty-five also produced a pair with a product of 360.

By thirty-nine I found what I believe is the smallest value with a three different partitions with the same product: 6 + 8 + 25 = 5 + 10+ 24 = 4 + 15 + 20; all three with a product of 1200.

I finally resorted to a computer search (it seemed quicker even with my poor programming skills than continuing to try to see a pattern in theses. Did I miss something easy?)to see if there was a quadruple before the 118 that Mario had found. By the time I reached the mid 70's I was repeatedly finding triple matches that had products of 5400, but no quadruples, and finally I concluded that 118 was indeed the smallest number that would have four partitions into three parts which had the same product.

So why is two partitions into three parts interesting? Well, for one thing, I now know that there are two different box shapes (rectangular Parallelepipeds) which have a volume of 36 for which the total sum of the edge lengths is 52 (4x13); and that for integer sides and volume, there is no smaller total side length for which that is true. So the L,W,H relation of 2,4,9 and 1,6,6 are special boxes.

The whole thing did get me off on another search. I wondered if there were distinct integer sided boxes for which the edge lengths had the same total and the surface areas are also equal. The short form would be, are there two partitions of a number n into x,y,z and a,b,c such that xy+xz+yz = ab+ac+bc?

Turns out they are pretty common. The smallest is for n = 9 (a total edge length of 36) with sides of 1,4,4 and 2,2,5. Each would have a surface area of 48 sq units.

With multiple examples of boxes that had the same total side lengths and either the same surface area or the same volume, as was left to wonder about the trifecta.... are there pairs, or triples of Boxes which share the same total edge lengths, surface area, and volume.  I already had decided they were obviously to be called Amicable Box Pairs (or triples, etc) ; but did they exist.

Certainly there were  boxes which had the surface area equal to the volume, such as the 6x6x6 cube.  In fact an infinite variety of them if we don't restrict ourselves to the integers.  Any box which has the volume more, or less than equal can be scaled up or down by some dimension to make them equal.  But could we get more than one of them, and could any two of them have the same sum of the edges.  We couldn't start with a cube, like 6,6,6 because no other three numbers which would have the same product with the sum of the edges equal to 72.

At this point I suspended my search due to the late hour, but wondered.  If I ask it, will they answer.... so it's on you know deep mathematical minds and computer programmers.  Do Amicable Box Pairs or Triples exist?

Chris Thompson writes on the Sequence Fanatics discussion list:

I take it that *Perfectly* Amicable Boxes refers to the question "are there pairs, or triples of Boxes which share the same total edge lengths, surface area, and volume". In which case, no there aren't any.

If the box is a X b X c, specifying the edge lengths fixes a+b+c, the surface ab+bc+ca, and the volume abc. Therefore you have fixed the cubic (x-a)(x-b)(x-c) = x^3 - (a+b+c)x^2 + (ab+bc+ca)x - abc, and therefore its three roots, up to a permutation.

Or to put it another way, if you have fixed all the elementary symmetric polynomials of a set of numbers, you have fixed the numbers themselves, up to a permutation.
Thanks To both Hans and Chris.

### On This Day in Math - January 24

* The art of photography

A mathematician's reputation rests on the number of bad proofs he has given.
~Abram S Besicovitch

The 24th day of the year; 24! (6.2044840173323943936 1023)is almost equal to Avogadro's Number, (6.022141×10^23).

24 may be the only number in the whole of the integers such that there are four distinct numbers which sum to 24, and whose reciprocals add up to one.  24= 2+4+6+12 and 1/2 + 1/4 + 1/6 + 1/12 = 1

Also 12 + 22 +...+ 242 = 702 the only pyramidal number that is also a square., that means 24 is the largest n such that the sum of the squares of the first n integers is a square number.

There are five regular polyhedra in three space, the Platonic solids. There are six regular 4-space polyhedra. Five of the 4-space polyhedra are analogs of the Platonic solids in 3-space, but there is also a 24-cell, with 24 octahedral faces w/o analog to the Platonic solids. Beyond 4-space, the number of regular polyhedra is always three.

