Pat'sBlog: 2024

Saturday 27 April 2024

Parabolas, Tangents, and the Wallace-Simson Line

Re-post from 2012, because of several visitors who ask questions that led me to refer them here. Thought it worth re-posting.

The oft-called Simson line was attributed to Simson by Poncelet, but is now frequently known as the Wallace-Simson line since it does not actually appear in any work of Simson. (Oh go on, ask your teacher, so WHY do we still call it the Simson line at all?)
The Wallace for whom the line should more probably be named is William Wallace FRSE (23 September 1768, Dysart—28 April 1843, Edinburgh; the Scottish mathematician and astronomer who invented the eidograph, a more complicated version of the pantograph used to make scale images of drawings. He was a protegee of John Playfair, and teacher to Mary Somerville. He wrote about the line in 1799. He is also not credited for his 1807 proof of a result about polygons with an equal area, which has become the Bolyai–Gerwien theorem. He was also one of the first in England/Scotland to promote the calculus as taught on the Continent.

The theorem says that if a triangle is inscribed in a circle, then if perpendiculars are dropped from a point on this circumcircle to the three sides of the triangle (extended as needed) the feet of these perpendiculars will lie on a straight line. It works the other way too. If you draw a straight line cutting all three sides of the triangle, perpendiculars drawn at these points of intersection will be concurrent at a point on the circumcircle. (With dynamic geometry software, it is relatively easy for students/teachers to create a single line through three sides of a triangle, then construct perpendiculars to the three intersections and make their intersection a traceable point, the rotate the line about ther middle point of the three to get the circumcircle.)

I mentioned recently in a description of David Well's new book, Games and Mathematics, that I keep finding out new stuff. Well, he pointed out a connection between the Wallace line (he uses Simson, but I believe he knows better) and tangents of a parabola.

If you find three tangents to parabola and construct the circumcircle to the triangle formed by their mutual intersections, the circumcircle will pass through the focus of the parabola.
Tricky and cool, but what does that have to do with the the Wallace line? Well if you drop a perpendicular from the focus to ANY tangent, the foot of the perpendicular will always fall on the line tangent to the parabola at the vertex. The tangent at the vertex is a Wallace line for any triangle formed by three tangents to a parabola.

On This Day in Math - April 27

I believe that we do not know anything for certain,but everything probably.
~ Christiaan Huygens

The 117th day of the year; 117 can be written as the difference of prime squares (11² - 2²) or prime cubes (5³ - 2³). *Prime Curios (Can you find another number which can be expressed as both the difference of squared primes and cubed primes?)

117 is the smallest possible length for the longest side of a Heronian tetrahedron (one whose sides are all integers, and all surface areas and volume are rational). The other edges are 51, 52, 53, 80, & 84. (Are the areas / volume integral?)

*Mathworld.Wolfram

An Euler brick, named after Leonhard Euler, is a cuboid whose edge and face diagonals all have integer lengths. A primitive Euler brick is an Euler brick whose edge lengths are relatively prime.The smallest Euler brick, discovered by Paul Halcke in 1719, has edges(a,b,c) = (44, 117, 240) and face diagonals 125, 244, and 267.

EVENTS

1521 In the Philippines, Magellan became involved in a tribal war in which he was killed. His remaining ships returned to Spain in September of 1522 without their leader. *VFR

1610 Martin Horky writes to Kepler saying that Galileo's telescope was admirable for terrestrial observations,but completely failed to show others what the latter claimed for the heavens. *John McCafferty

1657 Christiaan Huygens published De ratiociniis in ludo aleae. *VFR [Download of English version printed in London in pdf]

In 1655 he made his first visit to Paris. He informed the mathematicians in Paris including Boulliau of his discovery and in turn Huygens learnt of the work on probability carried out in a correspondence between Pascal and Fermat. On his return to Holland Huygens wrote a small work De Ratiociniis in Ludo Aleae on the calculus of probabilities, the first printed work on the subject.

1740 The French Academie des Sciences announced that their prize on the ebb and ﬂow of the tides would be shared between Leonhard Euler, Daniel Bernoulli, Antoine Cavalleri, one of the last of the Cartesians, and Colin Maclaurin, then Professor of Mathematics at the University of Edinburgh. [Niccol´o Guiciardini, The Development of Newtonian Calculus in Britain 1700–1800 (1989), p. 69.] *VFR

1783 In a letter to A. M. Lorgna, Gian Francesco Malfatti gave the polar equation concerning the squaring of the circle. [DSB 9, 55] Does this refer to the polar equation of the spiral of Archimedes, r = aθ? *VFR

Malfatti and Lorgna were two of the founders of the "Società Italiana delle Scienze" (1782) and Malfatti was active in academic reform, especially in the Napoleonic period. He is remembered for the Malfetti problem, finding three circles in a triangle that include the largest area. Malfatti applied the distinction that circles had to all be mutually tangent and tangent to two sides of the triangle. Later this solution was chipped apart as better solutions were found by not having the three circles all be cotangent.

*wik

In 1871, the American Museum of Natural History opened to the public in New York City. With a series of exhibits, the Museum's collection Went on view for the first time in the Central Park Arsenal, the Museum's original home, on the eastern side of Central Park. The museum began from the efforts of Albert Smith Bickmore, one-time student of Harvard zoologist Louis Agassiz, who was successful in his proposal to create a natural history museum in Central Park, New York City, with the support of William E. Dodge, Jr., Theodore Roosevelt, Sr., Joseph Choate, and J. Pierpont Morgan. The Governor of New York, John Thompson Hoffman, signed a bill officially creating the American Museum of Natural History on 6 Apr 1869. *TIS

1865 King George V of Hanover visited Gottingen and ordered a commemorative plate placed at the room in which Gauss had died ten years before.

