On This Day in Math - June 16

In a world in which the price of calculation

continues to decrease rapidly, but the price of theorem proving continues to hold steady or increase, elementary economics indicates that we ought to spend a larger and larger fraction of our time on calculation.

John Tukey

EVENTS

1497   Amerigo VESPUCCI (1454-1512) was born in one of the Vespucci houses in Borgo Ognissanti.  He is said to have made four voyages to the New World.  He reported sighting the South American mainland on 16 Jun 1497, a week before Cabot reached North America, which led to his name being attached to the New World in Martin Waldseemüller's Cosmographiæ Introductio of 1507 – but many authorities doubt that Vespucci ever made this voyage.  Waldseemüller realised that he had overrated Vespucci's accomplishments and removed the name from later versions of his map, but it was too late.  Simonetta Vespucci, who married his distant cousin was a celebrated beauty, immortalised in Botticelli's paintings – his 'Mars and Venus' in the Uffizi shows little wasps ('vespucci') circling the head of Mars.  Florence's airport, in the NW suburb of Peretola, is named Amerigo Vespucci.

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1641  In a letter to Fr. Marin Mersenne, Descartes states that no prime of the form 12n ± 1 will divide a number that is one more than a power of three. He adds that 12n ± 5 will always divide some 3^X +1.  He gives a similar rule for five, and states he has one for all primes.  (History of the theory of numbers,  By Leonard Eugene Dickson)

1657, the first pendulum clock was patented  by its inventor, Christiaan Huygens. Although others may have worked in this field before him, Huygens made major advances in building a practical clock. He needed time accuracy for his astronomical measurements.*TIS

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1799 Gauss awarded his Ph.D. at age 22, the usual requirement of an oral exam being dropped. His dissertation gave the first correct proof of the fundamental theorem of algebra. *VFR

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1833 Janos Bolyai was retired as Captain in the cavalry for dueling with thirteen other officers. He accepted their challenge on the condition that he be allowed to play his violin between duels. [Bonola, Non-Euclidean Geometry, Appendix 1, p. xxix]*VFR

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1854 For the first time in more than twenty years, Gauss left Gottingen. He went to see the railway between between Cassel and Gottingen that was under construction. VFR

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1867 A Memorial to Leonardo Bigolli (Fibonacci) was erected in Pisa.  The monument includes a 1241 decree by the commune of Pisa that bestowed an annual salary to Leonardo,  "In consideration of the honor brought to the city and its citizens and their betterment by the teaching and zealous cooperation of that discrete and learned man."  *The Man of Numbers, Keith Devlin

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1885 The first gravity-powered American roller coaster that was commercially successful was put in operation at Coney Island, N.Y., the invention of La Marcus Thompson (patent No. 310,966). Passengers rode a train on undulating tracks over a wooden structure 600-ft long. The train started at a height of 50-ft on one end and ran downhill by gravity until its momentum died. Passengers then left the train and attendants pushed the car over a switch to a higher level. The passengers returned to their sideways facing seats and rode back to the original starting point. Admission on the Thompson Switchback Railway was 5 cents and he grossed an average of $600 / day. Within 4 yrs he had built about 50 more across the U.S. and in Europe.

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1902 Bertrand Russell wrote Gottlob Frege that in his Grundgesetze der Arithmetik “there is just one point where I have encountered a difficulty.” The difficulty is the Russell Antinomy, a logical contradiction. See 22 June 1902.  Russell had found a class of contradictions to Frege's 1879 Begriffsschrift. This contradiction can be stated as "the class of all classes that do not contain themselves as elements".

1933   FDR signed the Banking Act, which separated commercial banking from investment banking and established the Federal Deposit Insurance Corporation. He also signed the Farm Credit Act, the Emergency Railroad Transportation Act, and the National Industrial Recovery Act (which created the Public Works Administration).

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1963 Valentina Tereshkova became the first woman in space. She was aboard the Soviet Union’s Vostok 6. See 18 June 1983.