The 24th dimension is the highest dimension for which the exact "kissing number", the number of spheres that can be placed around a central sphere so that they all are touching it, is known. For the 24th dimension, the "kissing number is 196,560. Beyond the fourth dimension, only the eighth (240) and twenty-fourth are known exactly.

EVENTS

1544, a solar eclipse was viewed at Louvain, which was later depicted in the first published book illustration of the camera obscura in use. Mathematician and astronomer Reinerus Gemma-Frisius viewed a solar eclipse using a hole in one wall of a pavillion to project the sun's image upside down onto the opposite wall. He published the first illustrationof a camera obscura, depicting his method of observation of the eclipse in De Radio Astronomica et Geometrica (1545). Several astronomers made use of such a device in the later part of the 16th century. Johannes Kepler viewed an eclipse Using the Camer Obscura in 1601 and again in 1605.It was Kepler who coined the name. *TIS *RMAT

1672 (14 Jan 1671 OS) In a letter from John Wallis to Oldenberg, he discusses his joint solution with Huygens of the area between the cissoid and it's asymptote, "Let mee know whether what I have inserted from Mr. Hugens ... be to his content." The cissoid (called Cissoid of Diocles) area was found to be 3πa2 , where a is the radius of the circle. *Wallis Corr. (cissoid means "ivy-shaped)

1801 Joseph Piazzi sends letters to Bode in Berlin, Oriani in Milan, and Lelande in Paris regarding a newly sighted "comet without tail and envelope".  To Oriani he admits that :

I have announced this star as a comet, but since it is not accompanied by any nebulosity and, further, since its movement is so slow and rather uniform, it has occurred to me several times that it might be something better than a comet. But I have been careful not to advance this supposition to the public.
*Wik

1870 Spectroscopy was a source of much excitement and interest in science in the last half of the 19th Century, but it created interest in the public sector as well. One example of the degree of the interest was the publication of a lengthy explanation of the topic in a seemingly unusual place, The Baptist Quarterly. The 20+ page article "claimed that the history, processes, achievements and possibilities of spectroscopy constitute "one of the marvels of the nineteenth century, and entitle it to the consideration of every thoughtful mind." *American History, Smithsonian

1889 Charles Darwin elected a Fellow of the Royal Society. *Friends of Darwin ‏@friendsofdarwin

1918 – The Gregorian calendar is introduced in Russia by decree of the Council of People's Commissars effective February 1 *Wik

1895 Karl Pearson first uses the term "Binomial Distribution" in Contributions to the Mathematical Theory of Evolution---II. Skew Variation in Homogeneous Material, Received by the Royal Society of London December 19, 1894, It would be read on January 24, 1895,” Philosophical Transactions of the Royal Society of London for the year MDCCCXCV. A footnote has: “This result seems of considerable importance, and I do not believe it has yet been noticed. It gives the mean square error for any binomial distribution, and we see that for most practical purposes it is identical with the value $\sqrt{npq}$ , hitherto deduced as an approximate result, by assuming the binomial to be approximately a normal curve.” Jeff Miller, Earliest Known Uses of Some of the Words of Mathematics

1925, a motion picture of a solar eclipse was taken by the U.S. Navy from the dirigible Los Angeles. The craft was at an elevation of about 4,500-ft and positioned about 19 miles east of Montauk Point, Long Island, NY. This give a view of a total eclipse of the sun that lasted just over 2-min. Four astronomical cameras and a spectrograph were used as well as two moving picture cameras. This was the first time in the U.S. that a dirigible had been used as a platform for observation of a total eclipse of the sun. The first U.S. attempt to photograph one from an aircraft 10 Sep 1923 was unsuccessful due to cloudy conditions, but on 28 Apr 1930, a flight over California sponsored by the U.S. Naval Observatory recorded a total solar eclipse. *TIS

