eclipse chasing

The last total solar eclipse visible from the mainland United Kingdom occurred on 11 August 1999, with the path of totality passing through part of Cornwall. I took a few days off work (at the time I was working on a short-term contract for a software company in Dublin) and joined some friends who were going to try to see it.  In the event it didn’t really work out: we saw some of the partial phase before the clouds thickened and it wasn’t until a few hours later, well after the end of the eclipse, that the sky cleared again. So apart from seeing the overcast sky go briefly dark, and trying to guess which topical pieces of music the local radio station would play (Moonlight Shadow by Mike Oldfield and Here Comes the Sun by the Beatles) it all came to naught. In retrospect, I’d have seen much more of the eclipse if I’d stayed in Dublin, which had clear skies and got about 90% totality. (Also, I wouldn’t have had to camp overnight in a tent, which I don’t remotely enjoy.)

Today’s eclipse, while partial, was a much better occasion. I heard from friends elsewhere that some parts of the UK were overcast, but Coventry had pretty clear skies throughout, apart from some thin, hazy clouds for a short while in the middle, and I was able to watch the whole thing from start to finish.  I dragged my telescope (a 130mm Newtonian reflector) out onto the drive and carefully set it up, projecting onto a pad of paper.

08:53am	09:06am

(If you look carefully at the second image you can see a sunspot.)
I also took the opportunity to try that thing with a colander. The holes in the colander act like a sort of unfocused pinhole camera, and project lots of slightly fuzzy copies of the partial eclipse:

The eclipse progressed over about the next hour and a half:

09:22am	09:26am
09:42am	10:34am

And finally finished at about 10:40am:

At about quarter past ten, I smelled burning plastic and noticed a thin wisp of smoke drifting out of the end of the telescope. The sun’s image had drifted half out of the field of view (where it would have remained if I’d ever got around to buying some batteries for the equatorial drive motor), and was now focused on the lens casing. Oops. The lens itself is ok, but the casing is now slightly melted.

And that, boys and girls, is why we must never, ever look directly at the sun, especially not through a telescope or binoculars.

Ramanujan’s construction for almost squaring the circle

I have ambivalent feelings about Pi Day. Certainly, anything that gets people talking about mathematics is broadly a Good Thing. And clearly a lot of people have tremendous fun, and really enter into the spirit of things, and jolly good for them. But, says a curmudgeonly voice in my head, it’s a little bit contrived. If you use base ten to represent numbers, and if you use the American month-then-day convention to represent dates, then the 14th of March gives actually a pretty rough approximation to the value of pi. (On the other hand, if you use the day-then-month ordering, the 22nd of July yields a more accurate approximation.) But this year, the sequence you get is 3/14/15, which is better than usual, especially if you happened to be looking at your clock at 9:26:53 precisely.
(I wasn’t – we went on holiday to the Peak District for the weekend, so at the numerologically significant second I was helping install a 14-month-old child into a car seat, ready to head off into one of the more scenic bits of Derbyshire for a couple of days.)
Anyway, the Aperiodical had a competition to find interesting methods of approximating pi. This looked great fun, but I totally failed to get around to doing anything about it. Many others were, as usual, better organised and motivated than me, and a list of some of their attempts can be found here.
But earlier today, in Ian Stewart‘s very readable textbook on Galois Theory, I found a nice ruler-and-compass construction, due to Ramanujan, of an approximate solution of the age-old, impossible problem of squaring the circle. This evening, I gave it a go with a moderately sharp HB pencil, a 30cm ruler, a pad of graph paper, and some compasses that were originally part of a geometry set and pencil case I got in the mid-1980s for opening a “Griffin Savers” account at the local branch of the Midland Bank (long since amalgamated into international tax-evasion facilitators HSBC).
And this is the result.

