Tuesday, 21 January 2020

Splitters, Cleavers Equalizers and... fractionalizers?

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

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




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

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

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

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

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

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

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

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

Restructuring the equations we arrive at the condition that a+b = 3ab.   So I quickly set up a geogebra sketch to test the range of possible r's that could occur. 
At a=b=2/3 I knew that 4/9=r, and as I adjusted the line on the x-axis to more than 2/3, the ratio increased toward 1/2 as the bottom (a side) approached a length of one, and the b side approached 1/2.   When I shortened side a, the same thing happened, as the b side became larger.  When a was 3/5 for example, the b side worked out to be 3/4, and r was 9/20. 

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

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

On This Day in Math - January 21


The whole of the developments and operations of analysis are now capable of being executed by machinery. ... As soon as an Analytical Engine exists, it will necessarily guide the future course of science.
C. Babbage

The 21st day of the year; To tile a square out of integer sided squares requires a minimum of 21 squares. (technically, this is true for what are called "simple" squared squares, one where no subset of the squares forms a rectangle or square. See the solution here) (btw: There are no cubed cubes!)
There are 21 possible ways to draw 5 circles that touch all the points on a 5x5 lattice.  *gotmath.com

21 repeated twenty-one times, following 1, forms a smoothly undulating palindromic prime
121212121212121212121212121212121212121 is prime

Blackjack primes are separated by exactly 21 consecutive composite numbers. Note that the pair {1129, 1151} is the smallest example.(Can you find more?) *Prime Curios


1472, the great daylight comet of 1472 passed within 10.5 million km of earth.*TIS (Johannes Müller von Königsberg (Regiomontanus) is said to have observed this comet)

1609 Modern astronomy dates all astronomical events using the Julian Day Count a system of dating that was first conceived by a Renaissance historian and Bible chronologist, Joseph Justus Scalier, who died on this day. The Julian Day Number (JDN) is the integer assigned to a whole solar day in the Julian day count starting from noon Greenwich Mean Time, with Julian day number 0 assigned to the day starting at noon on January 1, 4713 BC. At noon on the date of his death, the Julian Day 2308756 would have began. *Wik For a few details about his life see this post at the Renaissance Mathematicus.

1665 Samuel Pepys, having acquired a copy of Hooke’s Micrographia the day before, stays up to read it, “Before I went to bed I sat up till two o'clock in my chamber reading of Mr. Hooke's Microscopicall Observations, the most ingenious book that ever I read in my life.” *Pepy’s Diary

1807, the London Institution received a royal charter signed by King George III, to "promote the diffusion of Science, Literature, and the Arts, by means of Lectures and Experiments, and by easy access to an extensive collection of books, both ancient and modern, in all languages." The full name in the charter was the "London Institution for the Advancement of Literature and The Diffusion of Useful Knowledge." The first president was Sir Francis Baring. Its incorporation came after the Royal Society (1663) and Royal Institution (1800). The institution had an extensive lecture program. Instruction in practical chemistry was given in its laboratory, and significant chemistry research was done there through the 19th century. *TIS

1840, Charles Wheatstone and William F. Cooke were granted the earliest English alphabetic telegraph patent. Wheatstone made contributions to a broad range of fields in the mid 19th century. The ABC telegraph was popular in England and Europe because it did not require a trained telegraphist to read or send the messages. The operator simply rotates a wheel to the desired letter. During rotation the instrument sends out the proper number of electric pulses to an electromagnetically controlled pointer on a remote synchronized slave receiver with a similarly lettered wheel which moves to the sender's letter. Electric telegraphs of the 1840-50's are of special historic importance as the earliest practical application of serial binary coded digital communication. *TIS

1888 Babbage's Analytical Engine Passes the First Test
The Analytical Engine of Charles Babbage was never completed in his lifetime, but his son Henry Provost Babbage built the mill portion of the machine from his father's drawings, and on January 21, 1888 computed multiples of pi to prove the adequacy of the design. Perhaps this represents the first successful test of a portion of a modern computer. Recently a portion of his earlier machine, the Difference Engine, was sold at auction by Christies of London to the Powerhouse Museum in Sydney, Australia.*CHM

1954, the first atomic submarine, the U.S.S. Nautilus, was launched at Groton, Connecticut. Nautilus' nuclear propulsion system was a landmark in the history of naval engineering and submersible craft. All vessels previously known as "submarines" were in fact only submersible craft. Because of the nuclear power plant, the Nautilus could stay submerged for months at a time, unlike diesel-fueled subs, whose engines required vast amounts of oxygen. Nautilus demonstrated her capabilities in 1958 when she sailed beneath the Arctic icepack to the North Pole. Scores of nuclear submarines followed Nautilus, replacing the United States' diesel boat fleet. After patrolling the seas until 1980, the Nautilus is back home at Groton. *TIS

1979 Pluto moves closer to the sun than Neptune. *VFR Pluto is usually farther from the Sun. However, its orbit "crosses" inside of Neptune's orbit for 20 years out of every 248 years. Pluto last crossed inside Neptune's orbit on February 7, 1979, and temporarily became the 8th planet from the Sun. Pluto crossed back over Neptune's orbit again on February 11, 1999 to resume its place as the 9th planet from the Sun for the next 228 years (well, now it is now one of five known "dwarf planets").

