Tuesday, June 21, 2016

More Mathematicians & Miscellaneous Math Stuff

During my travels I gathered information and took pictures of anything and everything I came across that was related to mathematics, mathematicians and the history of mathematics. Most of these images found their way into full-length posts on the many individual mathematicians I've written about, but in some instances I don't have enough material to support a full post on a specific topic or person.  I don't want these extra images and visits to go unused, however, so in this post I present additional images relating to mathematicians, the history of mathematics and mathematical objects, hoping that my students may find them to be interesting and to be helpful resources - and that others might simply enjoy the images taken of places and objects in the USA, the UK, France and Germany.

Rene Descartes

The great Rene Descartes (1596-1650) - a towering figure in philosophy as well as in mathematics - is one of the mathematicians for whom I was not able to get to a large number of relevant sites.  I did get to visit the church where he is buried, St. Germain de Pres in Paris, and I also happened across a Parisian University named for him.

The picture heading this post, and the next eight pictures are of the Abbey of St. Germain de Pres; the original church at this site was conceived in 512 AD and completed and dedicated in 558 AD, making it the oldest church in Paris.  It's bell tower is one of the oldest in all of France.  Descartes' tomb is located in one of the side chapels near the altar; sadly, I could not view the actual tomb because significant construction is currently taking place at the eastern end of the church.
The following two pictures are of the University of Rene Descartes in Paris.

Henri Poincare

Henri Poincare (1854-1912) was the most prominent French mathematician of his time.  Along with David Hilbert (1862-1943) he was one of the two most prominent mathematicians in Europe of the time.  I was thankful that Montparnasse Cemetery in Paris included Henri Poincare on their map of celebrities buried here.  During my travels it was the case that finding the graves of mathematicians was rather a hit or miss proposition and quite the treasure hunt in most cases - sometimes successful, sometimes not.

W. W. Rouse Ball

Walter William Rouse Ball (1850-1925) entered Trinity College, Cambridge in 1874, earned the spot of Second Wrangler in the Tripos, became a fellow of Trinity in 1875 and remained one the rest of his life.  In 1927 he established chairs at both Cambridge and Oxford.  Among those who have held the Rouse Ball Professorship of Mathematics are J. E. Littlewood and Abram Besicovitch (both at Cambridge) and Sir Roger Penrose (at Oxford).  Plaques commemorating Rouse Ball, Littlewood and Besicovitch can be seen in the chapel of Trinity College Cambridge.  Penrose is currently Emeritus Rouse Ball Professor of Mathematics at Oxford, and his famous tiles can be seen outside Oxford's Mathematical Institute.
When I expressed surprise to the professor hosting me at Cambridge at finding the name of Besicovitch here (I knew he was from eastern Europe), he replied, "Oh, yes, Besi was here!"
Penrose Tiling outside Oxford's Institute of Mathematics

Sir Isaac Barrow

Sir Isaac Barrow (1630-1677) held the Lucasian Chair of Mathematics  just prior to Sir Isaac Newton (1642-1726).  This professorship was founded in 1663 by Henry Lucas and was recently held by famed physicist Stephen Hawking from 1979 to 2009.  

Isaac Barrow resigned the chair in 1669 and was appointed chaplain to King Charles II in 1670.  Three years later, Charles II appointed Barrow to the Mastership of Trinity College, Cambridge, stating that he was the best scholar in England.  He is buried in Westminster Abbey and memorialized with a statue in the chapel at Trinity College, Cambridge.  

I'd always heard that it was Barrow who came up with the Fundamental Theorem of Calculus (before Newton) and that he was Newton's teacher, but during my time in Cambridge I learned in no uncertain terms that this was not the case and that Barrow was nowhere near existing in the same intellectual universe as Newton.  I have that on good authority!  That said, he clearly made contributions and was recognized for those contributions by the king at the time and is worthy of remembrance - as we see in Trinity College and at Westminster Abbey.
Sir Isaac Barrow - Chapel of Trinity College, Cambridge
Westminster Abbey, London - burial place of Sir Isaac Barrow

Mathematical Shapes and Objects

As well as taking pictures of places and things related to specific mathematicians I always had my eyes open for mathematical objects wherever they might be.  The first two pictures are of a sculpture in the grounds of Holyrood Palace, Edinburgh, Scotland.  I think it's pretty cool that the monarch has an icosahedron in her garden.  The pictures following those are of labyrinths whose locations are given in the photo captions.

