*Chris is a maths teacher at Grange Academy in Kilmarnock. I’m grateful to him for permission to post this link and to Stephen Wilson bringing it to my attention!*

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We will reveal some of the many ways in which mathematics helps us understand and improve sporting performance. Running, throwing, cycling, jumping, and weightlifting are among the examples we will take a look at from a new perspective. Along the way we will also see how Usain Bolt can break his world 100m record, investigate some odd scoring systems and see how delay differential equations help us understand American football.

(Personally, I doubt that even delay differential equations will ever explain American football to me, but I’m happy to give Prof. Barrow the benefit of the doubt…)

The lecture will take place at 4.30 p.m., in Lecture Theatre 3 of Appleton Tower, University of Edinburgh; tea will be served outside from 4.00 p.m. All are welcome, and there’s no charge for admission.

Previous lectures in this series have included Dr David Bedford on “Ants, Bricks, Bugles and Infinity” and Dr Colin Wright on the mathematics of juggling. We’ve had very good reports from those who’ve attended these events — it’s worth making the trek along the M8 for the occasion!

(DP)

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The famous dead salmon experiment should make us sceptical about reading too much into neuroimaging, but in justice to these researchers they seem to have been very careful in their analysis and they’re also careful not to overstate what they claim. Two aspects of the work, apart from the headline result, might be of particular interest to mathematicians. One is the question of which equations were regarded as particularly beautiful; the other is the relationship between understanding an equation and appreciating its beauty.

The complete list of equations used in the experiments is available as “Data Sheet 1” from the sidebar of the online article, and their scores in the pre-experiment beauty contest are in the spreadsheet “Data Sheet 3”. (You mght like to take a look at these and see how you rate them…) The top-rated equation was an old classic, Euler’s identity

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Also jostling for positions on the podium were the Pythagorean identity

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Euler’s expression for the complex exponential

(of which Euler’s identity is just a special case!), and the Cauchy–Riemann equations

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Firmly in the relegation zone, on the other hand, were Euler’s polyhedron formula (sorry Leonhard: can’t win them all)

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and the spectral theorem

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while right at the bottom of the table was one of Ramanujan’s bizarre formulae for ,

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I found myself less surprised by the top of the league table than by the bottom of it: Euler’s identity regularly tops lists of the most beautiful equations out there, while it as well as Pythagoras’s theorem and the complex exponential connect elementary mathematical objects in a very fundamental (and useful) way. The beauty of the Cauchy–Riemann equations is perhaps less apparent until you’ve done a first course in complex analysis and come to realise just how strongly they structure the behaviour of analytic functions.

At the bottom, I guess Euler’s polyhedron formula just looks too simple to tickle many people’s fancy — which is a shame for the philosophers of mathematics, because the derivation of this formula is a classic (and much-debated) case study in how mathematical results are obtained. I was surprised to see Ramanujan’s formula down there, because for me such expressions pack a considerable punch due to their sheer unexpectedness… but it’s possible that Ramanujan’s “strange Indian melodies”, as Douglas Hofstadter called them, strike the brain rather more like one of Courtney Pine’s further-out solos than one of Mozart’s sonatas — that is to say, our initial reaction is puzzlement, followed by appreciation, rather than pure enjoyment from the outset.

The Gauss–Bonnet and spectral theorems, I suspect, suffered from a different handicap, which is that they are sufficiently “technical” that many mathematicians (myself included) simply don’t know their context well enough to appreciate them. This brings us to the second interesting aspect of the study: it seems that to appreciate mathematical beauty it really helps if we understand what we’re looking at.

When non-mathematicians were shown the equations, they identified some of them (though not many!) as more beautiful than others, but their judgement didn’t coincide closely with that of the mathematicians — the researchers conclude that they “did so on the basis of the formal qualities of the equations”, or to put it more colloquially, whether they made pretty shapes on the page. For mathematicians, on the other hand, there was a significant correlation between their understanding of an equation and how beautiful they rated it as being. The intellectual beauty that we experience in mathematics really does seem to be earned by our knowledge of mathematics — to borrow a line from Plato, quoted by the researchers, “nothing without understanding would ever be more beauteous than with understanding”.