1793 Théodore Olivier (21 Jan 1793 in France - 5 Aug 1853 in France) From the 1840's Olivier wrote textbooks. His greatest fame, however, is as a result of the mathematical models which he created to assist in his teaching of geometry. Some of the models were of ruled surfaces, with moving parts to illustrate to students how the ruled surfaces were generated. Others were designed to illustrate the curves of intersection of certain surfaces. In fact Olivier earned quite a good income from selling these models, particularly in the United States.
The United States Military Academy at West Point had 23 mathematical models made for them by Olivier to use as teaching aids:=
These models are built on wooden boxes as bases, have metal supports, and consist of strings suspended from movable arms and arranged to form a variety of geometrical figures. The strings are held in place by lead weights that are concealed by the bases. The models illustrate such things as the intersection of two half cones, the intersection of a plane, hyperbolic paraboloid and a hyperboloid of one sheet, and the intersection of two half cylinders.
Other institutions in the United States such as the Columbia School of Mines also purchased models from Olivier while Princeton had copies of Olivier's models made for them. In 1849 Olivier presented a full set of the range of models he had created to the Conservatoire National des Arts et Métiers. The models had been manufactured by the firm of Pixii, Père et Fils, and later by the firm of Fabre de Lagrange which took over their manufacture. In 1857, four years after Olivier died, Harvard University purchased 24 of Olivier's models from Fabre de Lagrange and after the university received the order Benjamin Peirce gave a series of lectures on the mathematics which they illustrated. These models are still in Harvard's collection of scientific instruments.
Even after giving a complete set of his models to the Conservatoire National des Arts et Métiers, forty models were still in Olivier's possession at the time of his death. These were sold in 1869 to William Gillispie from Union College in Schenectady, east-central New York, United States. Gillispie exhibited the models at Union College which was appropriate since, twenty years earlier, Union College had became one of the first liberal arts colleges in the United States to give engineering courses. When Gillispie died Olivier's models were sold to the college. *SAU

1846 Pieter Hendrik Schoute (January 21, 1846, Wormerveer–April 18, 1923, Groningen) was a Dutch mathematician known for his work on regular polytopes and Euclidean geometry. *Wik Schoute was a typical geometer. In his early work he investigated quadrics, algebraic curves, complexes, and congruences in the spirit of nineteenth-century projective, metrical, and enumerative geometry. Schläfli's work of the 1850's was brought to the Netherlands by Schoute who, in three papers beginning in 1893 and in his elegant two-volume textbook on many-dimensional geometry 'Mehrdimensionale Geometrie' (2 volumes 1902, 1905), studied the sections and projections of regular polytopes and compound polyhedra. ... Alicia Boole Stott (1870-1940), George Boole's third daughter (of five), ... studied sections of four- and higher-dimensional polytopes after her husband showed her Schoute's 1893 paper, and Schoute later (in his last papers) gave an analytic treatment of her constructions. *SAU

1860 David Eugene Smith, Ph.D., LL.D. (January 21, 1860 in Cortland, New York – July 29, 1944 in New York) was an American mathematician, educator, and editor. David Eugene Smith attended Syracuse University, graduating in 1881 (Ph. D., 1887; LL.D., 1905). He studied to be a lawyer concentrating in arts and humanities, but accepted an instructorship in mathematics at the Cortland Normal School in 1884. He also knew Latin, Greek, and Hebrew. He became a professor at the Michigan State Normal College in 1891, the principal at the State Normal School in Brockport, New York (1898), and a professor of mathematics at Teachers College, Columbia University (1901).
Smith became president of the Mathematical Association of America in 1920.[3] He also wrote a large number of publications of various types. He was editor of the Bulletin of the American Mathematical Society; contributed to other mathematical journals; published a series of textbooks; translated Klein's Famous Problems of Geometry, Fink's History of Mathematics, and the Treviso Arithmetic. He edited Augustus De Morgan's Budget of Paradoxes (1915) and wrote many books on Mathematics and Mathematics History. *Wik

1874 René-Louis Baire (21 Jan 1874; 5 Jul 1932) French mathematician whose study of irrational numbers and whose concept to divide the notion of continuity into upper and lower semi-continuity greatly influenced the French School of Mathematics. His doctoral thesis led to the solution of the problem of the characteristic property of limited functions of continuous functions and helped establish the theory of functions of real variables.*TIS

1897 Alexander Weinstein (21 Jan 1897 in Saratov, Russia - 6 Nov 1979 in Washington DC, USA) is famed for solving a variety of boundary value problems which have been used in a wide range of applications. Weinstein's method was developed to give accurate bounds for eigenvalues of plates and membranes. In examining singular partial differential equations he introduced a new branch of potential theory and applied the results to many different situations including flow about a wedge, flow around lenses and flow around spindles. *SAU

1908 Bengt Strömgren (21 Jan 1908; 4 Jul 1987) Bengt (Georg Daniel) Strömgren was a Danish astrophysicist who pioneered the present-day knowledge of the gas clouds in space. Researching for his theory of the ionized gas clouds around hot stars, he found relations between the gas density, the luminosity of the star, and the size of the "Strömgren sphere" of ionized hydrogen around it. He surveyed such H II regions in the Galaxy, and he also did important work on stellar atmospheres and ionization in stars. *TIS

1915 Yuri Vladimirovich Linnik (January 8, 1915 – June 30, 1972) was a Soviet mathematician active in number theory, probability theory and mathematical statistics.*Wik