Location - London Tube Station
Location - Paris - on the bank of the Seine
Location - Paris - on the bank of the Siene
Seattle - near the Space Needle - taken at the JMM Conference before I left for the European part of my sabbatical
In London, after visiting the Royal Society I was walking back toward Westminster Pier along the east side of St. James's Park.  As I was looking across toward Buckingham Palace, a conversation caught my attention.  Someone in a group of people approached another group and asked, "What do you call a seven-sided shape?"  Discussion ensued regarding "heptagon" vs. "septagon."  It caught my attention because this is something I talk about in certain classes I teach, but I'd never heard such a conversation just out and about before.  I thought, "Why on earth are people who are just walking down the street conversing about names of polygons?"  

It struck me later in the day that this must have had to do with British currency, for which, though most coins are circular, two of them are seven-sided polygons.

Friday, June 10, 2016

Carl Friedrich Gauss

Gottingen University Observatory (front)
Carl Friedrich Gauss (1777-1855) is known as the Prince of Mathematics and is generally considered to be one of the "top three" mathematicians of all time, along with Archimedes and Sir Isaac Newton. Gauss was the first director of the observatory in Gottingen, pictured above and below.  From 1807 until his death in 1855, Gauss lived in the rooms at the observatory - rooms later occupied by Dirichlet and then Riemann.
Gottingen University Observatory (back)
His aptitude for mathematics was evident early-on.  When he was 3 years old he discovered an error in his father's payroll calculations.  When he was 7 years old, his teacher, wanting to keep the students busy for a while, assigned the class to add the numbers 1 to 100.  Instead of doing this in the way most people would and in the way his teacher certainly must have expected, beginning with 1+2=3 and then adding 3 onto that result and then 4 and so on, Gauss somehow saw a faster, easier way of doing this and wrote down the correct answer 5050 almost immediately.  (Later in the post I'll explain how he did it, but I'm saving it for later in case you want to take a moment to see if you can find a short cut too!)

Gauss contributed to many branches of pure and applied mathematics and mathematical physics.  We, who love our phones, can thank Gauss for work he did very early on telegraphy.  Through the work of Gauss and his Gottingen colleague, physicist Wilhelm Weber, the first electromagnetic telecommunication was sent in 1833, predating Samuel Morse's telegraph by 4 years.  Below are photographs of memorials of this event on the grounds of the observatory - a monument and a plaque - as well as a statue elsewhere in town. These photos are followed by images relating to Weber - a street in Gottingen named for him, and his tombstone.
Wilhelm Weber tombstone - Stadtfriedhof Cemetery, Gottingen, Germany
Though Gauss made contributions to almost all branches of mathematics, his favorite seems to have been number theory.  He called mathematics the "queen of the sciences," but he called number theory the "queen of mathematics."  In 1798, at the age of 21, he wrote a textbook titled Disquisitiones Arithmeticae, which transformed number theory from a collection of isolated theorems and conjectures into a well-structured branch of mathematics.

He had a passion for prime numbers and would at times spend "an idle quarter of an hour" factoring a thousand consecutive numbers (a "chiliad" of numbers) in order to determine which were prime.  At the age of 19 he formulated his prime number theorem, which has to do with the distribution of prime numbers among the counting numbers.

In that same year he constructed a regular 17-sided polygon using a straight-edge and compass.  His proof that this could be done was the first progress in 2000 years in the area of constructing regular polygons.  The images below - window art and wall posters - celebrate this, but they are not from Gottingen, as the rest of the pictures in this post are, but rather from the Mathematical Sciences Research Institute in Berkeley, California.
Not only does MSRI have a display about Gauss's construction of the 17-gon, but it is the case that they are located at 17 Gauss Way!  How's that for homage to a great mathematician?
While Gauss worked in many areas of mathematics, he did not always publish his findings.  His motto was "Pacua sed matura" ("Few but ripe").  One example of this is his work in non-Euclidean geometry, and, unfortunately the tale has a bit of a sad ending.

A young man named Janos Bolyai, a Hungarian, also entered into the (at the time) strange and very new universe of non-Euclidean geometry, a geometry of curved space, which we now know to be the sort of geometry Einstein needed in order to formulate his theories of relativity.

Janos's father, Farkas, who was also a mathematician, warned him not to pursue this strange, new area - calling it an "endless night" that would consume all his leisure, his health, his peace and his joy in life.  But Janos couldn't leave this intriguing mathematics alone, and he ended up developing what we now call hyperbolic geometry.