In 1895, Professor Charles F. Marvin, a future chief of the Weather Bureau, began experimenting with kites for routine use in the Bureau. In 1896 he perfected his kite meteorograph, an instrument capable of measuring and recording temperature, pressure and humidity. These measurements were recorded by pens tracings on paper, or on a smoked copper sheet, which was attached to a clock rotated drum. i
n 1898, the first Weather Bureau kite was launched from Topeka, Kansas, and by the end of the year, 16 additional kite stations were attempting daily, early morning, simultaneous observations. The kites were large "box types" with dimensions of 8 feet long, 7 feet wide and 3 feet high. As many as seven kites would be attached to the kite wire during and observation. These kites were placed at regular intervals with the second 1500 feet behind the first, the third 2000 feet behind the second and from there on a spacing of 2500 feet.*TIS

1938 Lev D Landau, head of the Theoretical Division at the Institute for Physical Problems in Russia, was arrested for a leaflet which compared Stalinism to German Nazism and Italian Fascism. The remainder of the story I learned from Lautaro Vergara@VergaraLautaro on Twitter as presented below, although there is a one day difference in his date of the arrest: On 28 April 1938, Landau was arrested. Piotr Kapitsa reacted immediately by sending a letter to Joseph Stalin, but no reaction followed. On 6 April 1939, Kapitsa wrote a letter to Molotov, in which he interceded with him to pay NKVD’s (secret service) attention to “accelerate the Landau’s case.” The reaction was very rapid: in a few days Kapitsa was invited to NKVD, where he was received by a large group of Lavrentiy Beria deputies led by the Head of the NKVD Investigation Unit Kobulov. There were five large volumes of “The Case of Landau” lying on a table. Kobulov suggested that Kapitsa should look through these materials. Kapitsa instantly realized that a discussion would follow the reading of these volumes, without any guarantee of success. Then he made a counter move and asked Kobulov and everyone else present a question “Here you claim that Landau has been a German spy, which is a crime. But every crime has to have a motive. Explain it: what kind of motives could Jew Landau have to become a German spy?” At this point a silent scene followed in the spirit of Gogol’s “Government Inspector.” Kapitsa’s question nonplussed the generals, up to this moment they had not thought about motives of crimes and did not even quite grasp the meaning of this word Kobulov immediately suggested to interrupt the conversation, and in two days he himself asked Kapitsa for a letter to Beria with an appeal to “release from custody the arrested Professor of Physics Lev D. Landau, on a personal guarantee.” In two days, on 28 April 1939, exactly one year after he was arrested, Landau was released. Apparently the puzzled generals could not find an answer to Kapitsa’s question about “motives.” This story was told by Isaak M.Khalatnikov, a close collaborator of Landau, in "From the Atomic Bomb to the Landau Institute"

1961 Patent issued for multilayer circuit boards. Hazeltyne, a US firm, patented hole technology and its use in multi-layer printed circuit board assemblies. The result was that component density increased, and the newly close-spaced electrical paths changed the design of printed circuit board assemblies dramatically.

1962 The Netherlands issued a stamp showing Christiaan Huygens’ Pendulum

Clock as pictured by van Ceulen. [Scott #B365] *VFR

1994 U.S. Companies Get Aid From Government The Clinton administration unveils a multimillion-dollar program to aid U.S. companies that make flat-panel display screens as part of an effort to help the industry stay afloat in light of Japanese domination of 95 percent of the industry. The funding comes partly from the Defense Department, for use of flat screens on military equipment. The flat-panel display market had previously been limited to laptop computers. *CHM

Japanese Assembly Plant

2002 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. Pioneer 10 was 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 final, very weak signal from Pioneer 10 was received on January 23, 2003 when it was 12 billion kilometers (80 AU) from Earth. *Wik

Wave Goodby

BIRTHS

1791 Samuel Finley Breese Morse (27 Apr 1791; 2 Apr 1872 at age 81) was an American artist and inventor who is famous for developing the Morse Code (1838) and independently perfecting an electric telegraph (1832-35). He spent the first part of his life as a portrait artist, and did not turn to science until 1832, when he was past his 40th birthday. He was returning to America from a tour of Europe, when he met Charles T. Jackson on the boat, who inspired him about newly discovered electromagnets. From that point, Morse worked to develop apparatus for electrical communications. Backed by Congress, he erected a line spanning 40 miles between Baltimore, Maryland and Washington D.C. which had its first trial on 23 May 1843. It was ready for public use on 1 Apr 1845. *TIS

1837 Paul Albert Gordan,(27 April 1837 – 21 December 1912) king of the invariant theorists, (died: 1912). He found simpler proofs that π and e are transcendental. Emmy Noether, the ﬁrst woman to get a doctorate in Germany, was his student. *VFR

He was known as "the king of invariant theory". His most famous result is that the ring of invariants of binary forms of fixed degree is finitely generated. Clebsch–Gordan coefficients are named after him and

A famous quote attributed to Gordan about David Hilbert's proof of Hilbert's basis theorem, a result which vastly generalized his result on invariants, is "This is not mathematics; this is theology." The proof in question was the (non-constructive) existence of a finite basis for invariants. It is not clear if Gordan really said this since the earliest reference to it is 25 years after the events and after his death. Nor is it clear whether the quote was intended as criticism, or praise, or a subtle joke. Gordan himself encouraged Hilbert and used Hilbert's results and methods, and the widespread story that he opposed Hilbert's work on invariant theory is a myth (though he did correctly point out in a referee's report that some of the reasoning in Hilbert's paper was incomplete).

1843 Felix Muller He compiled the earliest mathematical calendar (that I know of)*VFR.His advisors were Weierstrass and Kummer.

In 1868 he entered Schellbach's mathematical-pedagogical seminar and in 1869 became an assistant teacher at the Friedrich-Wilhelm-Gymnasium and at a secondary school in Berlin and in 1870 a full teacher. Finally, in 1882 he became a senior teacher and in 1887 a professor at the Luisengymnasium in Berlin. He retired in 1897 and then lived as a private citizen in Oberloschwitz and on the Weißen Hirsch near Dresden . Around 1900 he lived briefly in Berlin-Steglitz .

In 1869, together with Carl Ohrtmann, he founded the Yearbook on the Advances in Mathematics based on the model of advances in physics , of which he was a member of the editorial board until 1906.

As a historian of mathematics, he was primarily concerned with the history of mathematical terminology and bibliographical work.