1923 Daniel E. Gorenstein (January 1, 1923 – August 26, 1992) was an American mathematician. He earned his undergraduate and graduate degrees at Harvard University, where he earned his Ph.D. in 1950 under Oscar Zariski, introducing in his dissertation a duality principle for plane curves that motivated Grothendieck's introduction of Gorenstein rings. He was a major influence on the classification of finite simple groups.
After teaching mathematics to military personnel at Harvard before earning his doctorate, Gorenstein held posts at Clark University and Northeastern University before he began teaching at Rutgers University in 1969, where he remained for the rest of his life. He was the founding director of DIMACS in 1989, and remained as its director until his death.[1]
Gorenstein was awarded many honors for his work on finite simple groups. He was recognised, in addition to his own research contributions such as work on signalizer functors, as a leader in directing the classification proof, the largest collaborative piece of pure mathematics ever attempted. In 1972 he was a Guggenheim Fellow and a Fulbright Scholar; in 1978 he gained membership in the National Academy of Sciences and the American Academy of Arts and Sciences, and in 1989 won the Steele Prize for mathematical exposition. *Wik

1609 Joseph Justus Scaliger (5 Aug 1540, 21 Jan 1609) French scholar who was one of the founders of the science of chronology. Like Roger de Losinga, Bishop of Hereford, centuries before, Scaliger recognized that combining the three cycles of the 28-year solar cycle (S), the 19-year cycle of Golden Numbers (G) and the 15-year indiction cycle (I) produced one greater cycle of 7980 years (28×9×15). Scalinger applied this fact, called a Julian cycle, in his attempt to resolve a patchwork of historical eras and he used notation (S, G, I) to characterize years. The year of Christ's birth had been determined by Dionysius Exigus to be the number 9 on the solar cycle, by Golden Number 1, and by 3 of the indiction cycle, thus (9, 1, 3), which was 4713 of his chronological era. Hence, the year (1, 1, 1) was 4713 B.C. (later adopted as the initial epoch for the Julian day numbers).*TIS A formula for converting days to Julian day numbers is here.

1800 Jean-Baptiste Le Roy (15 August 1720;Paris, France - 21 January 1800, Paris) Son of the renowned clockmaker Julien Le Roy, Jean-Baptiste Le Roy was one of four brothers to achieve scientific prominence in Enlightenment France; the others were Charles Le Roy (medicine and chemistry), Julien-David Le Roy (architecture), and Pierre Le Roy(chronometry). Elected to the Académie Royale des Sciences in 1751 as adjoint géomètre, Le Roy played an active role in technical as well as administrative aspects of French science for the next half-century. He was elected pensionnaire mécanicies in 1770 and director of the Academy for 1773 and 1778, and became both a fellow of the Royal Society and a member of the American Philosophical Society in 1773.
Le Roy’s major field of enquiry was electricity, a subject on which European opinion was much divided at mid-century. The most prominent controversy engaged the proponents of the Abbé Nollet’s doctrine of two distinct streams of electric fluids (outflowing and inflowing) and the partisans of Benjamin Franklin’s concept of a single electric fluid. This debate intensified in France in 1753 with an attack on Franklin’s views by Nollet. Le Roy, later a friend and correspondent of Franklin, defended his single-fluid theory and offered considerable experimental evidence in support thereof. He played an important role in the dissemination of Franklin’s ideas, stressing particularly their practical applications, and published many memoirs on electrical machines and theory in the annual Histoires and Mémoires of the Academy and in the Journal de Physique.
A regular contributor to the Encyclopédie, Le Roy wrote articles dealing with scientific instruments. The most important of these included comprehensive treatments of “Horlogerie,” “Télescope,” and “Électrométre” (in which Le Roy claimed priority for the invention of the electrometer). He also promoted the use of lightning rods in France, urged that the Academy support technical education, and was active in hospital and prison reform. After the Revolutionary suppression of royal academies, Le Roy was appointed to the first class of the Institut National (section de mécanique) at its formation in 1795. *Encycopedia.com

1892 John Couch Adams (5 Jun 1819, 21 Jan 1892) In 1878 he published his calculation of Euler’s constant (Euler-Mascheronie constant) to 263 decimal places. (he also calculated the Bernoulli numbers up to the 62 nd) *VFR The Euler-Mascheronie constant is the limiting value of the difference between the sum of the first n values in the harmonic series and the natural log of n. (not 263 places, but the approximate value is 0.5772156649015328606065...)
He also predicted the location of the then unknown planet of Neptune, but it seems he failed to convince Airy to search for the planet. Independently, Urbanne LeVerrier predicted its locatin in Germany, and then assisted Galle in the Berlin Observatory in locating the planet on 23 September 1846. As a side note, when he was appointed to a Regius position at St. Andrews in Scotland, he was the last professor ever to have to swear and oath of “abjuration and allegience”, swearing fealty to Queen Victoria, and abjuring the Jacobite succession. The need for the oath was removed by the 1858 Universities Scotland Act. Adams made many other contributions to astronomy, notably his studies of the Leonid meteor shower (1866) where he showed that the orbit of the meteor shower was very similar to that of a comet. He was able to correctly conclude that the meteor shower was associated with the comet. *Wik & *TIS

