Math Guys in Rome

The Villa Borghese Gardens form a giant park in Rome, and at the western edge of it are the Pincian Gardens, so named because they’re at the top of the Pincian Hill.  (Belated note to self: the fact that they were on top of a hill means it should not have been any sort of surprise that there were many many steps to get up to the Gardens.)

These were [this was?] the first public park opened in the city, and around 1850 a bunch of busts of prominent Italians were commissioned for the park.  Some of these were kept in the park, some were moved and then moved again, and some were altered to represent Italians who seemed more worthy of being commemorated.   Then through the 1950s more busts were added and there are now a total of 228, of which 225 are of men and 3 are of women.

There’s a map of all the busts online [here, along with all the history], so it was pretty easy to search out mathematicians.  Here’s Archimedes:

This was one of the original busts, but back then it was of Niccolò Machiavell; it got re-formed into Archimedes around 1860. (You might be wondering, too, at Archimedes Italian background.    A few of these busts were a little more liberal than others on what it meant to be Italian.)

This next one is of Giordano Bruno, born in 1548:

Bruno was a big fan of Copernicus’s still-unpopular view that the earth revolves around the sun, though he also thought that the sun was nothing unique either — just one of an infinite number of heavenly bodies.  Poor Bruno didn’t get along too well with the church of the time, and was burned at the stake in 1600.

On a lighter note, here’s Leonardo da Vinci, along with a rose that someone left for him:

(I just noticed the square around his face.  What’s that about?  It’s in the few other pictures that we took of him, too.)

Next up is Giuseppe Luigi Lagrangia, also known as Joseph-Louis Lagrange (though Wikipedia and Mactutor say his middle name was Lodovico originally).

He looks totally proud of everything named after him, like the Lagrangian and Lagrange Multipliers.

Here’s Pythagoras (another “Italian”) with two of his closest friends:

And finally, this is Niccolò Fontana, who became known as Tartaglia (stutterer) because the French invaded his hometown of Brescia when he was a teen and sliced his face. Ugh.

He translated Euclid into Italian and is also known for his role is finding a general solution to the cubic equation, which deserves a post all to itself someday.

And that’s it!  In theory Galileo should be in this group, but we couldn’t find him (we think he was hidden behind a construction fence), and so should Barnaba Tortolini (not sure why we missed him).  Oh, and there was also an obelisk and this really cool water clock, which was one of the main reasons that we went to this neck of the woods in the first place, but that will appear in the next post…

RIP Martin

Martin Gardner passed away yesterday (May 22) at the age of 95 — a number that is 0 mod 1, 1 mod 2, 2 mod 3, and 3 mod 4, as I’m sure he’d appreciate.    It seems like half the puzzles I hear about were either invented by him or popularized by him.  Falling into the latter category are the flexagons, which are, of course, a favorite of Godzilla and which were described in the first “Mathematical Games” column of Scientific American (you know, according to Wikipedia).

A quick search reveals lots of puzzles, most of which have a visual component.  As a treat, here’s a quick one that doesn’t (from the group Gathering for Gardner):

• Write out the alphabet starting at J (and ending at I):
  JKLMNOPQRSTUVWXYZABCDEFGHI

  Now erase all the letters that have vertical symmetry, like M.

  There will be 5 groups of consecutive letters left.  Write the number of letters in each group:

  ___   ___   ___   ___   ___

Cool, huh?

Scientific American itself just posted 3 Gardner puzzles here in tribute to the man who served them well for many years.

Martin, thanks for the fun.

The photo of Martin Gardner was taken by Konrad Jacobs of  Erlangen and is licensed under CC.  From Wikipedia.

Nicolaus Copernicus: lost and found

He was born 535 years ago as Mikolaj Kopernik or Nicolaus Koppernigk, and he died 70 years later. In between, he proposed that the sun and not the earth is at the center of the universe, which was a bit of a shock at the time.