1875 (6th duke) (Louis-César-Victor-) Maurice de Broglie (27 Apr 1875; died 14 Jul 1960 at age 85.) a French physicist who made many contributions to the study of X rays. While in the navy (1895-1908), he first distinguished himself by installing the first French shipboard wireless. From 1912, his chief interest was X-ray spectroscopy. His “method of the rotating crystal” was an application of Bragg's “focussing effect” to eliminate spurious spectral lines. De Broglie discovered the third L absorption edge (1916), which led to the exploration of “corpuscular spectra.” During 1921-22, he worked with his brother Louis to refine Bohr's specification of the substructure of the various atomic shells. He also did pioneer work in nuclear physics and cosmic radiation.*TIS

1913 Philip Hauge Abelson (April 27, 1913 – August 1, 2004) was an American physicist, scientific editor and science writer. He proposed the gas diffusion process for separating uranium-235 from uranium-238 which was essential to the development of the atomic bomb. In collaboration with the U.S. physicist Edwin M. McMillan, he discovered a new element, later named neptunium, produced by irradiating uranium with neutrons. At the end WW II, his report on the feasibility of building a nuclear-powered submarine gave birth to the U.S. program in that field. In 1946, Abelson returned to the Carnegie Institution and pioneered in utilizing radioactive isotopes. As director of the Geophysics Laboratory of the Carnegie Institution (1953-71), he found amino acids in fossils, and fatty acids in rocks more than 1,000,000,000 years old. *TIS

1920 Mark Alexandrovich Krasnosel'skii (April 27, 1920, Starokostiantyniv – February 13, 1997, Moscow) was a Soviet, Russian and Ukrainian mathematician renowned for his work on nonlinear functional analysis and its applications. *Wik

1932 Gian-Carlo Rota Rota (April 27, 1932 – April 18, 1999) worked on functional analysis for his doctorate and, up to about 1960, he wrote a series of papers on operator theory. Two papers in 1959-60, although still in the area of operator theory, looked at ergodic theory which is an area which requires considerable combinatorial skills. These papers seem to have led Rota away from operator theory and into the area of combinatorics. His first major work on combinatorics, which was to change the direction of the whole subject, was On the Foundations of Combinatorial Theory which Rota published in 1964.

Rota received the Steele Prize from the American Mathematical Society in 1988. The Prize citation singles out the 1964 paper On the Foundations of Combinatorial Theory as:-... the single paper most responsible for the revolution that incorporated combinatorics into the mainstream of modern mathematics. *SAU

DEATHS

1936 Karl Pearson, (27 March 1857 in London, England - 27 April 1936 in Coldharbour, Surrey, England) English mathematician, one of the founders of modern statistics. Pearson's lectures as professor of geometry evolved into The Grammar of Science (1892), his most widely read book and a classic in the philosophy of science. Stimulated by the evolutionary writings of Francis Galton and a personal friendship with Walter F.R. Weldon, Pearson became immersed in the problem of applying statistics to biological problems of heredity and evolution. The methods he developed are essential to every serious application of statistics. From 1893 to 1912 he wrote a series of 18 papers entitled Mathematical Contributions to the Theory of Evolution, which contained much of his most valuable work, including the chi-square test of statistical significance. *TIS l There is a plaque in the church at Crambe in No. Yorkshire where he was born and many of his family are buried.

1978 Guido Stampacchia (March 26, 1922 - April 27, 1978) was a 20th century mathematician. Stampacchia was active in research and teaching throughout his career. He made key contributions to a number of fields, including calculus of variation and differential equations. In 1967 Stampacchia was elected President of the Unione Matematica Italiana. It was about this time that his research efforts shifted toward the emerging field of variational inequalities, which he modeled after boundary value problems for partial differential equations.
Stampacchia accepted the position of Professor Mathematical Analysis at the University of Rome in 1968 and returned to Pisa in 1970. He suffered a serious heart attack in early 1978 and died of heart arrest on April 27 of that year *Wik

Gerard Kitchen O'Neill (February 6, 1927 – April 27, 1992) was an American physicist and space activist. As a faculty member of Princeton University, he invented a device called the particle storage ring for high-energy physics experiments. Later, he invented a magnetic launcher called the mass driver. In the 1970s, he developed a plan to build human settlements in outer space, including a space habitat design known as the O'Neill cylinder. He founded the Space Studies Institute, an organization devoted to funding research into space manufacturing and colonization.*Wik

As a leading advocate of space colonization, he wrote in his book The High Frontier (1978), that space colonies could be the ultimate solution to such terrestrial problems as pollution, overpopulation, and the energy shortage. He designed a 1-km long sealed cylindrical space station to be built primarily of processed lunar materials and using solar energy. It would be capable of sustaining a human colony indefinitely in space between the Earth and the Moon. *TIS

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

1999 Mark David Weiser (23 Jul 1952, 27 Apr 1999 at age 46) American computer scientist and visionary who was the chief technology officer at XEROX PARC, and is remembered for developed the pioneering idea for what he referred to as “ubiquitous computing.” He coined that term in 1988 to describe a future in which personal computers will be replaced with tiny computers embedded in everyday “smart” devices (everyday items such as coffeepots and copy machines) and their connection via a network. He said, “First were mainframes, each shared by lots of people. Now we are in the personal computing era, person and machine staring uneasily at each other across the desktop. Next comes ubiquitous computing, or the age of calm technology, when technology recedes into the background of our lives.” He died at age 46, only six weeks after being diagnosed as having gastric cancer. *TIS

2002 Ruth Marianna Handler (née Mosko; November 4, 1916 – April 27, 2002) was an American inventor who created the Barbie Doll (1959), a teenage doll with a tiny waist and slender hips, and Ken, a boy doll (1961), which she named after her children. She co-founded the Mattel company in 1942. The business originally sold picture frames, and later dollhouse furniture which shortly led to specializing in toys. With a blonde ponytail and a zebra-striped swimsuit, the first "Barbie Teen-Age Fashion Model" sold over 350,000 the first year. The company soon made $100m annually. After being diagnosed with breast cancer in 1970, resulting in a mastectomy, she founded Ruthton Corporation to manufacture and market a prosthetic breast for women with a similar need. *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

Friday 26 April 2024

Sang-Heronian Triangles and some History about Near Equilateral Triangles

The first Near Equilateral triangles with consecutive integer sides and integer area (sometimes called Brahamagupta triangles) was discovered over 2500 years ago. The discovery of the 3,4,5 right triangle seems lost in antiquity back before 500 BC. All Pythagorean triangles are Heronian, but lots (infinitely many) of other triangles that are not right triangles also are Heronian. The second near equilateral triangle, the 13, 14, 15; was known to Heron of Alexandra as early as 70 AD, almost 2000 years ago. Since then, they've grown in number, and to infinity, and been dissected and diagnosed repeatedly. They've even been generalized to three dimensions in Heronian Tetrahedra. Here is one part of their story.