1930 H(ugh) L(ongbourne) Callendar (18 Apr 1863, 21 Jan 1930) was an English physicist famous for work in calorimetry, thermometry and especially, the thermodynamic properties of steam. He published the first steam tables (1915). In 1886, he invented the platinum resistance thermometer using the electrical resistivity of platinum, enabling the precise measurement of temperatures. He also invented the electrical continuous-flow calorimeter, the compensated air thermometer (1891), a radio balance (1910) and a rolling-chart thermometer (1897) that enabled long-duration collection of climatic temperature data. His son, Guy S. Callendar linked climatic change with increases in carbon dioxide (CO2) resulting from mankind's burning of carbon fuels (1938), known as the Callendar effect, part of the greenhouse effect.*TIS

1931 Cesare Burali-Forti (13 August 1861 – 21 January 1931) was an Italian mathematician. He was born in Arezzo, and was an assistant of Giuseppe Peano in Turin from 1894 to 1896, during which time he discovered what came to be called the Burali-Forti paradox of Cantorian set theory.*Wik

1946 Harry Bateman FRS (29 May 1882 – 21 January 1946) was an English mathematician. He first grew to love mathematics at Manchester Grammar School, and in his final year, won a scholarship to Trinity College, Cambridge. There he distinguished himself in 1903 as Senior Wrangler (tied with P.E. Marrack) and by winning the Smith's Prize (1905). He studied in Göttingen and Paris, taught at the University of Liverpool and University of Manchester before moving to the US in 1910. First he taught at Bryn Mawr College and then Johns Hopkins University. There, working with Frank Morley in geometry, he achieved the Ph.D. In 1917 he took up his permanent position at California Institute of Technology, then still called Throop Polytechnic Institute.
Eric Temple Bell says, "Like his contemporaries and immediate predecessors among Cambridge mathematicians of the first decade of this century [1901–1910]... Bateman was thoroughly trained in both pure analysis and mathematical physics, and retained an equal interest in both throughout his scientific career."*Wik

1974 Arnaud Denjoy (5 January 1884 – 21 January 1974 in Paris) a French mathematician born in Auch, Gers. His contributions include work in harmonic analysis and differential equations. His integral was the first to be able to integrate all derivatives. Among his students is Gustave Choquet.*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

Monday, 20 January 2020

On This Day in Math - January 20

If you are afraid of something, measure it, and you will realize it is a mere trifle.
~Renato Caccioppoli

The 20th day of the year; 20 is the smallest number that cannot be either prefixed or followed by one digit to form a prime. (What is next smallest?)

The 20th palindromic prime (929) has a digit sum of 20. *jim wilder ‏@wilderlab ... There is no larger nth prime palindrome for which digit sum = n less than \(10^7\) *Derek Orr & *David ‏@InfinitelyManic

\( e^\pi - \pi = 20 \) Well, almost, it's 19.999099979...,  (The number \( e^\pi \) is often called Gelfond's constant, after the mathematician Aleksandr Gelfond, who proved that it was transcendental.

20 is involved in a challenge by Fermat in 1647 to "find a cube, that when increased by the sum of its aliquot parts, is a square."

And in Paul Halcke observed that 20 was one of three (all rather modestly sized) numbers for which the product of their aliquote divisors equaled their square.  Can you find others?


In 1633, Galileo, at age 68, left his home in Florence, Italy, to face the Inquisition in Rome. By 22 Jun 1633, he buckled under the threats and interrogation by the Inquisition, and renounced his belief that the Earth revolved around the Sun. *TIS

1748 In a surprising letter of January 20, 1748, D'Alembert wrote to Euler [Euler 1980] to suggest a new theory: perhaps the moon (or at least its distribution of mass) was not spherical. If, after all, we only see one side from the Earth, we can't know how far back it truly extends. And perhaps if it extends far enough back, the apsidal motion would indeed be 3 degrees, as observed. In an even more surprising response written less than four weeks later [Euler 1980], Euler says that he too had considered this idea, and had worked out the details! He found that moon would have to extend back about 2 1/2 Earth diameters in the direction away from us, which seemed untenable. *VFR

1896, X-rays were reported by Henri Poincaré reading a letter from Wilhelm Roentgen to the weekly meeting of the Academie des Sciences in Paris, and members viewed some of his X-ray photographs. Henri Becquerel was present, and took note that X-rays seemed to come from the phosphorescent patch on the glass tube where the cathode rays struck it. This inspired him to study if the phosphorescence of minerals was related to X-rays. (Instead, a few weeks later, he discovered the radioactivity of a uranium mineral.) *TIS

1969, the New York Times made public the news of the discovery a few days earlier of the first optical pulsar by astronomers at the University of Arizona on 16 Jan 1969. It was the result of a year's search using a stroboscopic technique. Flashes of light in the optical range were found coming from the same location in the Crab Nebula as a previously known pulsar emitting radio bursts. The rate of pulsation of the two signals was found to be the same, and thus presumed to be from a single star. Other observatories were immediately notified and the flashing was confirmed by the McDonald Observatory and by the powerful 84-inch reflector telescope at the Kitt Peak National Observatory in Arizona. The star was flashing at a rate of about 30 times per second, with intermediate flashes of lesser intensity. *TIS