When he died in 1543, he was buried in Frombork Cathedral in Northern Poland but his exact grave was never marked. Then four years ago the Bishop (Jacek Jezierski) requested help from archaeologist Dr. Jerzy Gassowski in finding the grave. It took a few years, but a grave was indeed found in an appropriate spot. But was this Copernicus? The body was the right age, but that’s hardly conclusive. What is fairly conclusive is DNA evidence, except that there weren’t exactly databases set up at the time. What they needed was something like a piece of his hair.

And that, it appears, is exactly what they had. Some of Copernicus’s books (that he himself owned, not that he wrote) are still around and on display. In one of those books were four hairs. Dr. Marie Allen tested the DNA, and it turned out that two of those hairs belong to the body under the cathedral.

Proof? Maybe not. But as CSI meets De revolutionibus, it’s pretty cool.

For more information, see the Post-Gazette or CNN.

Mathematician of the Week: Émilie du Châtelet

Émilie du Châtelet was born December 17, 1706, and died September 10, 1749. Her academic training came at home under the supervision of tutors and her parents. Voltaire commented that “her mind was nourished by reading good authors in more than one language…[and that] her dominant taste was for mathematics and philosophy”.

Many sources comment that her most significant mathematical contribution was her translation, into French, of Newton’s Principia. And while it is true that hers would be for quite some time the only translation of Newton’s work available in French, to characterize du Châtelet the mathematician as merely a translator of Newton does her a disservice, for she produced a significant amount of original work as well. Among her other publications:

• She coauthored [without attribution] Voltaire’s Elements de la philosophie de Newton (1736)
• She translated and added additional material to Mandeville’s The fable of the bees. Her preface includes the following passage:
  • I am convinced that many women are either unaware of their talents by reason of the fault in their education or that they bury them on account of prejudice for want of intellectual courage. My own experience confirms this. Chance made me acquainted with men of letters who extended the hand of friendship to me. … I then began to believe that I was a being with a mind …

• Her Dissertation su la nature et la propagation du feu was submitted to the Academie des Sciences in Paris, for the Grand Prix of 1737. While her submission did not win (she lost out to Leonhard Euler), her work was still published by the Academie in 1744.
• In 1740, she published a book Institutions de physique, an attempt at a unified treatment of Cartesian, Newtonian, and Leibnizian philosophy.

At the time of her death, she was working on her translation of, and commentaries on, Newton’s Philosophiae naturalis principia mathematica. Subsequent work on that translation was undertaken by Alexis Clairaut, and  completed in 1759.

The next French translation of Newton’s Principia, so far as I can determine, was published in 1985, which speaks to the influence and significance of Émilie du Châtelet’s contribution.

Mathematician of the week: Louis Antoine de Bougainville

Louis Antoine de Bougainville was born November 11, 1729, and died August 31, 1811. While his mathematical contributions were modest, he has surprisingly strong name-recognition for an eighteenth-century mathematician…

By 1756, Bougainville had published two volumes on the integral calculus, explicitly presented as a supplement to and extension of L’Hopital’s Analyse des infiniment petits pour l’intelligence des lignes courbes (published in 1696, the first textbook on the differential calculus). Bougainville’s work earned significant praise, including Bougainville’s election to membership in the Royal Society of London. However, this publication also marked the end of Bougainville’s mathematical career.

After joining the French Army in 1754, Bougainville served with some distinction in the French and Indian war. By the early 1760s, Bougainville had joined the French Navy. In 1764, he establishing the first European settlement on the Falkland Islands (Port St. Louis), and during 1766 – 1769, he became the 14th known Western navigator, and first Frenchman, to circumnavigate the globe. During that voyage, his ships came upon the heavy breakers of the Great Barrier Reef, and turned away to the north, toward the Solomon Islands. (Bougainville thus narrowly avoided sailing upon Australia, some three years before James Cook’s expedition which claimed New South Wales for Great Britain.) Bougainville Island (politically part of Paupa New Guinea) was apparently named by Bougainville during this voyage.

The flowering vine bougainvillea is also named for Louis Antoine de Bougainville. A plant native to South America, Bougainville wrote extensively about it for European readers following his circumnavigatory voyage.