Heron of Alexandria is known to have developed a method of finding the area of triangles using only the lengths of the three sides. It is known that it was proven in his Metrica around 60 AD. His proof was extended in the 7th century by Brahamagupta extended this property to the sides of inscribable quadrilaterals. Since around 1880, the triangular method of Heron has been known as Heron's formula, or Hero's Formula. It emerged in French, formula d'Heron (1883?) and German, Heronisch formel (1875?) and in George Chrystal's Algebra in 1886 in England.

L E Dickson's History of Number Theory states that Heron stated the 13, 14, 15 triangle and gave its area as 84, the height of 12 being the common side of a 5,12,13 triangle and a 9, 12, 15. The 5 and 9 combining to form the base of length 14. Brahmagupta is cited in the same work for giving an oblique triangle composed of two right triangles with a common leg a, stating that the three sides are $ \frac{1}{2}(\frac{a^2}{b}+ b)$ , $ \frac{1}{2}(\frac{a^2}{c}+ c)$, and $ \frac{1}{2}(\frac{a^2}{b}- b) + ( \frac{1}{2}(\frac{a^2}{c}- c)$

In 1621 Bachet took two Pythagorean right triangles with a common leg, 12, 35, 37 and 12, 16, 20 and produced a triangle with sides of 37, 20, 51. With an area of 306 if I did my numbers right.
Vieta and Frans van Schooten, both used the same approach of clasping two right triangles with a common leg; and by the first half of the 18th century, the Japanese scholar, Matsunago, realized that any two right triangles would work, by simply multiplying the sides of each by the hypotenuse of the other, he could juxtapose the two resulting triangles.

In the early 1800's through 1825 the problem was alive and hopping on the Ladies Diary and the Gentleman's Math Companion. One method created right triangles in another triangle to be reassembled into a rational triangle, similar in fact, to the problem that would appear in the 1916 American Mathematical Monthly. (Note; any triangle with rational sides and area can be scaled to become a Heronian triangle.)

In a letter of Oct 21, 1847; Gauss to H. C. Schumacher, he stated a method using circumscribed circles, and found lots of others chose the exact same solutions in their response. E. W. Grebe tabulated a set of 46 rational triangles in 1856. W. A. Whitworth noticed that the 13, 14, 15 triangle of antiquity, that had an altitude of 12, was the only one in which the altitude and sides were all consecutive. (1880)

Somehow, among all those, the contributions of a Professor from Scotland was not observed by Dickson.

The first modern western article I can find on the topic of Near Equilateral triangles with integer sides and area is from Edward Sang which appeared in 1864 in the Transactions of the Royal Society of Edinburgh, Volume 23. I find it interesting that this is only a small aside in a much larger article and that he begins with an approach to examining the angles. Then he arrives at the use of a Pell type equation for approximating the square root of three, $a^2 = 3x^2 + 1 $ and shows that every other convergent in the chain of approximations is a base of a Near Equilateral Triangle, using sides of consecutive integers. The alternate convergents we seek are given by 2/1, 7/4, 26/15, 97/56... each approaching the square root of three more closely, but also each with a numerator that is 1/2 the base of triangle with consecutive integers for sides and integer area. Perhaps it is easier to just use the recurent relation, $n_i=4n_{i-1} - n_{i-2}$ with $n_0=2$, and $n_1=4$ for the actual middle side,2, 4, 14, 52, 194.... The first few such triangles have their even integer base as2x1=2; (1, 2, 3) area 0; 2x2=4; (3, 4, 5); area 12; 2x7=14; (13, 14, 15); area 84; 2x26=52; (51, 52, 53); area 1170... etc. Throughout, he refers to "trigons" rather than triangles, and never invokes the name of Heron throughout.

The next paper using consecutive integer sides was in 1880 by a German mathematician named Reinhold Hoppe, who produced a closed form expression for these almost equilateral Heronian Triangles that was similar to, $ b_n =(2+2\sqrt{3})^n + (2-2\sqrt{3})^n $. His paper calls them "rationales dreieck" (rational triangles) I have not seen the entire paper, and don't know if the term Heronian appeared, or not.

The first American introduction to the phrase "Heronian Triangles", seemed to be an article in the American Mathematical Monthly which posed the introduction as a problem, to divide the triangle whose sides are 52, 56, and 60 into three Heronian Triangles by lines drawn from the vertices to a point within. The problem was posed by Norman Anning, Chillwack, B.C. It then includes a description that suggests it is introducing a new term, "The word Heronian is used in the sense of the German Heronische (with a German citation) to describe a triangle whose sides and area are integral.

The only other mentions of a Heronian triangle in English in a google search before the midpoint of the 20th century revealed a 1930 article from the Texas Mathematics Teacher's Bulletin. It credits a 1929 talk, it seems, by Dr. Wm. Fitch Cheney Jr. who, "discusses triangles with rational area K and integral sides a, b, c, the g.c.f of the sides 1, under the name Heronian triangles." (Dr Cheney published an article in the American Mathematical Monthly in 1929, The American Mathematical Monthly, Vol. 36, No. 1 (Jan., 1929), pp. 22-28) Since any such rational area can be scaled up to an appropriate integer area with integer sides these address the general Heronian Triangle, but still no Near Equilateral, or at least not revealed in the snippet view.

