Friday, July 29, 2016


Having concluded my mathematical travels, and desiring this blog to work as a reference for my students and to be comprehensible for others who come across it, I feel it's important to put up this final post in order to round things out and provide an overview.  This will serve as a conclusion for those of you who have followed me on my travels, and it will serve as an introduction for those of you just now finding this blog (including my future students).
Tram to from the airport into Edinburgh - my first stop in Europe
This blog covers my mathematical travels from spring 2016, which were undertaken as part of a sabbatical focused on the history of mathematics from the Renassaince to the present in northern Europe.  The map at the top of the post shows my stops.  These travels were profoundly impacting to me, and I believe the outcomes will resonate throughout my teaching over the rest of my teaching career.  Multiple goals were achieved through this travel, but one of the biggest is that of having walked in the footsteps of the famous mathematicians from the last 500 years in such a way that I will be able to bring them to life for my students.  I hope that through doing this I will be able to help make math more approachable and interesting to my students and to people in my community by truly putting a human face on mathematics.

I hope this final post relating to my mathematical travels in northern Europe will serve as a sampler of what is to be found in the previous posts of this blog.  I will include some links here to other posts, as well as a wide variety of pictures.  There is also a search bar on the top left, and a list of labels on the right that can be used to search for specific topics, places and people of interest.  ENJOY!
 Mathematician Mary Somerville (1780-1872), who eventually tutored Ada Byron (yes, that Byron), who was to become a mathematician herself.  Mary grew up in Burntisland, Scotland.  Part of what inspired Mary to study mathematics was the natural world around her - especially stars she could see out of her bedroom window and the Burntisland beach and tide pools near her childhood home.
 My intent was to walk in the footsteps of the mathematicians I was studying, and I did that very literally!  I wanted to experience what inspired them so I can use it to inspire others!
Mary had extended family who lived further south in Scotland, in the borderland town of Jedburgh.  Her uncle was the minister at Jedburgh Abbey (seen in ruins above) and lived on the Abbey grounds.  Mary and her cousins would swim in the Jedwater (also seen in the picture above).

As well as having walked in Mary's footsteps, I also accessed her papers and read correspondence between the Somerville and Byron/Lovelace families housed in Oxford's Bodleian Library.  The lives of Mary Somerville and Ada Byron Lovelace opened up before me in amazing ways as I read their letters to each other.
The Radcliffe Camera - one of the Bodleian Reading Rooms in Oxford
A folder containing correspondence between Mary Somerville and Ada Byron Lovelace
 While in Oxford I also accessed the archives at Queen's College Library.  Here I found the rumors that mathematicians are lazy to be confirmed for me!  In reading a first edition copy of Robert Recorde's Whetstone of Witte, I came across the place where the equals sign was used for the first time.  Recorde wrote that he decided to create the equals sign because he was tired of the "tedious repetition"of writing the words "is equalle to" between both sides of all of his equations. He created that small symbol = that we all recognize immediately today.  He decided to make the symbol of two parallel lines of the same length, as he put it, "bicaufe noe.2. thynges can be moare equalle." (The reading of 16th century English proved a bit challenging, but in a fun way, like a treasure hunt!)
The view from my table at Queen's College Library, Oxford
My table at Queen's College Library, Oxford, with the 1557 first edition Whetstone of Witte waiting for me
 I chose each of my destinations solely with regard to connections with mathematics and mathematicians.  Another mathematician I was interested in at Oxford is a person most of us know better as a literary figure: Lewis Carroll.  Actually, his "day job" was that of being an Oxford don of mathematics - on the side he was a writer and photographer - very accomplished in those areas!
 Carroll's Alice stories were inspired by a real person, Alice Liddell, the young daughter of Carroll's dean at Christ Church, Oxford.  Carroll often took Alice and her sisters boating and also to the then newly opened Oxford University Museum of Natural History (see above), where they encountered some of the "beasts" that make an appearance in his works (see below).
Lewis Carroll is not the only whimsical mathematician I came across in my study and travels, another such is Dr. John Dee (1527-1608).  Given what I've learned I can't believe I hadn't come across him before; he's everywhere in popular culture - appears frequently in modern fantasy, horror, and historical fiction novels, may have been the model for Prospero in Shakespeare's Tempest and Christopher Marlowe's Doctor Faustus.  He even shows up in an Iron Maiden song and a PlayStation video game!