Mathematician of the Week: Alonzo Church

Alonzo Church was born on June 14, 1903, and died August 11, 1995. Essentially his entire early academic career took place at Princeton University, having completed his AB (1924) and his PhD (1927, under Oswald Veblen) there, and then serving as a professor of mathematics from 1929 until 1967. (After retiring from Princeton in 1967, he taught at UCLA as a professor of mathematics and philosophy until 1990.)

Church’s most significant mathematical contribution was the creation (with Stephen Kleene) of the λ-calculus, a formal system in the language of functions.

Church is probably best remembered for Church’s Thesis, the claim that every effectively computable function is in fact a function that is definable in his λ-calculus. Kurt Gödel balked at this claim, and introduced the primitive recursive functions as a more natural alternative to model the notion of effective computability. Stephen Kleene, a student of Church, showed that in fact the functions definable in the λ-calculus exactly correspond to Gödel’s primitive recursive functions. By the late 1930s, another notion of computability had been put forward by Alan Turing, and it too had been shown to be equivalent to λ-definability.

The sets of λ-definable functions, primitive recursive functions, and functions implementable as Turing Machines, are identical sets of functions. This agreement of three diverse approaches to formalizing the vague notion of “effectively computable” is viewed as strong evidence that all three approaches have in fact captured that concept. At its most general, Church’s Thesis is the claim that effective computability is equivalent to these three formalizations. Given that “effectively computable” is unlikely to ever be formally defined, Church’s Thesis remains an unproven (and unprovable) claim.

Alas, Church’s Thesis first appeared in 1936, and was not a part of Church’s (doctoral) thesis of 1927 (Alternatives to Zermelo’s Assumption, an attempt to create a logic in which the axiom of choice is false).

Mathematician of the Week: Théodore Olivier

Théodore Olivier was born January 21, 1793, and died on August 5, 1853.  He was a student of Gaspard Monge, and indeed Monge’s influence seems apparent in Olivier’s most famous work: his models of the intersections of  three dimensional surfaces.  Olivier used strings arranged on a metal framework to model each individual surface.  As the surfaces move relative to one another, the strings allow one to study the curve where the two surfaces intersect.  (Monge had constructed static models depicting such intersections; by changing materials Olivier had a pedagogic breakthrough.)

Union College has a large collection of these models, and has created a web page devoted to their collection and its history.  One of their computer science students [now alumnus], Mike Pinch, created software to simulate the models, and give the user the opportunity to manipulate the models.  Union also has posted several .avi videos of this software in action.

Mathematician-of-the-Week: Pierre-Joseph-Étienne Finck

Running time of the Euclidean Algorithm

Pierre-Joseph-Étienne Finck was born October 15, 1797, and died July 27, 1870. Finck’s most significant mathematical contribution appears to have been his analysis of the running time of the Euclidean Algorithm, which he published in 1841.

One wonders if his own life experiences contributed to his interest in recursive algorithms. Upon graduating from the École Polytechnique in 1817, he was admitted to the Artillery School. However, he wasn’t satisfied with his studies there, and applied ( in March 1818 ) to transfer to the Royal Guard cavalry. Request denied.

4 months later, he applied to the cavalry again, this time saying that he would resign if his request was not honored. Request denied.

So he resigned from Artillery School…. But by early 1819 he had second thoughts, and applied for reinstatement to the Artillery School. Request denied.

At this point, he changed tactics, and began studying mathematics at the University of Strasbourg. He completed his doctoral dissertation (on movements of the terrestrial equator) in 1829. Ironically, by that time, he had been appointed as a mathematics instructor at the Artillery School of Strasbourg.

I suppose Finck’s life might provide a valuable lesson in the importance of sticking to your guns.

Source:  Mactutor History of Math archive

Mathematician of the week: Georg Pick

Georg Pick was born on August 10, 1859, and died on July 26, 1942, having spent much of his career as a professor at the German University of Prague.

He is most commonly remembered for a theorem concerning the area of lattice polygons, polygons in the plane whose vertices occur at points with integer coordinates: the area of such a polygon is numerically equal to one less than the number of lattice points in its interior plus half the number of lattice points on its boundary. Pick published his theorem in 1899, but it received scant attention until it was commented upon by Steinhaus in his 1969 text Mathematical Snapshots.