By the 1980's an article in the Fibonacci Quarterly found a way to produce a Fibonacci like sequence, a second order recursive relation to produce the even bases. Letting $B_0 = 2, and B_1 = 4$, the recursion was $ U_{n+2} = 4 U{n+1} + U_n$ . This paper by W. H. Gould of West Virginia University addresses the full scope of consecutive sided integer triangles and mentions Hoppe, but not Professor Sang. Gould's paper seems to be his solution to a problem he had posed earlier in the Fibonacci Quarterly, "of finding all triangles having integral area and consecutive integral sides." (H. W. Gould, Problem H-37, Fibonacci Quarterly, Vol. 2 (1964), p. 124. .)
Gould also mentions two other, seemingly earlier posed problems in other journals which I have yet to explore, and given the opportunity, will do so and return to this spot, If you are impatient, they are

7. T. R. Running, Problem 4047, Amer. Math. Monthly, Vol. 49 (1942), p. 479; Solutions by W. B. Clarke and E. P. Starke, ibid. , Vol. 51 (1944), pp. 102-104.

8. W. B. Clarke, Problem 65, National Math. Mag. , Vol. 9 (1934), p. 63

Gould's article is a wonderful read for the geometry of the incircles and Euler lines in such special triangles is well explored.

These are each candidates to be the first American proposal of these consecutive integer sided triangles, but it seems Gould's paper was the first to expand the full scope of the solutions in any detail.

Some of the characteristics of these I think would be found interesting to HS and MS age students I will spell out below.

As mentioned above, the length of the middle (even) side follows a 2nd order recursive relation $B_n = 4B_{n-1}-B_{n-2}$ so the sequence of these even sides runs 2, 4, 14, 52, 194, 724..... etc. ) is there to represent the degenerate triangle 1,2,3.

Interestingly, the heights follow this same recursive method giving heights of 0, 3, 12, 45, 168....

The height divides the even side into two legs of Pythagorean triangles that make up the whole of the consecutive integer triangle. They are always divide so that one is four greater than the other, or each is b/2 =+/- 2.

Of the two triangles formed by on each side of the altitude, one is a primitive Pythagorean triangle, PPT, and the other is not. The one that is a PPT switches from side to side on each new triangle, alternately with the shorter leg, and then the longer leg. Here are the triangles with the two subdivisions of them with an asterisk Marking the PPT:

Short Base Long small triangle large triangle
3 4 5 *3 4 5
13 14 15 *5, 12, 13 9, 12, 15
51 52 53 24, 45, 51 * 28, 45, 53
193 194 195 *95, 168, 193 99, 168, 195
723 724 725 360, 627, 723 *364, 627, 725

The pattern of the ending digits of 3, 4, 5 repeated twice, and 1,2,3 once by looking at the end number behavior of if the previous two numbers end in 4's or a four followed by a two.

In the 1929 article mentioned above, Dr. Cheney writes that he knows of no examples of Heronian triangles up to that time that were not made up of two right triangles, and then gives an example of one that is not decomposable, 25, 34, 39. He also points out that the altitudes of Heronian triangles are not always integers, and gives the example of 39,58,95 as an example which I calculate to be 4.8.

A paper by Herb Bailey and William Gosnell in Mathematics Magazine, October 2012 demonstrates Heronian triangles in other arithmetic progressions from the near-equilateral ones.

I mentioned that there are also Heronian Tetrahedra, although that use of Heronian seems even later than for triangles, perhaps as late as 2006. The earliest example of an exact rational tetrahedra with all integer edges, surfaces and volume was by Euler. He created a tetrahedron formed by three right triangles parallel to the xyz coordinate axes, and one oblique face connecting them. The triple right angle edges were 153, 104, and 672, and the three edges of the oblique face were 185, 680, and 697. These were each Pythagorean right triangles, the four faces of (104,672,680), (153,680,697), (153,104,185) and (185,672,697)

There are an infinite number of these Eulerian Birectangular tetrahedra, but they seem to get very large very quickly. Euler showed that they can be found by deriving the three axis-parallel sides a, b, and c by using four numbers that are the equal sums of two fourth powers. Euler found an example using

59^{4}+158^{4}=133^{4}+134^{4}

, and that's the easy part. Then he constructed the three monster lengths of 386678175, 332273368, and 379083360, Yes, those numbers are each in the hundreds of millions, and each pair had a larger hypotenuse to form a third side.
And as the near end of the Wikipedia discussion of these states, "A complete classification of all Heronian tetrahedra remains unknown."

On This Day in Math - April 26

Mathematics is like childhood diseases. The younger you get it, the better.

~Arnold Sommerfeld

The 116th day of the year; 116! + 1 is prime! *Prime Curios (Students might investigate how often n!+1 is prime)
And:
116^2 + 1 is prime

The number 1 appears 116 times in the first 1000 digits of pi. Thanks to *Math Year-Round ‏@MathYearRound

Impress your History teacher, the 100 Years war between France and England..... lasted 116 years.

and Jiroemon Kimura died in 2013 in Japan. He was 116 years old. Two years later his record was broken by an even older Japanese citizen who died.

And for a bit of Americana, from a British web site called *isthatabignumber.com.. It's about Hyperion, a tree that is 116 meters tall.

EVENTS

1514 Nicolaus Copernicus (1473-1543) made his first observation of Saturn. Copernicus later proposed that the sun is stationary and that the Earth and the planets move in circular orbits around it. *astronomy.wikia.com Saturn_Project

1760 Euler was asked to tutor the niece of Frederick the Great, the Princess of Anhalt-Dessau. Euler wrote over 200 letters to her in the early 1760s. On this date he sent the third of these letters. The letter covered the physics of sound and he gave a speed of one thousand feet per second. He closes by telling the Princess that we are incapable of hearing a string vibrating at less than 30 vibrations per second, or one that is more than 7552 vibrations per second. Euler started the first letter with an explanation of the concept of "size". Starting with the definition of a foot, he defined the mile and the diameter of the earth as a unit in terms of foot and then calculated the distance of the planets of the Solar System in terms of the diameter of the earth.