1948 IBM dedicates the Selective Sequence Electronic Calculator​ (SSEC). Later the SSEC was put on public display near the company's Manhattan headquarters so passers-by could watch its operational speed. Before its decommissioning in 1952, the SSEC produced the moon-position tables used for plotting the course of the 1969 Apollo flight to the moon. *CHM

1984 Steve Jobs presented the the very first Macintosh computer, which became the first commercially successful personal computer with a mouse as standard input device and a user friendly graphical user interface to the public. *SciHiBlog

1986 – Voyager 2 passes within 81,500 kilometres (50,600 mi) of Uranus. *Wik

2003 Last signal (in distinct) traced to Pioneer 10. Launched in 1972, Pioneer 10 crossed the orbit of Saturn in 1976 and the orbit of Uranus in 1979. On June 13, 1983, Pioneer 10 crossed the orbit of Neptune, the outermost planet at the time, and so became the first man-made object to leave the proximity of the major planets of the solar system.
The last successful reception of telemetry was received from Pioneer 10 on April 27, 2002; subsequent signals were barely strong enough to detect, and provided no usable data. The final, very weak signal from Pioneer 10 was received on January 23, 2003 when it was 12 billion kilometers (80 AU) from Earth. *Wik and a HT to Hansruedi Widmer ‏@HansruediWidmer

BIRTHS

1679 Christian Freiherr von Wolff (24 Jan 1679; 9 Apr 1754) (baron) German philosopher, mathematician, and scientist who worked in many subjects but who is best known as the German spokesman of the Enlightenment, the 18th-century philosophical movement characterized by Rationalism. Wolff's first interest was mathematics. Though he made no original contribution to the discipline, he was important in the teaching of mathematics and instrumental in introducing the new mathematics into German universities. Later, as a philosopher, he developed the most impressive coherent system of his century. Thoroughly eclectic, influenced by Leibniz and Descartes, yet he continued fundamental themes of Aristotle. His system was important in making the discoveries of modern science known in Germany.
*TIS

1798 Karl Georg Christian von Staudt (January 24, 1798 – June 1, 1867) was a German mathematician born in the Free Imperial City of Rothenburg, which is now called Rothenburg ob der Tauber in Germany. From 1814 he studied in Gymnasium in Ausbach. He attended the University of Göttingen from 1818 to 1822 where he studied with Gauss who was director of the observatory. Staudt provided an ephemeris for the orbits of Mars and the asteroid Pallas. When in 1821 Comet Nicollet-Pons was observed, he provided the elements of its orbit. These accomplishments in astronomy earned him his doctorate from University of Erlangen in 1822.
The book Geometrie der Lage (1847) was a landmark in projective geometry. As Burau (1976) wrote, "Staudt was the first to adopt a fully rigorous approach. Without exception his predecessors still spoke of distances, perpendiculars, angles and other entities that play no role in projective geometry."
Furthermore, this book uses the complete quadrangle to "construct the fourth harmonic associated with three points on a straight line", the projective harmonic conjugate. *Wik (TIS gives birthdate as Jan 23)

1863 August Adler (24 Jan 1863 in Opava, Austrian Silesia (now Czech Republic)-17 Oct 1923 in Vienna, Austria) In 1906 Adler applied the theory of inversion to solve Mascheroni construction problems in his book Theorie der geometrischen Konstruktionen published in Leipzig. In 1797 Mascheroni had shown that all plane construction problems which could be made with ruler and compass could in fact be made with compasses alone. His theoretical solution involved giving specific constructions, such as bisecting a circular arc, using only a compass.
Since he was using inversion Adler now had a symmetry between lines and circles which in some sense showed why the constructions needed only compasses. However Adler did not simplify Mascheroni proof. On the contrary, his new methods were not as elegant, either in simplicity or length, as the original proof by Mascheroni.
This 1906 publication was not the first by Adler studying this problem. He had published a paper on the theory of Mascheroni's constructions in 1890, another on the theory of geometrical constructions in 1895, and one on the theory of drawing instruments in 1902. As well as his interest in descriptive geometry, Adler was also interested in mathematical education, particularly in teaching mathematics in secondary schools. His publications on this topic began around 1901 and by the end of his career he was publishing more on mathematical education than on geometry. Most of his papers on mathematical education were directed towards teaching geometry in schools, but in 1907 he wrote on modern methods in mathematical instruction in Austrian middle schools. He produced various teaching materials for teaching geometry in the sixth-form in Austrian schools such as an exercise book which he published in 1908. *SAU