Part of the popularity of Pick’s Theorem is its elegance: it is simple to state, it is simple to discover (once told to expect a relationship involving area, lattice points in the interior, and lattice points on the boundary, initial explorations with rectangles suffice to generate linear relationships that suggest Pick’s result); and a formal proof [e.g., this one at Cut the Knot] is relatively straightforward.

Mathematicians with significant anniversaries during the week of July 20 – July 26:

July 20: Death anniversaries of John Playfair [1819] (Playfair’s Axiom), Bernhard Riemann [1866] (geometric foundations), and Andrei Andreyevich Markov [1922] (probability, stochastic processes)

July 21: Birthday of John Leech [1926] (Leech Lattice); death of Giovanni Frattini [1925] (group theory)

July 22: Birthday of Wilhelm Bessel [1784] (analysis), Gabriel Lamé [1795] (differential geometry, proof of FLT for exponent 7), and Konrad Knopp [1882] (analysis)

July 23: Death of Florence Nightingale David [1993] (statistics) [her parents were friends of "the" Florence Nightingale]

July 24: Birthday of Errett Bishop [1928] (Foundations of Constructive Analysis); death of Hans Hahn [1934] (Hahn-Banach Theorem)

July 25: Birthday of Johann Listing [1808] (topology)

July 26: Birthday of Kurt Mahler [1903] (p-adic numbers, geometry of numbers); deaths of Gottlob Frege [1925] (mathematical logic), Henri Lebesgue [1941] (measure and integral), Georg Pick [1942] (Pick’s Theorem), Raymond C. Archibald [1955] (history of mathematics), and John Tukey [2000] (mathematical statistics)

Source: MacTutor

Mathematician of the Week: Jean-Robert Argand

Jean-Robert Argand was born on July 18, 1768. He was a bookkeeper and amateur mathematician, and is remembered for having introduced a geometric interpretation of the complex numbers as points in the Cartesian plane (a discovery that had been anticipated by Caspar Wessel in a paper published in 1799). Argand’s treatment of the subject appeared in a self-published anonymous monograph in 1806, which found its way to Legendre and eventually Jacques Français, who published an article about the idea and some of its ramifications, and asked for help in identifying the originator of the idea. Argand came forward, and a debate ensued in print as to the validity of working with complex numbers as geometric quantities, not merely algebraic objects (Argand and Français arguing in favor of geometry, François-Joseph Servois arguing against). This exchange served to solidify the association between Argand’s name and the geometric representation, and henceforth the name has stuck.

Curiously enough, Servois and Argand nearly share a birthday.

Mathematicians with birth or death anniversaries during the week of July 13 – 19:

July 13: Birthday of John Dee [1527]

July 14: Birthday of Laurence C. Young [1905] (generalized curves); deaths of Lorenzo Mascheroni [1800] (analysis) , Augustin Fresnel [1827] (wave theory of light), and Benjamin Gompertz [1865] (mortality rates)

July 15: Birthdays of Adolph Pavlovich Yushkevich [1906] (historian of mathematics) and Stephen Smale [1930] (differential geometry, Field’s Medalist 1966)

July 16: Birthday of Siegfried Aronhold [1819] (theory of invariants)

July 17: Birthday of Wilhelm Lexis [1837] (time series); deaths of Henri Poincaré [1912] (algebraic topology, polymath) and Adolph Pavlovich Yushkevich [1993]

July 18: Birthdays of Robert Hooke [1635] and Jean-Robert Argand [1768]; death of Abraham Sharp [1742] (calculator, mathematical and astronomical tables)

July 19: Birthdays of François-Joseph Servois [1768] (operator theory) and Charles Briot [1817] (theory of functions)

Source: MacTutor

Mathematician of the Week: George Darwin

George Darwin (born July 9, 1845), the son of Charles Darwin, devoted much of his professional efforts to the study of the Sun-Earth-Moon system.  He conjectured that the Moon had formed from material pulled by tidal action of the Sun from the primordial Earth.  He studied the dynamics of rotating liquids (motivated by his theory of lunar formation), and is credited with being “the first to apply mathematical techniques to study the evolution of the Sun-Earth-Moon system.” [MacTutor]