The construction is as follows: First draw a circle with centre O, and let AB be a diameter. Now M is halfway along OA and T is 2/3 along OB. P is vertically above T, and then the length BQ is equal to that of TP. TR and OS are parallel to BQ, the length of AD is equal to that of AS, and then draw AC tangent to the circle with the same length as RS. Now mark E so that BE is the same length as BM, and mark X so that EX is parallel to CD.
Once you’ve done all that, says Ramanujan, the square with side BX should be very nearly equal to the area of the circle you first thought of.
So I did all this, very carefully measured BX (~15.8cm) and OB (9 cm), and got a value of about 3.082 for pi, which isn’t that far off the correct value. With a sharper pencil and a larger piece of paper, I reckon I could probably have got a bit closer to the correct answer. I need to think further about why this construction works, and I might even get around to writing more if I figure it out.
(Also, go and visit Derbyshire sometime, because it’s very pretty.)

In Defence of the Study of Mathematicks

Another 18th-century academic rant, this time from a correspondent who signs himself “Philo-Mathematicus”. From The Gentleman’s Magazine, vol. 6, no. 11, November 1736, pages 665-666.

So contemptible is the name of Arts and Sciences grown (especially those Arts that belong to the Mathematicks) among some Sort of People, whose Fortunes best qualify them to understand ’em, that they speak of them with Derision, and look upon the Mathematicks as a mere whimsical Invention or Jest, which they despise, because they will not take Pains to comprehend. Tell them of a Pack of Cards, or of any such fashionable Arts, whereby they can spend Time in a Laugh, (and so avoid the Trouble of thinking too nicely) and they’ll as zealously join with you, against Books and Study, as they’ll exclaim against Vices they often commit. They insist, that by spending Time over Books, a Man pores himself into Stupidity, and unqualifies himself for Company, and polite Conversation. But what can be said to a multitude of Men, who are above being sway’d by Reason, and are resolved to chuse their Delights according to their own vitiated, habitual, and confin’d Way of thinking? They admit of nothing for an Excellency, but what has Pomp or Fashion attending it, and directly suited to their own deprav’d Palate. There are Cavillers of all Degrees and Capacities, who make it their Business to degrade and find fault with the Performances of other Men; and would, if possible, reduce all Mens Capacities, above theirs, to the Level of their own: But such invidious Dispositions discover themselves by their own Pride and Weakness; and whilst they asperse others for a Vanity, in attempting Things, as they say, not worth the Enquiry, they appear Grovelings, who would suppress all Arts and Knowledge, whereby Men are lifted up above Men, or Nation above Nation. They encourage Flattery and Dissimulation, but hate to be thought Knowing or Wise. It is very hard, Truth and useful Knowledge should suffer so much Disgrace by so many Enemies they both have in the World, and that so little regard is had for them by Mankind in general! It is much to be desir’d, for the Honour of our British Nation and its Inhabitants, that the fashionable Arts of Carding, Drinking, Gaming, Powdering and Cringing, were neglected or more thrown aside; and the brave Scientifical, and noble Grecian and Roman Arts, introduced or revived among us. We should be more the Glory and Delight of ourselves, as well as the Esteem and Dread of our Neighbours, if we would endeavour to flourish in Arts, and grow wise in Knowledge: Whereas, we are now dwindled to a despis’d Race of Mortals, who are vers’d only in Mimickry, Foreignism and Luxury; and the Lord knows when we are to be deliver’d out of our Troubles.

— Philo-Mathematicus

Of Academical Education

From The Universal Spectator, no. 135, Saturday 8 May 1731; reprinted in The Gentleman’s Magazine, vol. 1, no. 5, May 1731, pages 196-197.

The Gentleman’s Magazine, May 1731

A Gentleman, who had been lately at one of our Universities, for his Diversion, gives our Author some Account of their Methods of Education.

Says, he found in the younger part of the University a generous and noble Spirit reigning, and good Sense improv’d and elevated by a valuable stock of choice and useful Learning, widely different from the usual run of young Fellows about London, whose utmost stretch of Learning is to repeat scraps out of Plays, or Poetry, or perhaps produce a few stale Arguments against Christianity.

But takes notice of one general fault among them, i.e. the distance observ’d by Tutors to their Pupils; whereby the paternal and filial Affection which should subsist between them, is prevented, and misunderstanding and distance occasion’d. For nothing wins more upon young People, than a good natur’d open Treatment.

To this Distance and Reserve may be attributed, that so few friendships are contracted between Tutor and Pupil. The Haughty and Dogmatical are substituted in the room of the Friendly, Benevolent and Obliging.