1983 The Department of Commerce officially withdrew the commercial standard for the sizing of women's apparel on January 20, 1983. In the mid-1940s, the Mail-Order Association of America, a trade group representing catalog businesses such as Sears Roebuck and Spiegel, asked the Commodity Standards Division of the National Bureau of Standards (NBS, now NIST )to conduct research to provide a reliable basis for industry sizing standards for women's clothing. Men's clothing had become somewhat standardized as early as the American Civil War due to demands for quick manufacture of uniforms. The resulting commercial standard was distributed by NBS to the industry for comment in 1953, formally accepted by the industry in 1957, and published as Commercial Standard (CS)215-58 in 1958.
However, with the passage of time, the standard became outdated. Both American men and women were becoming heavier. Whereas the average woman's figure once came a little closer to approaching the hourglass shape of the fashion magazines, she was now becoming more pear-shaped, with a thicker waist and fuller hips. At the same time that the average woman's body was changing shape, manufacturers discovered the advantage of appealing to women's vanity. They began selling bigger clothes labeled with smaller size numbers. Today only pattern makers consistently use the old commercial standard.*NIST

1989 Statistician and political scientist Edward R Tufte sent a letter to the New York Times with a copy of his book on statistical graphics. On a visit later to New York, the Times called it as "founding document of New York Times Graphics" *
Edward Tufte@EdwardTufte


1573 Simon Marius (20 Jan 1573, 26 Dec 1624) (Also known as Simon Mayr) German astronomer, pupil of Tycho Brahe, one of the earliest users of the telescope and the first in print to make mention the Andromeda nebula (1612). He studied and named the four largest moons of Jupiter as then known: Io, Europa, Ganymede and Callisto (1609) after mythological figures closely involved in love with Jupiter. Although he may have made his discovery independently of Galileo, when Marius claimed to have discovered these satellites of Jupiter (1609), in a dispute over priority, it was Galileo who was credited by other astronomers. However, Marius was the first to prepare tables of the mean periodic motions of these moons. He also observed sunspots in 1611 *TIS Galileo initially named his discovery the Cosmica Sidera ("Cosimo's stars"), but names that eventually prevailed were chosen by Simon Marius and were suggested by Johannes Kepler, in his Mundus Jovialis​, published in 1614. *Wik You can find a nice blog about the conflict with Galileo by the Renaissance Mathematicus.

1775 André-Marie Ampère (20 January 1775 – 10 June 1836) was a French physicist and mathematician who is generally regarded as one of the main discoverers of electromagnetism. The SI unit of measurement of electric current, the ampere, is named after him.*Wik

1820 Alexandre-Emile Beguyer de Chancourtois (20 Jan 1820; 14 Nov 1886) French geologist who was the first to arrange the chemical elements in order of atomic weights (1862). De Chancourtois plotted the atomic weights on the surface of a cylinder with a circumference of 16 units, the approximate atomic weight of oxygen. The resulting helical curve which he called the telluric helix brought closely related elements onto corresponding points above or below one another on the cylinder. Thus, he suggested that "the properties of the elements are the properties of numbers." Although his publication was significant, it was ignored by chemists as it was written in the language of geology, and the editors omitted a crucial explanatory table. It was Dmitry Mendeleyev's table published in 1869 that became most recognized.*TIS

1831 Edward John Routh FRS (20 January 1831–7 June 1907), was an English mathematician, noted as the outstanding coach of students preparing for the Mathematical Tripos examination of the University of Cambridge in its heyday in the middle of the nineteenth century. He also did much to systematise the mathematical theory of mechanics and created several ideas critical to the development of modern control systems theory.*Wik

1834 William Watson born in Nantucket, MA. In 1862 he earned his Ph.D. at the University of Jena, being the first American to receive a Ph.D. in mathematics at a foreign university. Later he taught at Harvard and MIT. In the same year Yale was the first American school to grant a Ph.D. in mathematics (to J. H. Worall).*VFR

1895 Gabor Szego, Professor Emeritus at Stanford. He co-authored with George (originally Gy¨orogy) P´olya the renown book Problems and Theorems in Analysis. *VFR worked in the area of extremal problems and Toeplitz matrices.*SAU

1904 Renato Caccioppoli ( 20 January 1904 – 8 May 1959) was an Italian mathematician. His most important works, out of a total of around eighty publications, relate to functional analysis and the calculus of variations. Beginning in 1930 he dedicated himself to the study of differential equations, the first to use a topological-functional approach. Proceeding in this way, in 1931 he extended the Brouwer fixed point theorem, applying the results obtained both from ordinary differential equations and partial differential equations. *Wik

1931 David M. Lee (20 Jan 1931, ) American physicist who, with Robert C. Richardson and Douglas D. Osheroff, was awarded the Nobel Prize for Physics in 1996 for their joint discovery of superfluidity in the isotope helium-3.*TIS


1590 Giovanni Battista Benedetti died. In one of his books he forecast his death for 1592. Hence, on his deathbed, he recomputed his horoscope and declared that an error of four minutes must have been made in the original data, thus evincing his lifelong faith in the doctrines of judicial astrology.*VFR "The essence of Galileo’s laws of fall can be found in the work of Giambattista Bendetti: *RMAT