1872 Morris William Travers (24 Jan 1872; 25 Aug 1961)
English chemist who, while working with Sir Willam Ramsay in London, discovered the element krypton (30 May 1898). The name derives from the Greek word for "hidden." It was a fraction separated from liquified air, which when placed in a Plücker tube connected to an induction coil yielded a spectrum with a bright yellow line with a greener tint than the known helium line and a brilliant green line that corresponded to nothing seen before.*TIS

1882 Harold Delos Babcock (24 Jan 1882(Edgerton,Wisconsin) - 8 Apr 1968) American astronomer who with his son, Horace, invented the solar magnetograph (1951), for detailed observation of the Sun's magnetic field. With their magnetograph the Babcocks measured the distribution of magnetic fields over the solar surface to unprecedented precision and discovered magnetically variable stars. In 1959 Harold Babcock announced that the Sun reverses its magnetic polarity periodically. Babcock's precise laboratory studies of atomic spectra allowed others to identify the first "forbidden" lines in the laboratory and to discover the rare isotopes of oxygen. With C.E. St. John he greatly improved the precision of the wavelengths of some 22,000 lines in the solar spectrum, referring them to newly-determined standards.*TIS

1891 Abram Samoilovitch Besicovitch (24 Jan 1891 in Berdyansk, Russia -2 Nov 1970 in Cambridge, Cambridgeshire, England) Besicovitch left Petrograd for Copenhagen in 1924 and there worked with Harald Bohr. He had been awarded a Rockefeller Fellowship but his applications for permission to work abroad had been refused. He escaped across the border with a colleague J D Tamarkin under the cover of darkness. He managed to reach Copenhagen where he was supported financially for a year with the Rockefeller Fellowship. His interest in almost periodic functions came about through this year spent working with Harald Bohr. After he visited Oxford in 1925 Hardy, who quickly saw the mathematical genius in Besicovitch, found a post for him in Liverpool. At Cambridge Besicovitch lectured on analysis in most years but he also gave an advanced course on a topic which was directly connected with his research interests such as almost periodic functions, Hausdorff measure, or the geometry of plane sets. Besicovitch was famous for his work on almost periodic functions, his interest in which, as we mentioned above, came from his time in Copenhagen with Harald Bohr. In 1932 he wrote an influential text Almost periodic functions covering his work in this area.
One of the achievements, with which he will always be associated, was his solution of the Kakeya problem on minimising areas. The problem had been posed in 1917 by a Japanese mathematician S Kakeya and asked what was the smallest area in which a line segment of unit length could be rotated through 2p. Besicovitch proved in 1925 that given any e, an area of less than e could be found in which the rotation was possible. The figures that resulted from Besicovitch's construction were highly complicated, unbounded figures.
Other areas on which Besicovitch worked included geometric measure theory, Hausdorff measure, real function theory, and complex function theory. In addition to this work on deep mathematical theories, Besicovitch loved problems, particularly those which could be stated in elementary terms but which proved resistant to attack. Often he showed that the "obvious solution" to certain problems is false. An example of such a problem is the Lion and the Man problem posed by Richard Rado in the mid 1920s. *SAU
On his 36th birthday, feeling that his most fertile years were behind him, mathematician Abram Besicovitch said, “I have had four-fifths of my life.”
At age 59 he was elected to the Rouse Ball Chair of Mathematics at Cambridge.
When J.C. Burkill reminded him of his earlier remark, he said, “Numerator was correct.” *Greg Ross, Futility Closet