Hereby likewise, he says, they frequently embroil themselves with their Pupils, to the great uneasiness and prejudice of both. Knew a sober ingenious Youth treated with the utmost severity, on no other account than his Tutor’s ignorance of his temper and genius.

As to the Objection that Familiarity may breed Contempt, he answers, it may be just with respect to those Tutors, whose only Qualifications lie in Form and Distance, but not to those of real Merit.

A Tutor, he thinks, should delight in the Conversation of his Pupils, and make their Studies agreeable, and endear himself by Gentleness and Courtesy, whereby he would let himself into the knowledge of their Temper, and thereby be ready to amend the bad, and cherish the good.

 

Loncon 3

From about lunchtime on Wednesday 13 to about the same time the following Tuesday, I’m going to be at Loncon 3, the 72nd World Science Fiction Convention, which is taking place at the ExCeL exhibition centre in London Docklands. It’s looking like it might end up as the largest ever such occasion, and certainly the largest outside the USA: at the time of writing, there are 9904 registered members from all around the world (including one from the Holy See), of whom just over seven thousand are listed as attending.

The programme comprises nearly 1200 items (talks, panel discussions, interviews, presentations, concerts, plays, award ceremonies, book signings, etc) of which 121 are about something scientific. The exhibits hall is packed full of a wide range of displays, as well as the art show, the dealers’ tables, and a number of special displays relating to the guests of honour (including a wasp factory and a bone chair in honour of the late Iain (M) Banks).

I’m helping organise an academic poster session showcasing current research in fields ranging from astrophysics to palaeogenetics. Dr Moira Harrison has kindly donated money for a prize for the best poster, in memory of her father, the renowned science fiction author Harry Harrison.

Some people from Wikimedia UK will be running an editathon on Thursday in the Library area of the Capital Hall, and I’m intending to go to as much of that as I can manage: I’ve been to a few editathons over the past few years and they’re splendid occasions.

I’m on three programme items:

• Friday 15 August, 4:30pm (Capital Suite 7+12): Interview with Ian Stewart: I’ll be interviewing Prof Ian Stewart about his career in mathematics and science communication.
• Friday 15 August, 6:00pm (Capital Suite 15): What’s New in Maths: At the same time as Loncon 3, the 2014 International Congress of Mathematicians will be taking place in Seoul. On Wednesday, up to four new Fields Medals will be awarded, to recognise stellar achievement by mathematicians aged 40 or under. I’ll be moderating and participating in a panel discussion with Ian Stewart, Hannu Rajaniemi and Alice Hedenlund, looking at the work of the new Fields Medallists in particular, and some of the latest developments in mathematics in general.
• Monday 18 August, 10:00am (London Suite 2): Knots in Non-Euclidean Space: The space around some (actually, in some sense, almost all) knots has a well-defined hyperbolic 3-dimensional structure. I’ll try to explain what this means, and how we can use it to find out some useful geometric information about the knot itself.

The rest of the programme includes a wide range of talks and panel discussions, plays (including The Cancelling and Re-Imagining of Captain Tartan by David Wake, a sequel to a play he wrote and directed for Reconvene in 1999, the first convention I attended; and a dramatisation of Tim Powers’ novel The Anubis Gates) the 2014 Hugo and 1939 Retro-Hugo Award Ceremonies, an orchestral concert by the Worldcon Philharmonic Orchestra, and talks by a large number of interesting people, including the cosmonaut Anatolii Artsebarski and the Astronomer Royal, Lord Rees.

London Calling for Posters

The 72nd World Science Fiction Convention (called Loncon 3) will take place from 14-18 August 2014, at the ExCeL conference centre in London.  We’re expecting a good few thousand people to be there, from all around the world: the current membership list includes people from the UK, the USA, Canada, Australia, Europe, and even one member from the Vatican.

A lot of science fiction fans are interested in science, many have first or higher degrees in a scientific subject, and some are active researchers in academia or industry.  So, many of the main conventions (certainly Worldcons and Eastercons) have a strong science programming stream, with talks and panel discussions on subjects as diverse as new developments in genetics and the latest news in the search for exoplanets.

