Archive for the ‘Featured Mathematician’ Category

Monday Morning Math: Omar Khayyam

April 18, 2022

Good morning!  Our mathematician today is Omar Khayyam.

Omar/Umar Khayyam was born in Nishapur, Persia, (modern day Iran) in 1048. Not much is known about his mother, but his father was a doctor who hired tutors to teach Omar.   Omar Khayyam is known for his mathematics, including writing down the laws of algebra that we know today.  He was able to make progress toward finding a general formula for ax^3+bx^2+cx+d=0 similar to the quadratic formula:  Greek mathematicians had come up with solutions to the quadratic formula that used a straightedge and compass, but Khayyam conjectured that it was not possible to solve the cubic equation with just those tools, and so developed other means of finding the solutions geometrically, using a parabola.  (It would be more accurate to say solutions to cubic equations: although we write it as a single equation, at that time the quadratic and cubic equations were written as several different cases depending on whether the coefficients were positive or negative.)  It was 500 years before anyone found a more general solution than his.

Omar Khayyam was one of the earliest people to describe the Arithmetic triangle (which is sometimes called Pascal’s triangle, although this was 500 years before Blaise Pascal).  He also contributed to the fields of non-Euclidean geometry and number theory.

In addition to mathematics, Khayyam wrote about astronomy, geography, and music.  He is largely remembered for his poetry, especially the rubaiyat (aka  Rubā‘iyyāt, or quatrains)

The Moving Finger writes, and, having writ,
Moves on: nor all thy Piety nor Wit
Shall lure it back to cancel half a Line,
Nor all thy Tears wash out a Word of it.

(Translated into English by Edward Fitzgerald)

Khayamm passed away in Nishapur in 1131, and a mausoleum stands over his tomb

Sources:

Monday Morning Math: The L’Hôpitals

March 21, 2022

Good morning! The math tidbit for today is a two-for-one special: the L’Hôpitals, who lived in France in the 1600s.  But we’ll start with the phrase that my brain jumps to when I see L’Hôpital, which is L’Hôpital’s Rule (also called L’Hospital’s Rule).  It’s about limits, so it shows up in Calculus.  Essentially, if you have a limit as x\to a of a fraction where both the numerator and denominator individually are approaching 0 or where both the numerator and denominator individually are approaching \pm\infty, then:

\lim_{x\to a} \frac{f(x)}{g(x)}=\lim_{x\to a} \frac{f'(x)}{g'(x)}

provided that second limit actually exists.  So, for example,

\lim_{x\to 0} \frac{\sin(x)}{x}=\lim_{x\to 0} \frac{\cos(x)}{1}=1.

Huzzah!

This rule was named after Guillaume François Antoine de l’Hôpital, where that last name was spelled different ways even in his lifetime, even by him.  He learned calculus by correspondence with Johann Bernoulli, who was busy figuring out calculus himself at that time, since the subject was only a few decades old.  In 1696 l’Hôpital published what is considered to be the first Calculus book: Analyse des infiniment petits pour l’intelligence des lignes courbes.  He thanked several people in the introduction, including Johann Bernoulli, but it wasn’t clear at that time how much of the book was really after l’Hôpital’s own work (none?) and how much was based on Bernoulli’s notes (all?).  Apparently Bernoulli was fine with l’Hôpital publishing the book, possibly because of the money l’Hôpital paid him, possibly because he was happy just to have these still-new ideas disseminated. L’Hôpital died in 1704 when he was about 43 years old.

A lot of that information  comes from the MacTutor biography, which also states, “L’Hôpital married Marie-Charlotte de Romilley de La Chesnelaye; they had one son and three daughters.”   The English Wikipedia page adds that his wife was “also a mathematician and a member of the nobility, and inheritor of large estates in Brittany” with a link to a page for her – in French – from a French biography that indicates that she worked in Geometry and Algebra and lived from 1671-1737.  The only other site online that mentions her is this dictionary, also in French, which indicates that she helped with the printing of the aforementioned Calculus book and impressed another math professor, Monsieur de la Montre, with her knowledge of Euclid.  So was she too involved in the creation of that first Calculus book but not mentioned?  It sounds like it, though the extent of her involvement is unclear.  