Farkas, who had been a student of Gauss, wrote a mathematics textbook and included his son's work as an appendix; he then sent this to Gauss, his former teacher and the greatest mathematician of the age.

Gauss responded that he could not praise the work of Janos because, as he wrote, to "praise it would be to praise myself, for the entire content of the work, the methods that your son used, and the results to which he was led coincide almost completely with my own meditations from 30 to 35 years ago  .  .  .  My intent was not to let any of my own work on this, of which till now very little has been put on paper, be known during my lifetime  .  .  .  I am therefore surprised to learn that I have been spared the effort, and it is very pleasing to me that it is the son of my old friend who has anticipated me in such a surprising manner."  (This was no false boast on Gauss's part.  He had, in fact, made these discoveries years before and never published them.)

Upon receiving this response Janos was greatly upset to the point of physical and mental illness.  This promising young man, whom Gauss called a "genius of the first order" in a letter to someone else but not to Janos himself, gave up mathematics and remained impacted throughout life by what he felt to be a severe blow.  Gauss hadn't meant to discourage him, nor to belittle him; Gauss was just stating facts, but Gauss was rather lacking in tact in how he did so.

I could, and probably should, balance the scales by telling the story of Gauss's encouraging mentoring of Sophie Germain, but I have written a bit about it in my post about her life, which you can find by clicking on her name and then scrolling about three-fourths of the way down that post.  Yet though he mentored her kindly, at one point he just stopped responding to her letters when other things took his attention.  And then there's the story of his comment when he was in the middle of a math problem and was told his wife was dying  .  .  .  so it seems that tact was truly not a strong point for him.
The tower in the three pictures above, seen from Plesse Castle near Gottingen, is known as Gauss's Tower.  Though the current tower dates from 1964 it is near a spot used by Gauss in his work surveying Hanover, as commissioned by King George IV of England.  While doing this work, Gauss invented the heliotrope, an instrument that reflects sunlight across great distances.  (At this link is a nice essay about his work as a surveyor.)

I've mentioned that Gauss was the greatest mathematician of his age.  He first came to wide-spread fame in 1801 when he was 24 years old when his mathematical ability allowed him to predict where the newly discovered dwarf planet Ceres would reappear after disappearing, in its orbit, behind the glare of the sun.  Astronomers were thrilled at being given the ability to find Ceres again, because mathematical tools that they had access to at the time were not sufficient for them to extrapolate its future position given the small amount of data (1% of the orbit).

SPOILER ALERT: I guess it's time that I should share GAUSS'S SHORTCUT for adding the numbers 1 to 100, as promised earlier.  Though most people think math is about making life hard; math is actually about making things easier, and this can often be done by finding a different viewpoint.  Instead of plowing through and adding the numbers in order, Gauss realized that if he added bigger numbers and smaller numbers it would simplify the work.

Considering 1+2+3+4+5+6+7+ . . . +94+95+96+97+98+99+100

He noticed that the first number plus the last, 1+100, is 101.

Also, the second number plus the second-to-last number, 2+99, is 101.

AND 3+98 is 101, as is 4+97 and 5+96 and 6+95 and so on and so on.

By adding smaller numbers to bigger numbers in this way, he ended up with 50 pairs that each added to 101.  Fifty 101s is fifty times one-hundred one, which is an easy mental multiplication.  Do 5x101 and stick a zero on the end to get 5050.
 As was Weber, Gauss was honored with a street in Gottingen being named after him, Gaussstrasse.  He has also been honored by being pictured on currency and on postage stamps.  His is buried in what was the Albani Cemetery of Gottingen; his tombstone is pictured below.
Near Gauss's grave is a pond with a fountain, which is what remains of the old moat outside the town wall.
What was the Albani Cemetery is no longer in use as a cemetery and has been re-purposed as a park for the people of Gottingen.  Given Gauss's temperament, I'm not sure how he would feel knowing people were picnicking around his tombstone.  I hadn't known this was the case until after I arrived, so it felt rather strange to me as I came reverently and on sort of a pilgrimage to pay homage to Gauss to arrive and find people sunbathing, playing Frisbee and barbecuing all around his grave, but time does move on, I suppose, and life is for the living.  Just don't be surprised if you go to see his grave as well and there's a big party going on!