1902 Oskar Morgenstern (24 Jan 1902; 26 Jul 1977) German-American economist and mathematician who popularized "game theory" which mathematically analyzes behaviour of man or animals in terms of strategies to maximize gains and minimize losses. He coauthored Theory of Games and Economic Behavior (1944), with John von Neumann, which extended Neumann's 1928 theory of games of strategy to competitive business situations. They suggested that often in a business situation ("game'), the outcome depends on several parties ("players"), each estimating what all of the others will do before determining their own strategy. Morgenstern was a professor at Vienna University, Austria, from 1931 until the Nazi occupation in 1938), when he fled to America and joined the faculty at Princeton University. His later publications included works on economic prediction and aspects of U.S. defence.*TIS

1912 Nils Aall Barricelli (January 24, 1912; Rome–January 27, 1993) was a Norwegian-Italian mathematician.
Barricelli's early computer-assisted experiments in symbiogenesis and evolution are considered pioneering in artificial life research. Barricelli, who was independently wealthy, held an unpaid residency at the Institute for Advanced Study in Princeton, New Jersey in 1953, 1954, and 1956. He later worked at the University of California, Los Angeles, at Vanderbilt University (until 1964), in the Department of Genetics of the University of Washington, Seattle (until 1968) and then at the Mathematics Institute of the University of Oslo. Barricelli published in a variety of fields including virus genetics, DNA, theoretical biology, space flight, theoretical physics and mathematical language. *Wik

1914 Vladimir Petrovich Potapov (24 Jan 1914 in Odessa, Ukraine - 21 Dec 1980 in Kharkov, Russia) In 1948 Potapov was invited to the Pedagogical Institute at Odessa. He soon became Head of Mathematics and, from 1952, Dean of the Faculty of Physics and Mathematics. He used his position to invite Livsic and others to the Institute.
During the 1950s Potapov worked on the theory of J-contractive matrix functions and the analysis of matrix functions became his main work. He published papers on the multiplicative theory of analytic matrix functions in the years 1950-55 which contain work from his doctoral thesis. He also worked on interpolation problems.
From 1974 Potapov lectured at Odessa Institute of National Economy, then he went to Kharkov to head the Department of Applied Mathematics at the Institute for low temperature physics. *SAU

1931 Lars V. Hörmander (24 Jan 1931 - ) Swedish mathematician who was awarded the Fields Medal in 1962 for his work on partial differential equations. Spending five years in writing, he produced a text The analysis of linear partial differential operators, in four volumes (1983-85). Between 1987 and 1990 he served as a vice president of the International Mathematical Union. In 1988 Hörmander was awarded the Wolf Prize. Hörmander's text, An Introduction to Complex Analysis in Several Variables, has become a classic dealing with the theory of functions of several complex variables. It developed from lecture notes of a course which he gave in Stanford in 1964 and published in book form two years later, with updates in 1973 and 1990.*TIS

1941 Dan Shechtman (January 24, 1941 in Tel Aviv - ) is the Philip Tobias Professor of Materials Science at the Technion – Israel Institute of Technology, an Associate of the US Department of Energy's Ames Laboratory, and Professor of Materials Science at Iowa State University. On April 8, 1982, while on sabbatical at the U.S. National Bureau of Standards in Washington, D.C., Shechtman discovered the icosahedral phase, which opened the new field of quasiperiodic crystals. He was awarded the 2011 Nobel Prize in Chemistry for "the discovery of quasicrystals". *Wik

