Archive for May, 2011

An easy physics problem…

Posted in Cute Problems with tags , , , on May 26, 2011 by telescoper

Based on the popularity of something I posted last week, I thought some of you might find this little problem amusing. It’s from a Physics A-level paper I took in 1981. The examination comprised two papers in those days (and a practical exam); one paper had long questions, similar to the questions we set in university examinations these days, and the other was short questions in a multiple-choice format. This is one of the latter type, from the mechanics section.

And here is a poll in which you may select your answer:

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Pump up the Volume

Posted in Music with tags , on May 26, 2011 by telescoper

In the course of an archaeological investigation into one of the cupboards in my study last night I unearthed a box full of old vinyl 12″ singles, including this, Pump up the Volume by MARRS. Clearly inspired by the theory of cosmic inflation, it was very popular in dance clubs way back when I was a graduate student living in Brighton, then it got into the charts and climbed to Number 1 thus endowing MARRS with the status of One Hit Wonder. I was shocked when I realised that all this happened in 1987, before most of my students were born. Sigh. Anyway, it was nice to see the cute video again so I thought I’d share it for the benefit of other oldies out there, with the excuse that it’s slightly space-related. Try playing “spot-the-sample” as the record plays – it’s entirely cobbled together from bits of other tracks..

The Cosmos according to Disney

Posted in The Universe and Stuff with tags , , on May 25, 2011 by telescoper

Not really time for a proper post today but I’m grateful to one of my PhD students for coming to the rescue by pointing out this clip in which our own Professor Mike Disney tell us everything he knows about cosmology. The video lasts 2 minutes and 48 seconds.

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In Memoriam – HMS Hood

Posted in History with tags , , , on May 24, 2011 by telescoper

I just realised that today is a solemn anniversary which surprisingly hasn’t been marked in the media. On this day 70 years ago, i.e. 24th May 1941, the Royal Navy battlecruiser HMS Hood was sunk by the German Battleship Bismarck in the Battle of the Denmark Strait. Of a ship’s complement of 1418 only three survived the sinking of HMS Hood; it was one of the greatest maritime disasters of the Second World War. I’m not one for dwelling excessively  on the past, but I think it’s a shame this event has not been remembered. We owe a lot to people like the 1415 who gave their lives that day, so I’m glad I remembered in time to pay my respects.

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Whatever happened to Euclid?

Posted in Education, The Universe and Stuff with tags , , , , , , on May 24, 2011 by telescoper

An interesting article on the BBC website about the innate nature of our understanding of geometry reminded me that I have been meaning to post something about the importance of geometry in mathematics education – and, more accurately, the damaging consequences of the lack of geometry in the modern curriculum.

When I was a lad – yes, it’s one of those tedious posts about how things were better in the old days – we grammar school kids spent a disproportionate amount of time learning geometry in pretty much the way it has been taught since the days of Euclid. In fact, I still have a copy of the classic Hall & Stevens textbook based on Euclid’s Elements, from which I scanned the proof shown below (after checking that it’s now out of copyright).

This, Proposition 5 of Book I of the Elements, is in fact quite a famous proof known as the Pons Asinorum:

The old-fashioned way we learned geometry required us to prove all kinds of bizarre theorems concerning the shapes and sizes of triangles and parallelograms, properties of chords intersecting circles, angles subtended by various things, tangents to circles, and so on and so forth. Although I still remember various interesting results I had to prove way back then – such as the fact that the angle subtended by a chord at the centre of a circle is twice that subtended at the circumference (Book III, Proposition 20) – I haven’t actually used many of them since. The one notable exception I can think of is Pythagoras’ Theorem (Book I, Proposition 47), which is of course extremely useful in many branches of physics.

The apparent irrelevance of most of the theorems one was required to prove is no doubt the reason why “modern” high school mathematics syllabuses have ditched this formal approach to geometry. I think this was a big mistake. The bottom line in a geometrical proof is not what’s important – it’s how you get there. In particular, it’s learning how to structure a mathematical argument.

That goes not only for proving theorems, but also for solving problems; many of Euclid’s propositions are problems rather than theorems, in fact. I remember well being taught to end the proof of a theorem with QED (Quod Erat Demonstrandum; “which was to be proved”) but end the solution of a problem with QEF (Quod Erat Faciendum; “which was to be done”).