And we’ll leave on that uncertain note.  Any new information would of course be welcome.

References:

Monday Morning Math: Maryam Mirzakhani

March 7, 2022

It’s Women’s History Month, and we’ll celebrate with the first (and so far only) woman to receive the Fields Medal in Mathematics – an award given every four years in recognition of outstanding mathematics. 

Maryam Mirzakhani was born in Tehran, Iran, in 1977.  She loved reading novels and writing as a child, and became interested in mathematics when her brother would come home from school and tell her what he learned – her earliest memory of math specifically was when he told her the story of a young Carl Friedrich Gauss being asked to add the numbers from 1 to 100 and surprising his teacher by seeing a pattern that allowed him to do the computation in moments.

Maryam Mirzakhani went on to participate in (and win) the Math Olympiads as a teen; she then went to Sharif University of Technology in Iran and then Harvard University in the United States, where she earned her doctorate.  

In 2014 Dr. Mirzakhani was awarded the Fields Medal. Her research is sometimes described as a complicated version of billiards, where a ball bounces along the edges of a table.  “You want to see the trajectory of the ball,” Dr. Mirzakhani explained in a video produced by the Simons Foundation and the International Mathematical Union. “Would it cover all your billiard table? Can you find closed billiards paths? And interestingly enough, this is an open question in general.” You can see a short (under 3 minute) video of Dr. Mirzakhani explaining her work here:

Mirzakahni passed away in 2017 from breast cancer.  She was 40 years old.

The 2022 Fields Medals will be given out at the  International Congress of Mathematicians this summer.  The conference was originally scheduled to be held in St. Petersburg, Russia, but due to the Russian invasion of Ukraine the organizers have announced that the conference will be held virtually, without any contributions from the Russian Government, and will be free for everyone who would like to attend.  

Sources:

Monday Morning Math: Voronoi Diagrams and Georg Voronoy

February 28, 2022

Good morning!  Today’s math is Voronoi Diagrams, but I’ll lead into it with some Geometry.  Suppose you have two points A and B, and wonder which points on the plane are closer to A and which are closer to B.  Maybe A and B are schools, and this is for figuring out districts, or maybe they are pizza places and you’re wondering where to order from.  It turns out that the border between the two regions is on the perpendicular bisector of A and B.

But what if there are three points? It’s a little more complicated, but the boundaries are still made up from the perpendicular bisectors of the different pairs of points.  And this idea continues even when there are more points, as this picture shows.

Diagrams like these are called Voronoi Diagrams named after Ukrainian mathematician Georgy Feodosevich Voronoy (also written as Georgii Voronoi).

Wikimedia Commons

He was born in the village of Zhuravka in the north central part of Ukraine on April 28, 1868, and while he was still in the equivalent of high school he solved and published the result of a problem in algebra.  He then went to the University of St. Petersburg in Russia, first as an undergraduate but eventually as a doctoral student under Andrey Markov (himself well known because of something called Markov Chains, which are ways of calculating probabilities).   Both his Master’s thesis and his doctoral thesis were awarded the Bunyakovsky prize for outstanding work in mathematics by the St Petersburg Academy of Sciences.

Voronoy became a professor at the University of Warsaw in Poland, where he continued to do research and also supervise students: one of his students was Wacław Sierpiński, for whom Sierpinski triangles are named.   When he was only 40 years old he developed severe gallstones, and passed away on November 20, 1908.

Sources:

Monday Morning Math: Gloria Ford Gilmer

February 7, 2022

Our mathematician today is Dr. Gloria Ford Gilmer, a pioneer in ethnomathematics (the study of the relationships between mathematics and culture).  Gloria Ford Gilmer was born in Baltimore Maryland, in 1928.  She studied mathematics at Morgan State University in Baltimore, where she was a student of Clarence Stephens, and where she earned her bachelor’s degree in 1949.  Two years later she earned her master’s degree in math at the University of Pennsylvania.