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1973 Afghanistan issued a postage stamp commemorating the millennium of the birth of Ab'u Rayhan Muhammad ibn Ahmad Al Bırunı (born 4 September 973, died after 1050), author of books on arithmetic, geometry, trigonometry, astronomy and geography. [Scott #881].

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BIRTHS

1640  Jacques Ozanam was born in Sainte-Olive, Ain, France.

All his books sold well and ran to many editions, especially his famous works Dictionnaire mathématique (1691), the five volume work Cours de mathématiques (1693) and Récréations mathématiques et physiques (1694). It is certainly for this last work on recreational mathematics that Ozanam will be most remembered. The precursor of books to follow for the next 200 years, he published it in four volumes in 1694 and it later went through at least ten editions. Ozanam based his book on earlier works by Bachet, Mydorge, Leurechon, and Schwenter. It was later revised and enlarged by Montucla, then translated into English by Hutton (1803, 1814).

Ozanam's original edition contained an early example of a problem about orthogonal Latin squares:-

Arrange the 16 court cards so that each row and each column contains one of each suit and one of each value.

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1801 Julius Plucker born in Elberfeld, Germany. He was a geometer who worked in analytic andprojective geometry, and on the theory of plane curves.*VFR  He  was a pioneer in the investigations of cathode rays that led eventually to the discovery of the electron. He also vastly extended the study of Lamé curves. *VFR (Lame curves are curves with equations  of the form (x/a)^n + (y/b)^n = 1.  He investigated n for both rational and irrational values.  Piet Hein's   "super-ellipse" is an example of a Lame curve.)

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1830 Alfred Enneper born. He worked on elliptic functions and differential geometry. *VFR

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1839  Julius Petersen was a Danish mathematician who worked on geometry and graph theory. He is best remembered for the Petersen graph. 

In the mathematical field of graph theory, the Petersen graph is an undirected graph with 10 vertices and 15 edges. It is a small graph that serves as a useful example and counterexample for many problems in graph theory. The Petersen graph is named for Julius Petersen, who in 1898 constructed it to be the smallest bridgeless cubic graph with no three-edge-coloring. Although the graph is generally credited to Petersen, it had in fact first appeared 12 years earlier, in a paper by A. B. Kempe (1886).
Donald Knuth states that the Petersen graph is "a remarkable configuration that serves as a counterexample to many optimistic predictions about what might be true for graphs in general. *Wik

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1888  Alexander Alexandrovich Friedmann Russian mathematician who was the first to work out a mathematical analysis of an expanding universe consistent with general relativity, yet without Einstein's cosmological constant. In 1922, he developed solutions to the field equations, one of which clearly described a universe that began from a point singularity, and expanded thereafter. In his article On the Curvature of Space received by the journal Zeitschrift für Physik on 29 Jun 1922, he showed that the radius of curvature of the universe can be either an increasing or a periodic function of time. In Jul 1925, he made a record-breaking 7400-m balloon ascent to make meteorological and medical observations. A few weeks later he fell ill and died of typhus. *TIS  (His date of birth is often given as 29 June. However this is an error which came about in converting the "Old Style" Russian date to the "New Style" date, which requires an addition of 12 days.)

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 1915  John Wilder Tukey (June 16, 1915 – July 26, 2000) was an American statistician.  He was awarded the IEEE Medal of Honor in 1982 "For his contributions to the spectral analysis of random processes and the fast Fourier transform (FFT) algorithm."
Tukey retired in 1985. He died in New Brunswick, New Jersey Tukey coined many statistical terms that have become part of common usage, but the two most famous coinages attributed to him were related to computer science.
While working with John von Neumann on early computer designs, Tukey introduced the word "bit" as a contraction of "binary digit". The term "bit" was first used in an article by Claude Shannon in 1948.
The term "software", which Paul Niquette claims he coined in 1953, was first used in print by Tukey in a 1958 article in American Mathematical Monthly, and thus some attribute the term to him;  He also is credited with the terms ANOVA, and boxplot. *Wik