Mathematicians with birth or death anniversaries for the week of July 6 through July 12:

July 6:  Birthdays of Alfred Kempe [1849] (four color theorem) and Lothar Collatz [1910] (differential equations, Collatz Conjecture)

July 7: Deaths of Gösta Mittag-Leffler [1927] (analysis), William Young [1942] (measure and integration), Anatoly Ivanovich Malcev [1967] (logic, group theory), and Raymond Wilder [1982] (foundations of mathematics)

July 8: Deaths of Johann Müller (Regiomontanus) [1476] (trigonometry), Christiaan Huygens [1695] (astronomy, probability), and Kurt Reidemeister [1971] (knot theory)

July 9: Birth of George Darwin [1845] (three body problem); deaths of Henri Padé [1953] (continued fractions) and Arend Heyting [1980] (intuitionism)

July 10:  Birth of Roger Cotes [1682] (logarithms, interpolation); death of Emory McClintock [1916] (actuarial math)

July 11: Birth of John WIlliam Scott Cassels [1922] (number theory); deaths of Nicole Oresme [1382] (infinite series, coordinate geometry) and Simon Newcomb [1909] (mathematical astronomy)

July 12: Birth of Ernst Fischer [1875] (the Riesz-Fischer Theorem)

Source: MacTutor

Mathematicians of the week: Eduard Cech and Witold Hurewicz

June 29 is the birthday of two topologists:  Eduard Cech [born in 1893] and Witold Hurewicz [born in 1904].  Curiously, in addition to sharing a birthday, they also share credit for the independent discovery of higher homotopy groups, a subject which Cech had spoken on at the 1932 ICM, and which Hurewicz  developed independently in the mid 1930s.

 

Mathematicians with birthdays or death anniversaries during the week of June 29 through July 5:

June 29: Birthdays of Eduard Cech [1893] (algebraic topology; Stone-Cech compactification) and Witold Hurewicz [1904] (higher homotopy groups)

June 30: Death of WIlliam Oughtred [1660] (slide rule)

July 1: Birthday of Gottfried Leibniz [1646]

July 2:  Birthday of William Burnside [1852] (abstract groups); death of Bartholomeo Pitiscus [1613] (coined term “trigonometry”) and Thomas Harriot [1621] (solutions of equations)

July 3: Death of William Jones [1749] (introduced symbol π)

July 4: Deaths of Guido Grandi [1742], Peter Tait [1901], Oscar Zariski [1986], and Marshall Hall [1990]

July 5: Deaths of René Baire [1932] and Oskar Bolza [1942]

Source: MacTutor

Mathematician of the week: Jules Lissajous

Jules Lissajous was born March 4, 1822.  His doctoral studies were on vibrations of bars “using Chladni’s sand pattern method to determine nodal positions”.  This method of viewing vibration patterns entails covering the object with flour or sand, and inducing vibrations, often by stroking with a violin bow (or in a modern lab using amplified sounds at variable frequencies). The vibrations cause the sand or flour to accumulate into a pattern, indicating nodes in the vibrations of the object, locations where the standing waves of the bar have least magnitude.  

Lissajous died on June 24, 1880.

Mathematicians with birthdays or death anniversaries during the week of June 22 through June 28:

June 22: Birthday of Hermann Minkowski [1864] (mathematical foundations of space-time theories); death of Felix Klein [1925] (algebraic geometry)

June 23: Birthday of Alan Turing [1912] (foundations of computation)

June 24: Birthday of Oswald Veblen [1880] (geometry, topology); death of Jules Lissajous [1880] (visual study of vibration and sound)

June 25: Death of Alfred Pringsheim [1941] (analysis)

June 26: Birthday of Leopold Löwenheim [1878] (Löwenheim-Skolem Theorem); death of George Udny Yule [1951] (statistics)

June 27: Birthday of Augustus de Morgan [1806] (mathematical induction); deaths of Sophie Germain [1831] (number theory, elasticity) and Max Dehn [1952] (group theory)

June 28: Birthday of Henri Lebesgue [1875] (measure theory)

Source: MacTutor

Mathematician of the week: Julius Petersen

Julius Petersen [1839 - 1910] wrote on a wide variety of mathematical topics throughout his career. His dissertation concerned geometric constructability, but he also published work on differential equations, analysis, number theory, algebra, and even such applied topics as mathematical economics and cryptography.