You can see what I mean by looking at the Pons Asinorum, which is a very simple theorem to prove but which illustrates the general structure:

  1. GIVEN
  2. TO PROVE
  3. CONSTRUCTION
  4. PROOF

When you have completed many geometrical proofs this way it becomes second nature to confront any  problem in mathematics (or physics) by first writing down what is given (or can be assumed), often including the drawing of a diagram. These are key ingredients of a successful problem solving strategy. Next you have to understand precisely what you need to prove, so write that down too. It seems trivial, but writing things down on paper really does help. Not all theorems require a “construction”, and that’s usually the bit where ingenuity comes in so is more difficult. However, the “proof” then follows as a series of logical deductions, with reference to earlier (proved) propositions given in the margin.

This structure carries over perfectly well to problems involving algebra or calculus (or even non-Euclidean geometry) but I think classical geometry provides the ideal context to learn it because it involves visual as well as symbolic logic – it’s not just abstract reasoning in that compasses, rulers and protractors can help you!

I don’t think it’s a particular problem for universites that relatively few students know how to prove the perpendicular bisector theorem, but it definitely is a problem that so many have no idea what a mathemetical proof should look like.

Come back Euclid, all is forgiven!

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Move

Posted in Jazz with tags , , , , on May 23, 2011 by telescoper

Well, it’s 1pm and my third-year students are just sitting down for two hours of fun with their Nuclear and Particle Physics examination. For my part I’m obliged to sit by the phone for the next two hours in case there’s a problem with the examination paper. Ideal excuse for a quick blog post while I eat my sandwich.

I also notice from my trusty wordpress dashboard that this is my 1000th post since I started blogging, way back in late 2008.  Time to indulge myself, then. I haven’t posted much jazz recently so I thought I’d share this classic recording with you. It’s from my favourite era of jazz – the late 1950s – and my favourite kind of jazz, bebop, which by then had matured, ripened and hardened considerably since its birth in the 1940s.

This gives me the excuse to mention a nice article in Saturday’s Grauniad about the poet Philip Larkin, his love for “trad” and his hatred for the “modern” jazz exemplified by bebop. It’s entirely a matter of personal taste, of course, but speaking for myself I can say that I’ve never had any problem loving jazz of all ages. For me, though, it reached a peak in the late 50s with musicians of the calibre of Miles Davis, John Coltrane, and Ornette Coleman.

This particular track features alto-saxophonist Lou Donaldson whom many jazz critics regarded as a pale imitation of the pioneering be-bop icon Charlie Parker but whose playing I’ve always admired. In my book, anyone brave enough to follow Charlie Parker deserves the highest esteem. In any case when Lou Donaldson walked into the Van Gelder studio in Hackensack, New Jersey on July 28th 2008 he clearly had fire in his belly.

The tune is entitled Move and it was written by drummer Denzil Best. It’s quite unusual for a drummer also to be a composer, but Best wrote a number of classic jazz tunes. I even managed to find the chords that make up this one’s 32 bar AABA structure…

Many bebop compositions are based on the chord progressions of standard tunes, such as How High the Moon or I Got Rhythm, but with the melody replaced by something much more intricate than the original tune. I don’t recognize the chords above from anywhere else so it may be an entirely original composition by Denzil Best. I’m sure there’s a jazz buff out there who will correct me if I’m wrong. In any case the jagged melody is archetypal bebop stuff – complex and angular, very difficult to play but intensely exciting to listen to.

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Can the CMB Alone Provide Evidence for Dark Energy? (via astrobites)

Posted in The Universe and Stuff with tags , , , on May 22, 2011 by telescoper

While I’m in reblogging mood I’ll try to send some traffic the way of this post, which is somewhat related to Friday’s one about the Wigglezeddy survey (or whatever it’s called)…

Can the CMB Alone Provide Evidence for Dark Energy? Paper Title: The Atacama Cosmology Telescope: Evidence for Dark Energy from the CMB Alone Authors: Blake D. Sherwin et al. 1st Author’s Affiliation: Dept. of Physics, Princeton University Introduction Continuing with Monday’s theme of cosmology, today’s astrobite features an ApJ Letter that describes new evidence for dark energy.  In the past decade a number of cosmological tests have been developed that show a need for a cosmological constant th … Read More

via astrobites