1760 John Colson (1680–20 January, 1760) was an English clergyman and mathematician, Lucasian Professor of Mathematics at Cambridge University.
Colson was educated at Lichfield School before becoming an undergraduate at Christ Church, Oxford, though he did not take a degree there. He became a schoolmaster at Sir Joseph Williamson's Mathematical School in Rochester, and was elected Fellow of the Royal Society in 1713. He was Vicar of Chalk, Kent from 1724 to 1740. He relocated to Cambridge and lectured at Sidney Sussex College, Cambridge. From 1739 to 1760 he was Lucasian Professor of Mathematics. He was also Rector of Lockington, Yorkshire.
In 1726 he published his Negativo-Affirmativo Arithmetik advocating a modified decimal system of numeration. It involved "reduction [to] small figures" by "throwing all the large figures 9, 8, 7, 6 out of a given number, and introducing in their room the equivalent small figures \(1\bar{1}, 1\bar{2}, 1\bar{3}, 1\bar{4}\) respectively".
He translated several of Isaac Newton's works into English, including De Methodis Serierum et Fluxionum in 1736. It was Colsen who mistranslated the name of the curve that Maria Gaetana Agnesi called 'versiera' to become the Witch of Agnesi.*Wik

1864 Giovanni Antonio Amedeo Plana (6 November 1781 – 20 January 1864) was an Italian astronomer and mathematician.
His contributions included work on the motions of the Moon, as well as integrals, elliptic functions, heat, electrostatics, and geodesy. In 1820 he was one of the winners of a prize awarded by the Académie des Sciences in Paris based on the construction of lunar tables using the law of gravity. In 1832 he published the Théorie du mouvement de la lune. In 1834 he was awarded with the Copley Medal by the Royal Society for his studies on lunar motion. He became astronomer royal, and then in 1844 a Baron. At the age of 80 he was granted membership in the prestigious Académie des Sciences. He died in Turin. He is considered one of the premiere Italian scientists of his age.
The crater Plana on the Moon is named in his honor.*Wik

1907 Agnes Mary Clerke (10 Feb 1842, 20 Jan 1907) Irish astronomical writer who was a diligent compiler of facts rather than a practicing scientist. Nevertheless, by 1885, her exhaustive treatise, A Popular History of Astronomy in the Nineteenth Century gained international recognition as an authoritative work. In 1903, with Lady Huggins, she was elected an honorary member of the Royal Astronomical Society, a rank previously held only by two other women, Caroline Herschel and Mary Somerville. Her publications included several books and 55 pieces in the Edinburgh Review. She contributed some astronomer biographies to the Dictionary of National Biography and some astronomical entries in the Encyclopaedia Britannica. *TIS

1921 Mary Watson Whitney (11 Sep 1847, 20 Jan 1921) American astronomer who trained with Maria Mitchell and succeeded her as professor and director of the Vassar College Observatory. As Mitchell had before her, Whitney championed science education the advancement of professional opportunities for women. She developed the astronomy department. Four years before her 1910 retirement, there were 160 students and eight different astronomy courses, including some of the first courses anywhere on astrophysics and on variable stars. During her tenure as director, the Observatory staff published 102 papers in major astronomical journals reporting their work on comets, asteroids, and variable stars. From 1896, photographic plates were used to study and measure star clusters.*TIS

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

1944 James McKeen Cattell (25 May 1860, 20 Jan 1944) American psychologist who had a talent in mathematics from a young age, and accordingly applied objective, quantitative methods with his career in experimental psychology. As a university professor, he was the first in America to teach a course in statistical analysis. From 1890, he termed his investigations “mental testing,” with such goals as measuring the amount of pressure required to produce pain, or reaction time to sounds. He developed the order of merit ranking method. Fields in which he applied psychology were broad, including education, business, industry, and advertising. The ideas in Galton's eugenics theory also interested him, and he did support such concepts as sterilization of persons of lower intelligence. He originated professional directories, published scientific periodicals and founded the Science Press (1923) *TIS

1971 Jan Arnoldus Schouten (28 Aug 1883 in Nieuwer Amstel (now part of Amsterdam), Netherlands - 20 Jan 1971 in Epe, The Netherlands) worked on tensor analysis and its applications. He produced 180 papers and 6 books on tensor analysis, applying tensor analysis to Lie groups, general relativity, unified field theory, and differential equations. Influenced by Weyl and Eddington, Schouten investigated affine, projective and conformal mappings. Klein's Erlanger Programm of 1872 looked at geometry as properties invariant under the action of a group. This approach had a large influence on Schouten's approach to his topic. *SAU

2001 Crispin St. John Alvah Nash-Williams (December 19, 1932 – January 20, 2001) was a British and Canadian mathematician. His research interest was in the field of discrete mathematics, especially graph theory. *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

Sunday, 19 January 2020

So You Thought You Knew Everything About Equilateral Triangles, Expanded Version

Catching up on some reading lately I came across an interesting book on Academia called Mysteries of the Equilateral Triangle.  The only mystery is why he chose to use that word in the title, but it does include a lot of obscure historical and mathematical tidbits about this basic geometric standard.  I will share a few I like and add a couple of things I've come across in other places to share as well.

The first is in the historical section where he includes an ancient geometric motif from a temple in Chennai, India.  It shows three interlocking triangles, but notice they are interlocked in a Brunnian link, like the famous Borromean Rings.  While the three can not be separated, no two are individually linked.

Along the way he enumerates lots of interesting things about history and property of equilateral triangles that you may well teach, Viviani's theorem, and one of my favorites, Morley's theorem 

Then he mentions two common recreational challenges.  The harder, in my mind, Find the side length and area of the largest square that can be inscribed in an equilateral triangle with unit side lengths. The easier, perhaps, find the side length and area of an the largest equilateral triangle that can be inscribed in a unit square. 

The two are interestingly tied together. 