1947 Michio Kaku (January 24, 1947 - ) is an American theoretical physicist, the Henry Semat Professor of Theoretical Physics in the City College of New York of City University of New York, the co-founder of string field theory, and a "communicator" and "popularizer" of science. He has written several books about physics and related topics; he has made frequent appearances on radio, television, and film; and he writes extensive online blogs and articles.*Wik

DEATHS

1860 James Pollard Espy (9 May 1785, 24 Jan 1860) American meteorologist who was one of the first to collect meteorological observations by telegraph. He gave apparently the first essentially correct explanation of the thermodynamics of cloud formation and growth. Every great atmospheric disturbance begins with a rising mass of heated, thus less dense air. While rising, the air mass dilates and cools. Then, as water vapour precipitates as clouds, latent heat is liberated so the dilation and rising continues until the moisture of the air forming the upward current is practically exhausted. The heavier air flows in beneath, and, finding a diminished pressure above it, rushes upward with constantly increasing violence. Water vapor precipitated during this atmospheric disturbance results in heavy rains.*TIS

1865 Samuel Hunter Christie (22 March 1784 – 24 January 1865) was a British scientist and mathematician.
He studied mathematics at Trinity College, Cambridge, where he won the Smith's Prize and was second wrangler. It may help to understand the difficulty of this exam by looking at some of the great achievers who did NOT win Senior Wrangler. A short list of Second Wranglers, include Alfred Marshall, James Clerk Maxwell, J. J. Thomson, and Lord Kelvin.
Those who have finished between third and 12th include Karl Pearson and William Henry Bragg (third), George Green and G. H. Hardy (fourth), Adam Sedgwick (fifth), John Venn (sixth), Bertrand Russell and Nevil Maskelyne (seventh), Thomas Malthus (ninth), and John Maynard Keynes (12th).

Christie was particularly interested in magnetism, studying the earth's magnetic field and designing improvements to the magnetic compass. Some of his magnetic research was done in collaboration with Peter Barlow. He became a Fellow of the Royal Society in 1826, delivered their Bakerian Lecture in 1833 and served as their Secretary from 1837 to 1853. In 1833 he published his 'diamond' method, the forerunner of the Wheatstone bridge, in a paper on the magnetic and electrical properties of metals, as a method for comparing the resistances of wires of different thicknesses. However, the method went unrecognized until 1843, when Charles Wheatstone proposed it, in another paper for the Royal Society, for measuring resistance in electrical circuits. Although Wheatstone presented it as Christie's invention, it is his name, rather than Christie's, that is now associated with the device.
Christie taught mathematics at the Royal Military Academy, Woolwich, from 1838 until his retirement in 1854.[1] He died at Twickenham, on 24 January 1865. *Wik