She did ballistics research for the US Army, but soon turned to teaching.  She taught both high school and college students, eventually earning a PhD in curriculum and instruction at Marquette University.  Most of Dr. Gilmer’s research was in ethnomathematics.  She was particularly interested in finding mathematics in everyday places, and is known for her mathematical analyses of the braiding patterns in African American women’s hair. 

Dr. Gilmer was active in many professional organizations, and was a “first” for many of them. She was the first African American woman on the board of governors for the Mathematical Association of America, and the first woman to give the Cox-Talbot Address for the National Association of Mathematicians. In 1985 she, along with Ubiratan D’Ambrosio, Gil Cuevas and Rick Scott, co-founded the International Study Group on Ethnomathematics (ISGEm); she served as the organization’s president for 11 years. 

Dr. Gilmer passed away only a few months ago, on August 25, 2021.  The recently established American Mathematical Society’s Claytor-Gilmer Fellowship is named in her honor.

No picture here because I couldn’t find one without copyright restrictions, but you can see one on the site Mathematically Gifted & Black, where she was an honoree last year.  Every day in the month of February the site recognizes a mathematician – check out the 2022 honorees!

Sources:

Monday Morning Math: Pythagoras

January 31, 2022

This week’s mathematician is someone you have probably heard about, although it turns out very little is known for certain. 

The mathematician is Pythagoras.  

Pythagoras was probably born in Greece, on the island of Samos, over 2500 years ago (570 BCE, plus or minus a few decades).  His mother was from that island, and his father Mnesarchus was a merchant.  Pythagoras would travel with him sometimes when he was a child, and when he was an adult he studied mathematics with Thales, another now-famous mathematician.   

After many years (decades) of study, Pythagoras formed a group known as the Pythagoreans, who followed a strict vegan diet and believed that everything was essentially a number.  There was an inner circle of mathematicians and an outer circle, and there is some indication that the groups were equally welcoming to women and men. It is not possible to distinguish who proved any one result (because secrecy was the name of the game rather than publishing), but there were important results from this group related to music and geometry.  

Two of the most significant results attributed to Pythagoras are the Pythagorean Theorem (written in geometric terms that a square on the hypotenuse of a right triangle has the same area as the sum of squares on each of the two legs) and that the square root of 2 is irrational (written in geometric terms that the side and the diagonal of a square are incommensurable, meaning there’s no teeny tiny amount that fits into both an integer number of times.  There are stories that someone figured this out and was killed, either for figuring this out or for telling people outside the Pythagoreans, but like the rest of what I’ve written this was a story – everything we supposedly know about Pythagoras comes from reports well after his death, so at best this is educated guesswork.

Pyth.jpg
Etching on the wall of Peckham Hall, our math and science building. Not seen is the QED in the lower right corner, as it was also in darkness when I took the photo just now

Finally, the video below, from BBC learning, is under 5 minutes and fun to watch, though it does start off with an excited “Pythagoras!” that will get the attention of anyone around you.

Other sources:

Monday Morning Math: Dorothy Vaughn

November 29, 2021

Dorothy Vaughn was born on September 20, 1910 in Kansas City. Missouri. She graduated at the age of 19 from Wilberforce University, a historically black college in Wilberforce, Ohio. Dorothy Vaughn supported her family as a math teacher for 14 years prior to working at NASA as part of the National Advisory Committee for Aeronautics’ (NACA) West Area Computing unit in 1943.

The West Area Computing unit was a group of black women who, as a result of Jim Crow Laws, were segregated at NASA while they performed mathematical calculations on slide rules and graph paper to support the space race and the NASA astronauts’ flight missions to space. Dorothy Vaughn was an expert in the computer programming language FORTRAN and she became NASA’s first black supervisor of the group in 1949 where she taught the women programming to prepare them for the future which she believed would be machine computers.