The Figure above showed up on the wonderful Futility Closet blog a while back. It shows that two simple maximization problems are curiously related.  Problem one, shown in the equilateral triangle at top is the solution to what is the largest square that can be inscribed in an equilateral triangle with unit side lengths. Problem two, shown in the bottom square is the solution to what is the largest equilateral triangle that can be inscribed in a square with unit side lengths.  The coincidence?  The length of the side of the square in the triangle is \( 2 *\sqrt3 - 3 \) units; and the area of the equilateral triangle in the bottom square, is  \( 2 *\sqrt3 - 3 \) square units. Greg Ross, the mastermind behind the Futility Closet blog credits John Conway with discovering the proof of this relationship.

Digressing a moment from equilateral triangles, another couple of relationships I came across that, somehow, lead me back to the same topic in a roundabout way.
 I'm strolling through my twitter feed, and Colin Beveridge AKA @icecolbeveridge posted one of his always entertaining "Math Ninja" blogs and it was about Ailles Rectangle. He just showed that it was a great memory device to figure out the trigonometric values of 15o and 75o. It is easy to construct and a nice way to verify directly the sum and addition formulas.
 I had seen the Ailles (pronounced like the beverage, by coincidence) rectangle several years before, (it's been around even before I was a teacher, but seems not very well known) and had to do a little research to figure out why Colin's looked different.
I found an article by Jack S. Calcut at Oberlin College. He gave the original Ailles rectangle from the 1971 article, and sure enough it was different. (Chris Maslanka pointed out that the segment in the upper left should be \( \sqrt  3 - 1 \)

Where Colin had a 30-60-90 triangle inscribed in the rectangle, Ailles had used an isosceles right triangle. Both however, contain the three essential triangles necessary to demonstrate, what I believe is it's great power as a classroom demonstration, they contain ALL of the right triangles that exist with rational angles and each side length containing at most one square root.  There is a 30-60-90, a 45-45-90, and a 15-75-90 triangle. That's it, that's all of them, there are no others. And that seems to be impressive as heck to high school students.  "Here they are, memorize this image and you have the whole set!"  And all those Pythagorean triples you know how to create.... None of them have rational angles in degree measure or as multiples of \( \pi \).

Since the theme of the day is coincidences, I noticed that the diagrams of Ailles contained another somewhat well known historical result.; If you look  remove the 15-75-90  triangle  at the top, you are left with a trapezoid that is used in the proof of the Pythagorean Theorem by President James Garfield of the US . Garfield was a professor of mathematics at Hiram College in Ohio for several years before being elected to the Ohio Senate in 1859. He was in congress, not president,when he did the proof which was published in the New England Journal of Education in 1876.

As I was studying Conway's curiosity,  I realized that there was one more coincidence between the diagram and Ailles rectangle.   Using Colin's illustration, if you reflect the 30-60-90 triangle about its longest leg and extend the other lines it will look like this.  

Constructing the Equilateral Triangle at the bottom side of the figure and we have Conway's figure rotated 180 degrees.  So Garfield's Pythagorean proof, with an additional triangle becomes the Ailles Rectangle showing all the rational right triangles with sides with a single quadratic root; and reflecting part of that gives us the square that demonstrates a curious equality between two classic maximization problems.  I imagine you could assemble the whole thing out of tiles available for the elementary school. 

Another little know truth the book reveals is that if you find the Euler Line in a triangle with a 60 degree angle, the Euler line will cut off an equilateral triangle.

Ok, how about creating a parabolic arc with equilateral triangles?  Simply done by erecting equilateral triangles along a straight line with progressive side lengths of the odd numbers.

Another novelty to let your students discover, take an equilateral triangle and circumscribe it in a rectangle.  (there are several ways to do that, so they should do a couple each.  Now find the area of the three new triangles created inside the rectangle but outside the equilateral triangle.  Make an observation.  One they may pick up after some thought, if we call the three area A, B, and C, then there will always be one that is the sum of the other two.  

Two of my favorite equilateral triangle ideas are the arithmetic triangle and Sierpinski's gasket.  

And of course the second can be arrived by coloring in all the odd numbers in one color and the evens in another.   

Ok, How about a mathematical quadrilateral you never (probably) heard of, the equlic quadrilateral.  A quadrilateral ABCD such that AD and BC are of equal length, and angles A and B have a sum of 120 degrees.  

The two lower images show two relations of this shape with Equilateral triangles.  I an equilatera PCD is erected outside the quadrilateral, then PAB is also equilateral.   The bottom shows the midpoints of diagonals P, and Q, and the midpoint of CD will always form an equilateral triangle at all.  Toss "equalic quadrilateral" out at the next math dept meeting and rule the crew.  

Two non-associated pieces  not from the book, I'm writing this on November 7th and just printed up my post for On This Day in Math for November 8th, and it includes a note about the Death of Gino Fano on that date in 1952.  Fano's relation to the equilateral Triangle, of course, is his famous illustration of a finite geometry in which every line had three points and every point lies on three lines.  
One last hat tip to John D Cook who pointed out on Linkedin a connection between the Fermat Primes and the Sierpinski Gasket.  If you've forgotten, the only Fermat numbers now known to be prime are  3, 5, 17, 257, and 65537.  You may also know that these numbers are related to what regular polygons are constructable by straight edge and compass.  Only numbers of the form \(2^k F\) are constructable by classic geometry methods, where F is a product of distinct Fermat Primes .  