1877 Johann Christian Poggendorff (29 December 1796 – 24 January 1877), was a German physicist and science historian born in Hamburg. By far the greater and more important part of his work related to electricity and magnetism. Poggendorff is known for his electrostatic motor which is analogous to Wilhelm Holtz's electrostatic machine. In 1841 he described the use of the potentiometer for measurement of electrical potentials without current draw.
Even at this early period he had conceived the idea of founding a physical and chemical scientific journal, and the realization of this plan was hastened by the sudden death of Ludwig Wilhelm Gilbert, the editor of Gilbert's Annalen der Physik, in 1824 Poggendorff immediately put himself in communication with the publisher, Barth of Leipzig. He became editor of Annalen der Physik und Chemie, which was to be a continuation of Gilbert's Annalen on a somewhat extended plan. Poggendorff was admirably qualified for the post, and edited the journal for 52 years, until 1876. In 1826, Poggendorff developed the mirror galvanometer, a device for detecting electric currents.
He had an extraordinary memory, well stored with scientific knowledge, both modern and historical, a cool and impartial judgment, and a strong preference for facts as against theory of the speculative kind. He was thus able to throw himself into the spirit of modern experimental science. He possessed in abundant measure the German virtue of orderliness in the arrangement of knowledge and in the conduct of business. Further he had an engaging geniality of manner and much tact in dealing with men. These qualities soon made Poggendorff's Annalen (abbreviation: Pogg. Ann.) the foremost scientific journal in Europe.
In the course of his fifty-two years editorship of the Annalen Poggendorff could not fail to acquire an unusual acquaintance with the labors of modern men of science. This knowledge, joined to what he had gathered by historical reading of equally unusual extent, he carefully digested and gave to the world in his Biographisch-literarisches Handworterbuch zur Geschichte der exacten Wissenschaften, containing notices of the lives and labors of mathematicians, astronomers, physicists, and chemists, of all peoples and all ages. This work contains an astounding collection of facts invaluable to the scientific biographer and historian. The first two volumes were published in 1863; after his death a third volume appeared in 1898, covering the period 1858-1883, and a fourth in 1904, coming down to the beginning of the 20th century.
His literary and scientific reputation speedily brought him honorable recognition. In 1830 he was made royal professor, in 1838 Hon. Ph.D. and extraordinary professor in the University of Berlin, and in 1839 member of the Berlin Academy of Sciences. In 1845, he was elected a foreign member of the Royal Swedish Academy of Sciences.
Many offers of ordinary professorships were made to him, but he declined them all, devoting himself to his duties as editor of the Annalen, and to the pursuit of his scientific researches. He died at Berlin on 24 January 1877.
The Poggendorff Illusion is an optical illusion that involves the brain's perception of the interaction between diagonal lines and horizontal and vertical edges. It is named after Poggendorff, who discovered it in the drawing of Johann Karl Friedrich Zöllner, in which he showed the Zöllner illusion in 1860. In the picture to the right, a straight black line is obscured by a dark gray rectangle. The black line appears disjointed, although it is in fact straight; the second picture illustrates this fact.*Wik

1914 Sir David Gill (12 Jun 1843, 24 Jan 1914) Scottish astronomer known for his measurements of solar and stellar parallax, showing the distances of the Sun and other stars from Earth, and for his early use of photography in mapping the heavens. His early training in timekeeping as a watchmaker led to astronomy and he designed, equipped, and operated a private observatory near Aberdeen. To determine parallaxes, he perfected the use of the heliometer, a telescope that uses a split image to measure the angular separation of celestial bodies. In 1877, Gill and his wife measured the solar parallax by observing Mars from Ascension Island. He was appointed Her Majesty's Astronomer at the Cape of Good Hope (1879-1906). Gill also made geodetic surveys of South Africa. In fact he carried out all of the observations to measure the distances to stars in terms of the standard meter. His precise redetermination of the solar parallax was used for almanacs until 1968. *TIS

1930 Adolf Kneser (19 March 1862 in Grüssow, Mecklenburg, Germany - 24 Jan 1930 in Breslau, Germany (now Wrocław, Poland)) He is remembered most for work mainly in two areas. One of these areas is that of linear differential equations; in particular he worked on the Sturm-Liouville problem and integral equations in general. He wrote an important text on integral equations. The second main area of his work was the calculus of variations. He published Lehrbruch der Variationsrechnung (Textbook of the calculus of variations) (1900) and he gave the topic many of the terms in common use today including 'extremal' for a resolution curve, 'field' for a family of extremals, 'transversal' and 'strong' and 'weak' extremals *SAU

1955 Percy John Heawood (8 September 1861 Newport, Shropshire, England - 24 January 1955 Durham, England) was a British mathematician. He devoted essentially his whole working life to the four color theorem and in 1890 he exposed a flaw in Alfred Kempe's proof, that had been considered as valid for 11 years. With the four color theorem being open again he established the five color theorem instead. The four color theorem itself was finally established by a computer-based proof in 1976. *Wik