1766 D’Alembert after writing to Frederick II in praise of Lagrange writes to Lagrange about an offer to move to Berlin:

My dear and illustrious friend, the king of Prussia has charged me to write you that, if you would like to come to Berlin to occupy a place in the Academy, he would give you a pension of 1,500 crowns, which are 6,000 French pounds … Mr Euler, unhappy for reasons of which I do not know the details, but in which I see that everyone thinks him wrong, requests permission to leave and wants to go to St. Petersburg. The king, who was not too anxious to grant it, would definitely give it to him if you accept the proposition that he has made

Frederick II of Prussia had more than once invited both d’Alembert and Lagrange to move to Berlin. The encyclopaedist had declined the offer and suggested the name of his Turinese friend. But Lagrange, even though he was on good terms with Euler, did not relish a "cohabitation" with him in the Berlin Academy. *Mauro ALLEGRANZA, Stack Exchange

D'Alembert

1826 The first class of 10 students graduated from Renssalaer Polytechnic Institute on 26 Apr 1826. The Renssalaer School was founded in 1824 in Troy, N.Y., by Stephen van Renssalaer becoming the first engineering college in the U.S. It opened on 3 Jan 1825, with the purpose of instructing persons, who may choose to apply themselves, in the application of science to the common purposes of life." The first director and senior professor was Amos Eaton who served from Nov 1824 - 10 May 1842. The name of Renssalaer Institute was adopted on 26 Apr 1832, and Renssalaer Polytechnic Institute on 8 Apr 1861. *TIS

1861 Richard Owen gives the longest ever discourse at a Royal Institution lecture, ‘On the Scope and Appliances of a National Museum of Natural History’.
Discourse speakers were supposed to aim to speak for exactly one hour but Owen kept talking for two. (It may be coincidence but this is the last discourse he gave.) *Royal Institution web page

1882, the photophone was demonstrated by Alexander Graham Bell and Charles Sumner Tainter. In their device, a mirrored silver disc was made to vibrate by speech from a speaking tube. Light reflected off the disc was focused by a parabolic dish onto a selenium photocell. The variations in the reflected light were converted into electrical signals carried to headphones.

It was invented jointly by Alexander Graham Bell and his assistant Charles Sumner Tainter on February 19, 1880, at Bell's laboratory at 1325 L Street in Washington, D.C. Both were later to become full associates in the Volta Laboratory Association, created and financed by Bell.

While honeymooning in Europe with his bride Mabel Hubbard, Bell likely read of the newly discovered property of selenium having a variable resistance when acted upon by light, in a paper by Robert Sabine as published in Nature on 25 April 1878. In his experiments, Sabine used a meter to see the effects of light acting on selenium connected in a circuit to a battery. However Bell reasoned that by adding a telephone receiver to the same circuit he would be able to hear what Sabine could only see.

A photophone receiver and headset, one half of Bell and Tainter's optical telecommunication system of 1880

1892 Hermite to Stieltjes: “You state this result and then try to mortify me by saying that it is easy to prove. Since I can’t succeed in doing it I appeal to your good nature to help me out of this difficulty.” [Two Year Journal, 11, 49] *VFR (Boy, haven't we all been there?)

Charles Hermite

1920 Shapley and Curtis debate the nature of the nebulae. In astronomy, the Great Debate, also called the Shapley–Curtis Debate, was an influential debate between the astronomers Harlow Shapley and Heber Curtis which concerned the nature of spiral nebulae and the size of the universe.

Shapley was arguing in favor of the Milky Way as the entirety of the universe. He believed that "spiral nebulae" such as Andromeda were simply part of the Milky Way. He could back up this claim by citing relative sizes—if Andromeda were not part of the Milky Way, then its distance must have been on the order of 108 light years—a span most contemporary astronomers would not accept.

Curtis, on the other hand, contended that Andromeda and other such as "nebulae" were separate galaxies, or "island universes" (a term invented by the 18th-century philosopher Immanuel Kant, who also believed that the "spiral nebulae" were extragalactic). He showed that there were more novae in Andromeda than in the Milky Way. From this, he could ask why there were more novae in one small section of the galaxy than the other sections of the galaxy, if Andromeda were not a separate galaxy but simply a nebula within Earth's galaxy.

Later in the 1920s, Edwin Hubble showed that Andromeda was far outside the Milky Way by measuring Cepheid variable stars, proving that Curtis was correct. It is now known that the Milky Way is only one of as many as an estimated 200 billion (2×1011)[6] to 2 trillion (2×1012) or more galaxies in the observable universe. more here.

Shapley

Curtis

1921 the first U.S. broadcast of the weather was made from St. Louis, Missouri, over station WEW for the federal government. *TIS
Radio Station WEW, the original radio station of Saint Louis University, played an important role in the history of early radio. In 1921 it became only the second radio station in the U.S. and the first station west of the Mississippi River. In 1939 it became the first station to broadcast Sacred Heart Radio, a Catholic religious program which eventually grew to include over a thousand stations around the world. Finally, in 1947 WEW became the first FM radio station in St. Louis.

1962 The UK became the world's third spacefaring country, after the US and the USSR, with the launch of the satellite Ariel 1. It was built by Nasa in collaboration with British scientists to study the properties of the upper atmosphere and cosmic rays, and formed the first of six missions. "The big legacy is that, despite the fact we are a relatively small country, we are a major international player in space research," said Martin Barstow, an astrophysicist and head of the college of science and engineering at the University of Leicester. *The Guardian

*NASA

1968 Time magazine (p. 41) reports a “Trial by Mathematics” in which a couple was convicted on the basis of mathematical probability. Later the reasoning was found to be incorrect. The discussion there is of interest. See also Journal of Recreational Mathematics, 1(1968), p. 183. *VFR See details here.

1985 A 22-cent commemorative stamp for Public Education in America issued in Boston.

1986 Nuclear reactor number 4 at Chernobyl, USSR, exploded and released a large amount of radioactive material into the atmosphere. [A. Hellemans and B. Bunch. The Timetables of Science, p. 597].