Dr. Dee was an adviser to Queen Elizabeth I.  At the time in which he lived science and magic were not well-distinguished, and scholars such as himself could be expected to do such things as draw up horoscopes and cast out demons.  Though this was expected, it put men such as he in danger of being associated with the occult, which did happen to him.  He then fell out of favor with the Queen and died in poverty and obscurity.  And yet, as we've seen, he lives on in literature, music, games and even an opera!

Pictured below are some of his mystical implements, including his crystal ball and obsidian scrying mirror.  These items were on display at London's Royal College of Physicians, on loan from the British Museum.
Dee is not the only mathematician of this era to have been considered by some to be a sorcerer.  Another such is John Napier (1550-1617), a Scottish nobleman and elder in the Presbyterian Church, whose castle tower is pictured further down in this post.

As well as visiting beaches, museums, libraries and universities where mathematicians lived, worked and gained inspiration, I also visited gardens, woodlands, crypts, castles, cemeteries, cathedrals, homes and classrooms associated with mathematicians and their lives - some photographs of which follow this paragraph.  Also included in the posts on this blog are commentary about the very real joys and very significant struggles of traveling abroad for the first time and traveling solo.  Although the travels were independent and solo there were three cities in which I met with fellow mathematicians who proved to be of tremendous help to me, so this post closes with pictures of each of them to whom I feel tremendous gratitude!
Tuileries Garden, Paris, where Mary Somerville spent time convalescing
Forest of Gottingen, Germany where David Hilbert walked with colleagues and students
Crypt of the Pantheon in Paris - tomb of Joseph Louis LaGrange (top, left, back)
Merchiston Tower - which remains of the castle of John Napier
Pere Lachaise Cemetery, Paris - burial place of Germain and Fourier
Window dedicated to George Goole in Lincoln Cathedral
View out of Sir Isaac Newton's bedroom window of the famous apple tree that he said inspired him
Cavendish Laboratory Classroom of James Clerk Maxwell - Cambridge
Dr. Piers Bursill-Hall & doctoral student Richard Chapling - Centre for Mathematical Sciences - Cambridge
Dr. Samuel James Patterson  - Mathematisches Institut - Gottingen, Germany
Dr. Manfred Stern of Halle, Germany and I having lunch and talking math at the famous Auerbachs Kellar, Leipzig

Thursday, July 28, 2016

Bletchley Park and Alan Turing

Bletchley Park "Mansion" from across the pond
My sabbatical travels were such a whirlwind that I didn't realize until long after I got home that I hadn't posted about my visit to Bletchley Park, the main site for British code-breaking during World War II, and its most famous code-breaker, mathematician Alan Turing.  The photographs above and below this paragraph are of "The Mansion," which was built in about 1880 as a modest gentleman's residence but was expanded in the late 1800s and early 1900s as a "country house."  It was bought by the Secret Intelligence Service (M16) in 1938 - one reason for this choice of location was that it was on the "Varsity Railroad Line" which ran between Oxford and Cambridge - providing easy access for mathematicians and other code-breakers from these universities.
A bit of the grounds at Bletchley Park
The Mansion

I found both the exterior and the interior to be absolutely exquisite!  Though it has been called by some a "maudlin and mounstrous pile" because it combines Victorian Gothic, Tudor and Dutch Baroque styles.  Though construction on this mansion didn't begin until 1878, residence on this site goes back at least as far as the Domesday Book of 1086, when it was part of the Manor of Eaton.
It's hard to estimate the full value of the work that was done here.  Until fairly recently it was Britain's "best kept secret."  Secrecy surrounding the activity done here was vital to national security and to victory in WWII.  As the plaque below expresses the intelligence work done here saved countless lives and helped significantly shorten the war.