The exhibition hall in the ExCeL is quite big, and one of the things we want to put in it is a multidisciplinary academic poster session, where researchers in sciences and social sciences can come and tell us all what they’re working on at the moment, and what the latest developments in their chosen fields are.  Something similar was run, on a smaller scale, a few years ago and proved very popular, so we’d like to try it again.

This is an opportunity for active researchers to explain their current and recent work to an interested and educated lay (and in some cases not so lay) audience. There are more details on the Loncon 3 site, but if you have any questions please contact me (Nicholas.Jackson@warwick.ac.uk) and I’ll try to answer them.

 

Up in the hills above Bradford, outside the napalm factory

Last weekend I caught the train up to Bradford for EightSquaredCon, the 64th Eastercon (the main UK national SF convention, which has taken place almost every Whitsun or Easter bank holiday weekend since 1948).  The guests of honour this year were the writers Walter Jon Williams, Freda Warrington, the artist Anne Sudworth and the historian and SF critic Edward James.  I started going in 1999, having been instructed to do so by a friend of a friend who was on the organising committee that year (she said “Hello Nick, pleased to meet you. You’re going to Eastercon” and then I found myself handing over a cheque for the membership fee).  It was tremendous fun and I’ve been back every year since; I also go to a couple of other conventions most years as well, usually Picocon and Novacon, and am very much looking forward to Loncon 3, next year’s World SF Convention.

Many science fiction fans, in addition to their fondness for SF and fantasy books, television and films, are also interested in science (or, in fact, anything else that people are willing to tell them about) and so there’s usually a strong programme of science talks and panel discussions on offer over the four days of the convention.  In fact, many SF fans also have first or higher degrees (including PhDs) in science or other subjects, and are a wonderfully engaged and intelligent audience. This year I went to a fascinating panel discussion about recent developments in microbiology and genetics, another in memory of the late astronomer and broadcaster Sir Patrick Moore, as well as the annual George Hay Lecture, which this year was given by the palaeontologist and Tolkien scholar Henry Gee, who spoke entertainingly about his day job as a senior editor at Nature.

For the last few years I’ve also given a talk on something mathematical, and have been delighted by the positive feedback I’ve received.  (At last year’s talk, on unsolvable problems like squaring the circle, the room was full, with people standing at the back, and I was astonished to find out later that this was despite the fact that I’d been scheduled opposite an item where George R R Martin had been talking about the Game of Thrones television series and some of the cast had been doing sword-fighting demonstrations. If I’d known at the time, I think even I might have gone to that instead.)

This year, I talked about Pure and Applied Mathematics, and argued that the distinction between them is not very well-defined (if indeed it really means anything at all). I gave three main examples: applications of knot theory (which started out as a failed 19th century Theory of Everything) to genetics, the use of symmetry groups in molecular chemistry, and the importance of the representation theory of Lie groups and Lie algebras in particle physics.  I only finished writing the slides the day before, and the whole thing felt a bit ramshackle and disorganised to me, but a lot of people said they found it interesting.

The other programme item I participated in was a panel discussion on the Clay Mathematics Institute‘s seven Millennium Prize Problems in Mathematics.  This was moderated by Michael Abbott, a mathematics graduate who now works in software engineering (and who also did a heroic job organising the programme for the entire convention) and also included Susan Stepney, a former astrophysicist who is now professor of computer science at the University of York.  We got through the seven problems in an hour, although some of them proved easier to talk about than others: the Riemann Hypothesis is relatively straightforward to state (although obviously none of the seven are in any sense easy to solve) and Susan did a very good job of explaining P vs NP, but explaining the Hodge Conjecture or the Birch and Swinnerton-Dyer Conjecture to a nonspecialist audience (especially when I don’t really understand the details myself) isn’t very easy.  But I think we at least managed to communicate that all of these problems are very difficult and involve some extremely advanced and technical bits of mathematics.

All in all, a jolly enjoyable convention.  Next year’s Eastercon is in Glasgow and I’m quite looking forward to it already.

(The title of this post, by the way, is a line from this song, by the British band The Mekons. In addition to the Bradford and space travel references, it turns out that Jon Langford, one of the band’s founding members, is the younger brother of the Hugo-award-winning writer and SF critic David Langford.)