He is best remembered for his pioneering work in graph theory; his 1891 paper on the theory of regular graphs has been cited as the beginning of graph theory as a discipline. His 10 vertex graph, known simply as The Petersen Graph, is “the smallest bridgeless cubic graph with no three-edge-coloring“, and has become one of the standard (counter)examples studied in introductory graph theory courses.

Mathematicians with birthdays or death anniversaries during the week of June 15 through June 21:

June 15: Birthday of Nikolai Chebotaryov [1894] (generalized Dirichlet’s theorem on primes in arithmetic progressions); death of Giovanni Ceva [1734] (Ceva’s Theorem)

June 16: Birthdays of Julius Plücker [1801] (analytic geometry), Julius Petersen [1839] (Petersen graph), and John Tukey [1915] (fast fourier transform); death of Julius Weingarten [1910] (theory of surfaces)

One wonders if John Tukey’s parents had considered the name “Julius”.

June 17: Birthday of Maurits Escher [1898] (artist); death of Frank Yates [1994] (statistics; design of experiments)

June 18: Birthdays of Frieda Nugel [1884] (one of the first German women to earn a PhD in Mathematics [in 1912]) and Alice Schafer [1915] (differential geometry; founding member of AWM); death of Kazimierz Kuratowski [1980] (topology)

June 19: Birthday of Blaise Pascal [1623] (pioneer of probability)

June 20: Birthday of Helena Rasiowa [1917] (interactions of algebra and logic)

June 21: Birthday of Siméon-Denis Poisson [1781] (influential work on definite integrals and trigonometric series); death of Gaston Tarry [1913] (combinatorics; results on magic squares)

Source: MacTutor

Mathematician of the Week: Pierre Wantzel

Pierre Wantzel was born on June 5, 1814, in Paris. At the age of 15, he “edited a second edition of Reynaud’s Treatise on arithmetic, giving a proof of a method for finding square roots which was widely used but previously unproved”. [MacTutor]

His interest in radicals continued throughout his career; his crowning achievement was the development of a criterion for determining which geometric problems were solvable by straight-edge and compass construction. (Equivalently, as Mascheroni had shown, constructible by compass alone.)

Wantzel died at a fairly early age (just prior to his 34th birthday), reportedly a victim of overwork, lack of sleep, irregular eating habits, and abuse of coffee and opiates.

Mathematicians with significant dates during June 1 – June 7:

June 1: Death of Kurt Hensel (1941) [invented p-adic numbers]

June 2: Birth of Tibor Radó [1895] (solution of Plateau Problem, existence of minimal surface bounded by a given contour); death of Otto Schreier [1929] (combinatorial group theory; theory of real closed fields)

June 3: Deaths of Leopold Gegenbauer [1903] (orthogonal polynomials) and Heinz Hopf [1971] (algebraic topology)

June 4: Deaths of Eugenio Beltrami [1900] (non-euclidean geometry on a pseudosphere), Ernst Lindelöf [1946] (behavior of power series near singular points), and Maurice Fréchet [1973] (introduced notion of abstract topological space)

June 5: Births of Pierre Wantzel [1814] (proofs of nonconstructability), and John Adams [1819] (predicted orbit of planet beyond Uranus); death of Roger Cotes [1716] (logarithms, continued fractions)

June 6: Births of Johann Regiomontanus [1436] (astronomy and trigonometry), Aleksandr Lyapunov [1857] (dynamics of rotating liquids), and Max Zorn [1906] (Zorn’s Lemma); death of Guido Fubini [1943] (analysis, calculus of variations)

June 7: Death of Alan Turing [1954] (foundations of computer science)