In addition to her work at NASA, Dorothy Vaughan raised her family of 6 children, one of whom also worked for NASA.

She retired from NASA in 1971 and died on November 10, 2008. She was featured in Margot Lee Shetterly’s book Hidden Figures and portrayed in the film based on the book by Octavia Spencer.

Sources:
https://www.nasa.gov/content/dorothy-vaughan-biography
https://scientificwomen.net/women/vaughan-dorothy-103
https://blog.sciencemuseum.org.uk/nasas-overlooked-star/

Written by Tracy Lyn Lause

Monday Morning Math: Grace Murray Hopper

November 8, 2021

In honor of Veterans Day, our mathematician this week is Grace Murray Hopper.

Grace Murray was born in New York in 1906. She earned her BA in math and physics from Vassar College. Over the next few years she married, earned an MA and PhD in math from Yale University, and became a professor at Vassar.

In 1943, at the age of 37, she joined the Navy in response to World War II and began working with computers. She worked on the Harvard Mark I (which was over 50 feet long, 8 feet tall, and 2 feet deep) and later the Mark II and III. She learned programming and was instrumental in both conceptualizing and creating the first compiler.

At the time of her retirement in 1986 she was at the rank of Rear Admiral and the oldest active military officer. She continued to work even after her retirement and died in 1992. She is buried in Arlington Cemetery.

One of her legacies is the popularization of the term computer bug. She invented the term “debugging” in response to an actual bug (shown in the photo below!)

Hopper found the first computer “bug” a dead moth that had gotten into the Mark I [possibly Mark II] and whose wings were blocking the reading of the holes in the paper tape. The word “bug” had been used to describe a defect since at least 1889 but Hopper is credited with coining the word \debugging” to describe the work to eliminate program faults.

(From computersciencelab.com)
Courtesy of the Naval Surface Warfare Center, Dahlgren, VA., 1988, public domain

Sources:

Monday Morning Math: Adolphe Quetelet

October 25, 2021

Adolphe Quetelet (pronounced Ket-eh-lay) was a Flemish Scientist who was the first to use the normal curve.

Born in Ghent, France in1796, Adolphe’s father died when he was just seven years old. At the age of 17, after his own schooling at Lyceum in Ghent where he excelled in mathematics, he took a job teaching mathematics at a school in 1813 to support his family. He was appointed a mathematics instructor at the College in Ghent in 1815 at the age of 19.

While at the College of Ghent, Adolphe was influenced by Garnier who encouraged Quetelet’s deeper studies in mathematics. He went on to earn a doctorate from the University of Ghent in 1819 with a dissertation on conic sections. After graduating and at the age of 23, he was appointed chair of elementary mathematics at the Athenaeum in Brussels. While he taught mathematics, Quetelet had a strong interest in astrology and lobbied for an observatory in Brussels. While visiting Paris on a fact-finding mission for the observatory, Quetelet learned the importance of statistical methods in astronomy.

As a result of his “zeal for statistics,” Quetelet identified society as a topic and studied and wrote papers on social statistics and in the course of that work was the first to use the normal curve/distribution and used what astronomers knew as the error law or bell curve on human populations. He also introduced the height/weight measure that we know today as the body mass index (BMI). He used the idea of an average as a central value. He collected statistics on crime and mortality and improved census taking for the government.

In 1855 Quetelet suffered a moderate stroke and never fully recovered suffering from a poor memory which negatively impacted his writings. Quetelet died in 1874.

Sources:
https://mathshistory.st-andrews.ac.uk/Biographies/Quetelet/
https://www.famousscientists.org/adolphe-quetelet/

Compiled by Tracy Lyn Lause

Math Guys in Rome

November 1, 2010

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

May 23, 2010

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

November 25, 2008

copernicus_krakowHe 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

September 9, 2008

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

August 31, 2008

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

August 12, 2008

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).