Happy birthday, Paul Erdős

A conjecture both deep and profound
is whether the circle is round.
In a paper by Erdős,
written in Kurdish,
a counterexample is found.

Today is the 100th birthday of the inordinately prolific, itinerant Hungarian mathematician Paul Erdős (26 March 1913 – 20 September 1996). During his long and illustrious career, he published just over 1500 papers in collaboration with 511 other mathematicians, on a variety of topics (particularly combinatorics, probability, number theory and analysis). It was perhaps inevitable that someone would analyse the interconnectedness of all these collaborations, and thus was born the Erdős number: the minimal path length between an author and Erdős in the collaboration graph.

So, Erdős has an Erdős number of 0, someone who has written a paper with Erdős but isn’t themselves Paul Erdős has an Erdős number of 1, someone who’s written a paper with one of them, but who hasn’t written a paper with Erdős himself (and isn’t themselves Paul Erdős) has an Erdős number of 2, and so on. This metric seems to have been first proposed by Caspar Goffman (1913–2006) in an article in a 1969 issue of the American Mathematical Monthly.

The American Mathematical Society provides a handy calculator to work out the collaboration distance between any two authors, at least via publications listed in their MathSciNet review database. My own Erdős number is 4, via the following path:

N M Dunfield, S K Friedl, N J Jackson, Twisted Alexander polynomials of hyperbolic knots, Experiment. Math. 21 (2012) 329–352
N M Dunfield, D Ramakrishnan, Increasing the number of fibered faces of arithmetic hyperbolic 3-manifolds, Amer. J. Math. 132 (2010) 53–97
V Kumar Murty, D Ramakrishnan, Period relations and the Tate conjecture for Hilbert modular surfaces. Invent. Math. 89 (1987) 319–345
P Erdős, M Ram Murty, V Kumar Murty, On the enumeration of finite groups, J. Number Theory 25 (1987) 360–378

Further details about the Erdős number can be found here, and biographies of Paul Erdős himself (who was a fascinating person and dedicated mathematician) can be found here, here, or in the book The Man Who Loved Only Numbers by Paul Hoffman.

A related concept is that of the Bacon Number, the minimal distance between oneself and the actor Kevin Bacon in the appropriate collaboration graph. So, for example, Kevin Bacon’s Bacon number is 0, Jack Nicholson’s is 1 (because they both appeared in A Few Good Men (1992)) and so on. The Oracle of Bacon provides a useful interface for finding shortest paths between actors listed in the IMDB database.

Depending on how strict you are about these things, my own Bacon number might be 4.

I was in a short amateur film called G103, as was a chap called Patrick Niknejad, who at the time was studying for a mathematics degree at Warwick, but who has also acted professionally. In particular, he was in 63 episodes of a children’s television series called My Parents Are Aliens which ran from 1999–2006. One of the other regular actors in that series was Tony Gardner, who appeared in Restoration (1995) with Meg Ryan, who appeared in In the Cut (2003) with Kevin Bacon himself.

Whether or not this actually counts depends on whether or not G103 is allowed. I argue that it is, because it has been shown on the big screen more than once (at about three of the Warwick Student Cinema‘s regular All-Nighter screenings, as well as a couple of departmental open days). (Also, this is my blog and I’m in charge. So there.)

Maths! And Jam!

A while ago, Colin Wright came to Warwick to give a talk on the mathematics of juggling.  Colin has a PhD in combinatorics, and is one of the inventors of the siteswap or Cambridge notation for describing juggling patterns.  This was a highly entertaining and interesting talk, and if you get an opportunity to hear it, I encourage you to do so.

Last month, I went to see Festival of the Spoken Nerd at the Warwick Arts Centre.  This was an evening of amusing and entertaining science-based comedy from Helen Arney (who writes and performs science-related songs), Steve Mould (who does spectacular experiments on stage) and Matt Parker (a stand-up mathematician).  Also excellent, and if you like Radio 4’s Infinite Monkey Cage (which starts a new series today) they’re definitely worth looking out for.