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DEATHS

1910 Julius Weingartnen died. He worked on differential geometry.*VFR

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1948  Marcel Brillouin worked on topics ranging from history of science to the physics of the earth and the atom. *SAU

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1970 Sydney Chapman English mathematician and physicist noted for his research in geophysics. After graduation (1910) he worked at the Greenwich Observatory, but returned to Cambridge upon the outbreak of WW I. Between 1915 and 1917 he completed a series of important papers on thermal diffusion and the fundamentals of gas dynamics. He developed systematic approximations to the Maxwell-Boltzmann formulation for the velocity distribution function for interacting particles under general force laws. During WW II he worked on military operational research and incendiary bomb problems. Chapman's main area of research was geomagnetism, beginning in 1913 and extending to terrestrial and interplanetary magnetism, the ionosphere and the aurora borealis.*TIS

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1977 Wernher Magnus Maximilian von Braun (23 Mar 1912; 16 Jun 1977 at age 65) was a German-American rocket engineer who was one of the most important developers of rockets and their evolution to applications in space exploration. His interest began as a teenager in Germany, and during WW II he led the development of the deadly V–2 ballistic missile for the Nazis (which role remains controversial). After war, he was taken to use his knowledge to produce rockets for the U.S. Army. In 1960, he transferred to the newly formed NASA and became director of Marshall Space Flight Center and chief architect of the Saturn V launch vehicle used to put men on the moon. His contributions include the Explorer satellites; Jupiter, Pershing, Redstone and Saturn rockets, and Skylab. *TIS

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Credits:
*VFR = V Frederick Rickey, USMA
*TIS= Today in Science History
*Wik = Wikipedia
*SAU=St Andrews Univ. Math History

Gauss' Missing Blackboard

I just came across a document on the web that I can't find the source of but it has a really nice story about Gauss and the constructable 17-gon story. It seems to be related to someone named Alex Anderson...(if you know that person, he has a picture I want a copy of very much)..
"Gauss was visiting Braunschweig (Brunswick) and still lying in bed on 29 Mar 1796 when he realised the connection of the cyclotomic polynomial to the construction of the n-gon and how to construct the 17 gon. Ahrens [p. 11] says Gauss sent the slate on which he had done the calculation to Wolfgang Bolyai, who preserved it." Does anyone know where that slate is today?

More Gauss

Just updated a couple of mild corrections to Dave Renfro's paper on x^17=1.  While I was at it I found this little quote about Gauss telling his professor that he had done so.

The brilliant mathematician Karl Friedrich Gauss once visited his professor and claimed to have constructed a heptendecagon (a seventeen-sided figure). "Nonsense," the professor replied. "That is impossible." "Well, then," Gauss persisted. "I have just figured out how to resolve a seventeenth degree polynomial." "Bah, trivial," the professor replied. "I've done it myself."  

Gauss later repaid this professor, an amateur poet, with a dubious compliment: "He is the finest poet among mathematicians, and the finest mathematician among poets."

Gauss and Constuctable Polygons

By special request, for Steven Colyer who wrote, "I WILL point out one thing I wish you'd mentioned, and that was Gauss' accomplishment with the number 17. "

I assume he is talking about  Gauss' discovery that the heptadecagon was constructable with the classic tools of Greek Geometry.... hope that's it anyway. 

So here is my version of that story, with many clips from Wolfram and Wikipedia...

One of the great problems of antiquity, along with doubling the cube and squaring the circle, was construction of a  heptagon, a seven-sided polygon, with a straight edge and compass.  Euclid had demonstrated the construction of the pentagon, the hexagon was easy, and so the early geometricians focused on the "next one". The early geometers knew that if an n-gon could be done, a 2n-gon was easy, so the octagon was easy, the decagon was easy, butt the nonagon also caused problems.  Over the years it became an open question in mathematics, "Which polygons are constructable with straight edge and compass?"