So that means there can only be \(2^5=32\) values for F if we allow the empty set to be represented by 1, If you print out those values in binary, and he did, you get a perfect Gasket.  John aligned his on the left margin so his gasket is a right triangle, but it's the same beast.  
So go check out this book, especially if you are a student of geometry, a parent of a geometry student, or a teacher of geometry, or just like me, and you love looking at beautiful math.  And check out John Cook's blogsite as well.  
Oh, and you might drop back by this blog some time just to see that I'm staying busy and out of trouble.  

One application of equilateral triangles is the use of the Warren Truss.  The Warren Truss was designed in 1848 by James Warren and Willoughby Theobald Monzani. This truss consists of longitudinal members joined only by angled cross-members, forming alternately inverted equilateral triangle-shaped spaces along its length. One example of the use is in the new Bridge across the Tennessee River between Paducah and Ledbetter Kentucky on Hwy US 60.

How about some other ideas related to equilateral triangles and circles: 

One blog I follow regularly is Antonio Gutierrez's gogeometry.  If you teach/study/like plane geometry he should be one of your regular references.  

Recently among his posts have been a couple with a related theme, circles inscribed or circumscribed about an equilateral triangle.  I'm listing these because they are each a wonderful relationship, and together give these otherwise somewhat mundane seeming triangles a luster students?teachers/others might miss.  
I will post the problems, but not the proofs, which (if you can't/won't work them out yourself you can find at the links provided to Antonio's site.  

So on we go...
1)  draw a circle and inscribe an equilateral triangle. Now pick any point on the circumference and construct segments from this point to the three vertices.  The sum of the lengths of the two shorter segments will equal the third. The problem, and solution is here

2)  OK:  
Same triangle, same circle, but now we sum the square of the three distances ...????? and they sum to twice the square of a side of the equilateral triangle.  That proof is here.  

3)  And now one with the circle on the inside.  Again, from any point on the circle construct segments to the three vertices of the equilateral triangle.  Again the sum of the squares is related to a side length, but I'll let you chase that down for yourself.  Or you can go to the site here.  

Addendum: John Golden sent a comment with a link to a GeoGebra sketch showing all three.

With that same  diagram, we should point out an interesting and almost trivial relationship that I   think is often overlooked. If you draw a median from either side to  the opposite vertex, the encircle will trident that  median into three equal parts.  The proof for any student in elementary geometry  involves only two relationships. The first is that the incenter divides each of the medians in a ratio of 2:1.  Now use the fact that the distance from the incenter to the encircle  is equal on both sides. 

And let's add one beautiful proof about infinite series that is approachable to clever students at the geometry level.  Analysis by elementary geometry.  Try it with your students.  

I have stated previously how much I like "napkin" techniques that give quick calculations or estimates of a problem. I also like things like visual displays which essentially prove some mathematical idea. The one at the top of the page is from the cover of Roger Nelsen's "Proofs without words II.." which is way too expensive for a paperback, but I will probably break down and buy it.

The problem, is that, except for mathematicians who already know the proof, it seldom convinces "without words". Most of my high school students will not look at this image and be able to explain easily and clearly why it shows that the limit as n goes to infinity of 1/4 + (1/4)2 + (1/4)3+ ... +(1/4)n is equal to 1/3. If I'm wrong, not generally, but in your particular case, then stay with me and read what's written, and tell me if you see it the way I do. The amazing thing in my experience, is that a question about analysis can be illustrated with simple plane geometry.

I think they will be able to see that the triangle is divided into thirds... by the colors, 1/3 purple, 1/3 orange, 1/3 white. But I don't know if they can see the series of powers of 1/4 going off to infinity. That's why they have high school teachers... and so here are some words to help make it more "visual"
Look at the largest white triangle... can you see it is 1/4 of the largest (outside) triangle?... Ok, now look at the line across the top of the biggest white triangle, it connects the midpoints of two legs of an equilateral triangle, so the triangle above this medial segment, the one with multiple smaller triangles in it, is exactly congruent to the Biggest white triangle and is 1/4 of the total area of the outside triangle also. This upper triangle is a scale model of the original outside triangle,with all the same colors in the same positions and the white triangle in it is 1/4 of the area of this upper replica. So the second largest white triangle is 1/4 of the area of the Largest white triangle.... its area is 1/16 or (1/4) 2. Now the line above the second smallest white triangle is a medial segment of the upper triangle, and so the triangle above it, which is also a scale model of the original biggest triangle, is also 1/16 of the total area... and the third smallest white triangle, is 1/4 of that, so its area is 1/4 of 1/16 or (1/4)3... OK, now you see it, and as you move out each white triangle is 1/4 of the previous one... and sure enough, all the white triangles add up to 1/3 of the total area.

And from a slightly different approach, making equilateral triangles from matchsticks.  I posed this question on the appropriate day of the year on my "On This Day in Math " blog.
If you build an equilateral triangle with nine matchsticks on each side, then subdivide into additional equilateral triangles, there will be a total of 235 triangles of several different sizes. The image shows the subdivision of a equilateral triangle with three matchsticks on a side. Can you find the thirteen triangles in it?

Students will see that there is obviously a single triangle when there is only one matchstick per side.  At 2 per side, they get four small triangles and one larger, or four triangles in all.  Above I've given away the number of triangles for lengths of 3 on a side, and 9 on a side.  Can students determine the relationship for the number of triangles with n toothpicks on a side?