A couple of years back, Matt started up a movement called MathsJam, where people interested in recreational mathematics gather together in pubs around the UK once a month to chat about maths and related things in a friendly and enthusiastic environment.  Colin and Matt, together with a small group of other people also organise an annual conference, partly inspired by the biennial Gathering 4 Gardner conference. I’d been vaguely tempted to go – Colin had mentioned it after his juggling talk, and then I got chatting to Matt after the FotSN show and he also waxed enthusiastic about it, so I decided to go.

And it was absolutely splendid. Roughly a hundred professional and amateur mathematicians (researchers, schoolteachers, engineers, lecturers, and a whole host of other people including at least one archaeologist and a professional magician) gathered together in a hotel near Crewe, and spent the weekend discussing a range of fun aspects of maths. Participants were encouraged to give a short talk (strictly limited to five minutes) on something interesting and vaguely mathematical, and this comprised the bulk of the programme on the Saturday and Sunday. (I gave a very quick talk about Poincaré dodecahedral space.)

Matt had devised a clever method of ensuring everyone stuck to their allotted five minutes: a countdown timer that chimed quietly when there was a minute left, buzzed increasingly urgently after the five minute mark, and then displayed this at the six-minute point.

Tweeting was very much in evidence, and a live feed of the #MathsJam tag was projected up onto the main screen throughout the conference (I’ve collected them all here).

I got to meet lots of splendid people and catch up with a few people I already knew, I learned lots of interesting stuff I didn’t know before, and had my enthusiasm for mathematics reinvigorated. I plan to go again next year.

Emmy Noether (1882-1935)

Over the last couple of days I’ve had cause to read a little bit about the life of the German mathematician Emmy Noether, who was responsible for some of the most important work in abstract algebra during the first third of the 20th century; in addition, Noether’s Theorem is important in theoretical physics, where it describes the connection between symmetry and conservation laws.

Born in Erlangen in 1882 into a Jewish family (to the algebraic geometer Max Noether and his wife Ida Kauffmann) she displayed an early talent for languages, and qualified as a teacher of French and English, but decided she wanted to study mathematics at university.  At the time (c.1900) women weren’t permitted to formally enrol at German universities, although with the individual lecturers’ permission they could sit in on lectures.  Nevertheless, she succeeded in persuading Erlangen to admit her as a student in 1904 (after the restrictions were lifted) and successfully completed her PhD in 1907.  For the next eight years she taught at Erlangen, essentially as her father’s unpaid teaching assistant, before Felix Klein and David Hilbert succeeded in arranging a more substantive post at Göttingen (at the time, one of the most prestigious mathematics departments in Europe).  Initially (due largely to complaints from the philosophy department) she taught under Hilbert’s name, but in 1919 she was awarded her habilitation diploma and promoted to the rank of Privatdozent (lecturer).

She worked at Göttingen for the next couple of decades, along the way making some really important discoveries in algebra: amongst other things, she formulated and proved the First, Second and Third Isomorphism Theorems which are a standard component of pretty much every undergraduate group theory textbook; also Noetherian rings are an important class of algebraic structures which she studied and were subsequently named after her.  She supervised 14 doctoral students (according to the Mathematics Genealogy Project) and in 1932 was invited to give a plenary address at the International Congress of Mathematicians.

The following year, she was dismissed from Gottingen along with several other colleagues (including Max Born and Richard Courant) as a result of legislation enacted by the new Nazi government, forbidding Jewish academics from working at German universities.  She reacted with stoicism and continued unofficially lecturing to students at her house. Her colleague Hermann Weyl later remarked that “her courage, her frankness, her unconcern about her own fate, her conciliatory spirit, was in the midst of all the hatred and meanness, despair and sorrow surrounding us, a moral solace”.  Shortly afterwards, she was invited to take up a visiting professorship at Bryn Mawr College, Pennsylvania.  She worked there and at the Institute for Advanced Study in Princeton, for the next year and a half, until her sudden death in April 1935 as a result of complications from surgery.

I’d been aware of some of the details of her life, and obviously I learned the Isomorphism Theorems as an undergraduate (although I only learned of her involvement relatively recently) but reading about her in more detail over the last few days I’ve been struck by how particularly impressive her achievements were given that she was Jewish and a woman working in an environment that was often unsympathetic to both.