BIRTHS

1711 David Hume, (7 May[O.S. 26 April]1711,– 25 August 1776) was a Scottish philosopher, historian, economist, and essayist, known especially for his philosophical empiricism and skepticism. He was one of the most important figures in the history of Western philosophy and the Scottish Enlightenment. Hume is often grouped with John Locke, George Berkeley, and a handful of others as a British Empiricist *Wik

1832 Robert Tucker (26 April 1832 in Walworth, Surrey, England - 29 Jan 1905 in Worthing, England) A major mathematical contribution made by Tucker was his work as editor of William Kingdon Clifford's papers. Fifty-seven of Clifford's papers were collected and edited by Tucker and published in 1882 as Mathematical Papers. Tucker also wrote many biographies including those of Gauss, Sylvester, Chasles, Spottiswoode, and Hirst, all of which appeared in Nature. But, like a number of schoolmaster's at this time, he also made a contribution to research in geometry. He wrote over 40 research papers which were published in leading journals. These papers, although sometimes not of the highest quality, do contain a number of interesting ideas. Hill specially singles out for special mention his work on the Triplicate-Ratio Circle, the group of circles sometimes known as Tucker Circles, and the Harmonic Quadrilateral. *SAU

1874 Edward Vermilye Huntington (April 26 1874, Clinton, New York, USA – November 25, 1952, Cambridge, Massachusetts, USA) . This enthusiastic and innovative teacher was professor of mechanics at Harvard from 1919 to 1941. He made many contributions to the logical foundations of mathematics. His book, The Continuum (1917), was the standard introduction to set theory for many years. In 1928 he recommended the “method of equal proportion” for the apportionment of representatives to Congress; in 1941 this method was adopted by Congress. *VFR (now often called the Huntington-Hill method)

1879 Sir Owen Willans Richardson (26 Apr 1879; 15 Feb 1959 at age 79) English physicist who was awarded the Nobel Prize for Physics in 1928 for “his work on the thermionic phenomenon [electron emission by hot metals] and especially for the discovery of the law named after him.”This effect is why a heated filament in a vacuum tube releases a current of electrons to travel an anode, which was essential for the development of such applications as radio amplifiers or a TV cathode ray tube. Richardson's law mathematically relates how the electron emission increases as the absolute temperature of the metal surface is raised. He also conducted research on photoelectric effects, the gyromagnetic effect, the emission of electrons by chemical reactions, soft X-rays, and the spectrum of hydrogen.*TIS

1889 Ludwig Josef Johann Wittgenstein (26 April 1889 – 29 April 1951) was an Austrian-British philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language.*Wik This noted philosopher introduced the word “tautology” in his Tractatus Logico Philosophicus of 1921. *VFR

1900 Charles Richter(April 26, 1900, Hamilton, Ohio - September 30, 1985, Pasadena, California ) This American seismologist developed the earthquake magnitude scale which bears his name. *VFR The scale is logarithmic (base ten). When an earthquake occurs, the maximum amplitude of the shake is measured on a seismometer and assigned a Richter number. A quake with a value of 5 on the Richter scale is 10 times more powerful than a quake with a value of 4. The choice of a log scale seems to have come from his associate, Beno Gutenberg,

1922 Asger Hartvig Aaboe (April 26, 1922 – January 19, 2007) was a historian of the exact sciences and mathematician who is known for his contributions to the history of ancient Babylonian astronomy. He studied mathematics and astronomy at the University of Copenhagen, and in 1957 obtained a PhD in the History of Science from Brown University, where he studied under Otto Neugebauer, writing a dissertation "On Babylonian Planetary Theories". In 1961 he joined the Department of the History of Science and Medicine at Yale University, serving as chair from 1968 to 1971, and continuing an active career there until retiring in 1992. In his studies of Babylonian astronomy, he went beyond analyses in terms of modern mathematics to seek to understand how the Babylonians conceived their computational schemes.*Wik

1933 Arno Allan Penzias (26 Apr 1933, ) is a German-American astrophysicist who shared one-half of the 1978 Nobel Prize for Physics with Robert Woodrow Wilson for their discovery of a faint electromagnetic radiation throughout the universe. Their detection of this radiation lent strong support to the big-bang model of cosmic evolution. (The other half of the prize was awarded to Pyotr Kapitsa for unrelated research.)*TIS

1938 Manuel Blum (26 April 1938; Caracas, Venezuela -) is a computer scientist who received the Turing Award in 1995 "In recognition of his contributions to the foundations of computational complexity theory and its application to cryptography and program checking".
Blum attended MIT, where he received his bachelor's degree and his master's degree in EECS in 1959 and 1961 respectively, and his Ph.D. in Mathematics in 1964 under professor Marvin Minsky.
He worked as a professor of computer science at the University of California, Berkeley until 1999. In 2002 he was elected to the United States National Academy of Sciences.
He is currently the Bruce Nelson Professor of Computer Science at Carnegie Mellon University, where his wife, Lenore Blum, and son, Avrim Blum, are also professors of Computer Science. *Wik

DEATHS

1600 Cunradus Dasypodius ((c. 1530–1532 – April 26, 1600) whose fame is based on the “construction of an ingeneous and accurate astronomical clock in the cathedral of Strasbourg, installed between 1571 and 1574.” *VFR The Strasbourg astronomical clock is located in the Cathédrale Notre-Dame of Strasbourg, Alsace, France. The current, third clock dates from 1843. Its main features, besides the automata, are a perpetual calendar (including a computus), an orrery (planetary dial), a display of the real position of the Sun and the Moon, and solar and lunar eclipses. The main attraction is the procession of the life-size figures of Christ and the Apostles which occurs every day at 12:30pm,(not sure if I read this right, but that seems to be when the clock reads noon (corrections anyone?))*Wik
[A minor point on language, the "orrery" was proabably not so-named in that period, according to a post at the Univ of Penn Library, "The name Orrery comes from the following train of facts. When George Graham, the celebrated London mechanic and watchmaker, employed one Rowley to construct his planetarium, said Rowley retained a model, and was afterward patronized by Charles Boyle, Earl of Orrery, in making a large machine which, though only representing one or two of the heavenly bodies, was sold to George the First for a thousand guineas. Sir Richard Steele in the work entitled "A New and General Biographical Dictionary", published in 1761, attributed this invention to the Earl of Orrery. Hence compilers of the British Encyclopaedia, which was republished in Philadelphia, followed his lead and such machines have since been known as Orreries. ]