One benefit of this location is that it includes extensive grounds that could be used for recreation from their intense work by the mathematicians and other intelligence workers and staff that were housed here.  The pond, seen in the first image in this post, would freeze over in winter and be used for ice-skating.  There are also tennis courts on the grounds.
Below is a view of the cottages where Turing and others made breakthroughs on the German Enigma Code.  Also included are photographs of some informational signage from this area.
The Cottages, Bletchley Park
Turing, a fellow of Trinity College, Cambridge, was instrumental in developing The Bombe which functioned to discover daily settings of the German Enigma machines.  Among other influences he was inspired in his work in logic and computing by the work of Charles Babbage and Ada Byron Lovelace (whose powerful story I have blogged about here).  Pictured below is a replica of The Bombe, which is on display in The Mansion.  It is a prop from the 2014 movie about these events and this era "The Imitation Game," starring Benedict Cumberbatch.
As well as the mansion and the cottages, which already existed on the site when M16 bought the estate, many huts were built to house specific aspects of the intelligence work going on at B.P. (as it was affectionately known by those who worked there).  For instance, Hut 4 dealt with Naval Intelligence, Hut 7 dealt with the Cryptanalysis of Japanese Naval Codes, Hut 2 was for "beer, tea, and relaxation," and Hut 11 was for Bombe Building. 
Alan Turing was a brilliant mathematician who contributed much to mathematics, to the foundations of computer science, and to the war effort.  He is widely acknowledged as the Father of Modern Computing and also as the Father of Artificial Intelligence.    His "Turing Machine"can be considered a model of a general purpose computer.  In 1951 he was elected a member of the Royal Society of London.  Yet for all that he contributed, because of the secrecy of the war effort his contributions could not be recognized in his time.  Additionally, his life ended tragically.

In 1952 Turing, who was homosexual, met a young man with whom he began a relationship.  An acquaintance of his partner robbed Turing's house.  Turing reported the robbery to the police, and during questioning it came out that he was in a homosexual relationship.  At that time homosexual acts were considered criminal offences in England, and Turing was charged with "gross indecency."  He plead guilty and was given a choice between imprisonment or probation which would include hormonal treatments with estrogen to suppress his libido.  The conviction also led to the loss of his security clearance, and it barred him from further cryptographic consulting for the British government.  He was also denied entry into the United States after this conviction.

Two years after his conviction his housekeeper found him dead in his home.  He had died of potassium cyanide poisoning.  There is some doubt as to whether his death was accidental or if it was suicide.  He was working with electrolysis experiments at the time and may have accidentally ingested it.  His mother maintained that his death was accidental.  But a half eaten apple was found by his bedside when his body was discovered, and, according to his biographer David Leavitt, his favorite fairy tale was Snow White and the Seven Dwarfs, and, in Leavitt's words, Turing took "an especially keen pleasure in the scene where the Wicked Queen immerses her apple in the poisonous brew."

Mathematician and artist Lidia Luquet, captured this tragedy well in her painting, pictured below, a Magritte homage which brought me to tears when I saw it displayed at a conference at U.C. Davis early in my sabbatical semester (more about that at this link).

One last look at The Mansion
There are a number of excellent movies about the life of Alan Turing and/or the work at Bletchley Park.  One of these is the 2014 film The Imitation Game starring Benedict Cumberbatch, another is the 2001 film Enigma starring Dougray Scott and Kate Winslet, and my favorite is the 1996 BBC production Breaking the Code: Biography of Alan Turing starring Derek Jacobi (which can be found at this link on youtube).

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.