 On April 30, 1796, Gauss, at the age of nineteen, proved the constructability of the regular 17-gon (heptadecagon).  He did this by showing, (I think) that it could be factored into equations involving no more than quadratics.    I have been looking on line for a copy of Disquisitiones Arithmeticae  in my price range (literally-free) but have not found one to see how he explains.  Within five years he had developed a theory that described exactly which polygons were, and which were not, constructable. He didn't actually construct one, but he proved it could be constructed. The proof relies on the property of irreducible polynomial equations that roots composed of a finite number of square root extractions only exist when the order of the equation is the product of powers of two and Fermat Primes.  Gauss showed that the 17-gon is constructable since the sine and cosine of  ^2π⁄₁₇ can be expressed with basic arithmetic and square roots alone (which can be formed with a compass and straightedge).   The first actual method of construction was devised by Johannes Erchinger, a few years after Disquisitiones Arithmeticae was published. In the book Gauss supposedly writes cos(2pi/17) as an expansion of square roots which would look like this in modern notation

 

(thank you wikipedia)

(wikipedia)


Guass was so  proud of this, among all his great discoveries, that he told his close friend, Farkas  Bolyai, that the regular 17-gon should adorn his tombstone, but this was not done. There is a 17 pointed star on the base of a monument to him in Brunswick because the stonemason felt everyone would mistake the 17-gon for a circle.

 Five years later, he developed the theory of Gaussian periods in his Disquisitiones Arithmeticae. This theory allowed him to formulate a sufficient condition for the constructability of regular polygons:

A regular n-gon can be constructed with compass and straightedge if n is the product of a power of 2 and any number of distinct Fermat primes.

Gauss stated without proof that this condition was also necessary, but never published his proof. A full proof of necessity was given by Pierre Wantzel in 1837.

Detailed results by Gauss' theory

Only five Fermat primes are known:

F₀ = 3, F₁ = 5, F₂ = 17, F₃ = 257, and F₄ = 65537 (sequence A019434 in OEIS)

The next twenty-eight Fermat numbers, F₅ through F₃₂, are known to be composite.
Thus an n-gon is constructable if

n = 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, … (sequence A003401 in OEIS),

while an n-gon is not constructable with compass and straightedge if

n = 7, 9, 11, 13, 14, 18, 19, 21, 22, 23, 25, … (sequence A004169 in OEIS).

Connection to Pascal's triangle

There are 31 known numbers that are multiples of distinct Fermat primes, which correspond to the 31 odd-sided regular polygons that are known to be constructable. These are 3, 5, 15, 17, 51, 85, 255, 257, … , 4294967295 (sequence A001317 in OEIS). As John Conway commented in The Book of Numbers, these numbers, when written in binary, are equal to the first 32 rows of the modulo-2 Pascal's triangle (see below), minus the top row. This pattern breaks down after there, as the 6th Fermat number is composite, so the following rows do not correspond to constructable polygons. It is unknown whether any more Fermat primes exist, and is therefore unknown how many odd-sided constructable polygons exist. In general, if there are x Fermat primes, then there are 2^x−1 odd-sided constructable polygons.

Pascal's triangle mod-2
1  1   = 3
1  0  1  = 5
1  1  1  1  = 15
1  0  0  0   1 = 17
1  1  0  0    1   1  = 51
1  0  1  0   1   0  1 = 85
1  1  1  1  1    1   1  1  = 255   etc...


This biography of Gauss, by far the most comprehensive in English, is the work of a professor of German, G. Waldo Dunnington, who devoted most of his scholarly career to studying the life of Germany's greatest mathematician. The author was inspired to pursue this project at the age of twelve when he learned from his teacher in Missouri that no full biography of Gauss existed at the time. His teacher was Gauss's great granddaughter, Minna Waldeck Gauss. Long out of print and almost impossible to find on the used book market, this valuable piece of scholarship is being reissued in an augmented form with introductory remarks, an expanded and updated bibliography, and a commentary on Gauss's mathematical diary, by the eminent British mathematical historian, Jeremy Gray.