1815 Carsten Niebuhr (March 17, 1733 Lüdingworth – April 26, 1815 Meldorf, Dithmarschen), German mathematician, cartographer, and explorer in the service of Denmark. Niebuhr's first book, Beschreibung von Arabien, was published in Copenhagen in 1772, the Danish government providing subsidies for the engraving and printing of its numerous illustrations. This was followed in 1774 and 1778 by the two volumes of Niebuhr's Reisebeschreibung von Arabien und anderen umliegenden Ländern. These works (particularly the one published in 1778), and most specifically the accurate copies of the cuneiform inscriptions found at Persepolis, were to prove to be extremely important to the decipherment of cuneiform writing. Before Niebuhr's publication, cuneiform inscriptions were often thought to be merely decorations and embellishments, and no accurate decipherments or translations had been made up to that point. Niebuhr demonstrated that the three trilingual inscriptions found at Persepolis were in fact three distinct forms of cuneiform writing (which he termed Class I, Class II, and Class III) to be read from left to right. His accurate copies of the trilingual inscriptions gave Orientalists the key finally crack the cuneiform code, leading to the discovery of Old Persian, Akkadian, and Sumerian. *Wik

1876 Osip Ivanovich Somov (1 June 1815 in Otrada, Moscow gubernia (now oblast), Russia - 26 April 1876 in St Petersburg, Russia) Somov was the first in Russia to develop a geometrical approach to theoretical mechanics. He studied the rotation of a solid body about a point, studying examples arising from the work of Euler, Poinsot, Lagrange and Poisson. Other topics Somov studied included elliptic functions and their application to mechanics. *SAU

1902 Lazarus Immanuel Fuchs (5 May 1833 – 26 April 1902) was a German mathematician who contributed important research in the field of linear differential equations. He was born in Mosina (located in Grand Duchy of Poznań) and died in Berlin, Germany.
He is the eponym of Fuchsian groups and functions, and the Picard–Fuchs equation; Fuchsian differential equations are those with regular singularities. Fuchs is also known for Fuchs's theorem. *Wik

1920 Srinivasa Aaiyangar Ramanujan died at age 32. This self educated mathematician, who was discovered by G. H. Hardy of Cambridge, is remembered for his notebooks crammed with complicated identities. *VFR
Although self-taught, he was one of India's greatest mathematical geniuses. He worked on elliptic functions, continued fractions, and infinite series. His remarkable familiarity with numbers, was shown by the following incident. While Ramanujan was in hospital in England, his Cambridge professor, G. H. Hardy, visited and remarked that he had taken taxi number 1729, a singularly unexceptional number. Ramanujan immediately responded that this number was actually quite remarkable: it is the smallest integer that can be represented in two ways by the sum of two cubes: 1729=1³+12³=9³+10³ *TIS
I later learned from a blog at John D. Cooks The Endeavour blog that there is a little more to the story. Here is how John writes it:

This story has become famous, but the rest of the conversation isn’t as well known. Hardy followed up by asking Ramanujan what the corresponding number would be for 4th powers. Ramanujan replied that he did not know, but that such a number must be very large.

Hardy tells this story in his 1937 paper “The Indian Mathematician Ramanujan.” He gives a footnote saying that Euler discovered 635318657 = 158^4 + 59^4 = 134^4 + 133^4 and that this was the smallest number known to be the sum of two fourth powers in two ways. It seems odd now to think of such questions being unresolved. Today we’d ask Hardy “What do you mean 653518657 is the smallest known example? Why didn’t you write a little program to find out whether it really is the smallest?”

His readers seem to find that Euler was correct. No surprise there.

1946 Louis Bachelier,(March 11, 1870 – April 28, 1946);the French mathematician, is now recognized internationally as the father of financial mathematics,..Bachelier was ahead of his time and his work was not appreciated in his lifetime. In the light of the enormous importance of international derivative exchanges (where the pricing is determined by financial mathematics) the remarkable pioneering work of Bachelier can now be appreciated in its proper context and Bachelier can now be given his proper place. *SAU

1951 Arnold (Johannes Wilhelm) Sommerfeld (5 Dec 1868, 26 Apr 1951 at age 82) was a German physicist whose atomic model permitted the explanation of fine-structure spectral lines. His first work was on the theory of the gyroscope (with Klein), and then on wave spreading in wireless telegraphy. More significant was his major contribution to the development of quantum theory, generally, and in its application to spectral lines and the Bohr atomic model. He evolved also a theory of the electron in the metallic state valuable to the study of thermo-electricity.*TIS

1976 Carl Benjamin Boyer (November 3, 1906 – April 26, 1976) was a historian of sciences, and especially mathematics. David Foster Wallace called him the "Gibbon of math history". He wrote the books History of Analytic Geometry, The History of the Calculus and Its Conceptual Development, A History of Mathematics, and The Rainbow: From Myth to Mathematics. He served as book-review editor of Scripta Mathematica. *Wik
His History of analytic Geometry is excellent.

1980 Stanisław Gołąb (July 26, 1902 – April 30, 1980) was a Polish mathematician from Kraków, working in particular on the field of affine geometry.
In 1932, he proved that the perimeter of the unit disc can take any value in between 6 and 8, and that these extremal values are obtained if and only if the unit disc is an affine regular hexagon. *Wik

1988 Guillermo Haro Barraza ( 21 March 1913 – 26 April 1988) was a Mexican astronomer who was working as a newspaper reporter, when he interviewed (1937) Luis Erro of Tonantzintla Observatory. By 1943, Haro’s increasing interest in astronomy was rewarded with a staff position there, despite no formal training. His name remains associated with Herbig-Haro objects, that he and George Herbig discovered independently. These seemed to be stars much younger than the rest of the stars in the sky, and had distinquishing anomalies in their spectra which remained unexplained for many years. Haro’s career of contributions marked the emergence of serious astronomy in Mexico, recognized when he was elected (1959) as the first foreign associate of the Royal Astronomical Society from a developing country. *TIS

2006 Yuval Ne'eman (14 May 1925, 26 Apr 2006 at age 80) Israeli theoretical physicist, who worked independently of Gell-Mann but almost simultaneously (1961) devised a method of grouping baryons in such a way that they fell into logical families. Now known as the Eightfold Way (after Buddha's Eightfold Path to Enlightenment and bliss), the scheme grouped mesons and baryons (e.g., protons and neutrons) into multiplets of 1, 8, 10, or 27 members on the basis of various properties. He had served as the head of his Israel's atomic energy commission, and founded the country's space program.*TIS

Pat'sBlog