1961 Albert Carlton Gilbert (15 Feb 1884, 24 Jan 1961) was an American inventor who patented the Erector set after he founded the A.C. Gilbert Co. New Haven, Connecticut (1908) to manufacture boxed magic sets. In 1913, he introduced Erector Sets. Similar construction toys then existed, such as Hornby's Meccano set made in England. Meccano sets included pulleys, gears, and several 1/2" wide strips of varying length with holes evenly spaced on them. Gilbert needed something unique for his Erector sets, so he created the square girder, made using several 1" wide strips with triangles cut in them. These had their edges bent over so 4 strips could be screwed together to form a very sturdy square girder. Over the next 40 years, some 30 million Erector Sets were sold.*TIS
Gilbert was also an accomplished athlete, he broke the world record for consecutive chin-ups in 1900 and distance record for running long dive in 1902. He invented the pole vault box and set two world records in the pole vault including a record for 12' 3" (3.66 meters) at the Spring meet of the Irish American Athletic Club, held at Celtic Park, Queens, New York, in 1906. He tied for gold with fellow American Edward Cook at the 1908 Summer Olympics in London for pole vaulting. *Wik

1982 Karol Borsuk (8 May 1905 in Warsaw, - 24 Jan 1982 in Warsaw) Borsuk introduced the important concept of absolute neighbourhood retracts in his doctoral dissertation, published in 1931, which was to lead to new and fruitful ideas in metric differential geometry. In 1936 he introduced the notion of cohomotopy groups, which could be said to mark the beginning of stable homotopy theory.   Shape theory grew up at the same time as infinite-dimensional topology and the interaction between the two fields was of great mutual benefit. He was important for the many deep questions which Borsuk posed which stimulated most of the top mathematicians working in the area. *SAU

1988 Moritz Werner Fenchel (3 May 1905 – 24 January 1988) was a mathematician known for his contributions to geometry and to optimization theory. Fenchel established the basic results of convex analysis and nonlinear optimization theory which would, in time, serve as the foundation for Nonlinear programming. A German-born Jew and early refugee from Nazi suppression of intellectuals, Fenchel lived most of his life in Denmark. Fenchel's monographs and lecture notes are considered influential. *Wik

2016 Marvin Minsky (August 9, 1927 - January 24, 2016 (aged 88)) Biochemist and the founder of the MIT Artificial Intelligence Project. Marvin Minsky has made many contributions to AI, cognitive psychology, mathematics, computational linguistics, robotics, and optics. He holds several patents, including those for the first neural-network simulator (SNARC, 1951), the first head-mounted graphical display, the first confocal scanning microscope, and the LOGO "turtle" device. His other inventions include mechanical hands and the "Muse" synthesizer for musical variations (with E. Fredkin). In recent years he has worked chiefly on imparting to machines the human capacity for commonsense reasoning. *TIS He died in Boston of a cerebral hemorrhage .

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

## Saturday, 23 January 2021

### On This Day in Math - January 23

 Me Considering Pascal Considering the Roulette

Nature creates curved lines while humans create straight lines.
~Hideki Yukawa

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

There are only two values of $x>0$ so that ${x}^{3}+\left(x+1{\right)}^{3}+\left(x+2{\right)}^{3}$is a perfect square. The largest is 23.  *@BenVitale

decimal # 11111111111111111111111 is prime (23 ones) . *

EVENTS

1571 The Royal Exchange in London was founded by the merchant Thomas Gresham to act as a center of commerce for the City of London. *SciHiBlog

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

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

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

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

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

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

1883 The first session of the Edinburgh Mathematical Society,  *Proceedings of the Edinburgh Mathematical Society, Volumes 1-4

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

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

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

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

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

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

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

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

2013 On January 23rd Dr. Curtis Cooper of Central Missouri University discovered the 48th known Mersenne prime, 257,885,161-1, a 17,425,170 digit number. The GIMP site records this as the 25th of January, so I shall use both dates until I figure out why two different dates reported.

BIRTHS

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

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

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

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

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

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

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

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

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

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

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

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

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

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

DEATHS

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

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

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