Nevertheless, the correlation between beauty and understanding wasn’t perfect. Our experience of mathematical beauty, it seems, is more than just a sense of satisfaction that we’ve understood something: understanding, perhaps, acts only as a gateway to appreciation, and exactly what else contributes to that appreciation lies beyond the ken of neuroimaging. It seems fitting that the paper ends, like all honest discussions of aesthetics, on an open note:

… that there was an imperfect correlation between understanding and the experience of beauty and that activity in the mOFC [the relevant bit of the brain] cannot be accounted for by understanding but by the experience of beauty alone, raises issues of profound interest for the future. It leads to the capital question of whether beauty, even in so abstract an area as mathematics, is a pointer to what is true in nature, both within our nature and in the world in which we have evolved…

Hence the work we report here, as well as our previous work, highlights further the extent to which even future mathematical formulations may, by being based on beauty, reveal something about our brain on the one hand, and about the extent to which our brain organization reveals something about our universe on the other.

And possibly that is all we can know for the moment, even if it’s not quite all we need to know…

(DP)

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Perhaps the best way to express this is to say that what crossword solving shares with mathematics is the eerie power of a good heuristic. This shows up most obviously in anagrams. By my reckoning, there are 9! = 362880 ways to arrange the letters in a nine-letter anagram (not at all uncommon in an everyday grid) and 15! = 1307674368000 ways to arrange the letters in a fifteen-letter anagram (rarer but hardly unknown). Despite this, many solvers have had the experience of looking at a lengthy anagram and spotting the answer within a second or so. It seems barely plausible that the brain has had time to implement a brute-force strategy such as working through all the possible combinations of letters, or even (more plausibly) thumbing through a mental dictionary of nine- or fifteen-letter words and checking each for compliance with the clue. Rather, what seems to happen is that we spot certain combinations of letters (like a T, an I, an O and an N), and consider the working hypothesis that our solution ends in -TION: this gives us a massively reduced brute-force problem to solve, and because of the highly restricted set of letter combinations to be found in any natural language, the strategy works pretty frequently. (The hardest anagrams to spot tend to be those of words which contain unusual letter combinations and don’t contain common prefixes or suffices — the well-known anagram of CARTHORSE is a case in point.)

Paradoxically, the longer an anagram becomes the easier it can be to solve: with some of the late and irreplaceable Araucaria’s colossal quotation anagrams, the most successful solution strategy was simply to look at the pattern of letters and wait for a plausible sequence of words to present itself. (To take a very sub-Araucarian example, if the pattern of letters is (2, 2, 2, 3, 2, 2) and you have a suspicion that the crossword is themed around lines from Shakespeare, then a possibility will rapidly present itself and render the clue “i.e.?” comprehensible!) In a cryptic clue, of course, the problem is rendered even easier by the promise that the clue will contain, somewhere, a “straight” definition of the solution, so another effective strategy is to consider the possible definitions in turn, and for each one riffle through your mental thesaurus looking for plausible candidates for the anagram.

The reason why heuristics work, in crosswords as in mathematics, is that we are playing a game in a highly conventionalised world. This is obvious in mathematics, where there are formal rules that determine whether a particular step is admissible or not, and where, less formally, our developing mathematical maturity gives us a sense of which claims are credible and which methods could plausibly lead to particular results. (A nice illustration of these informal rules is given by Scott Aaronson’s *Ten signs a claimed mathematical breakthrough is wrong*.) Crosswords, less obviously, don’t just work within a restricted “language” (there are, after all, only so many ways to indicate that a clue contains an anagram) and a set of formal rules (usually the Ximenean rules); they also have a surprisingly restricted field of reference. It’s hard, in my experience, to score a reasonable percentage against the broadsheet prize crosswords without some vague knowledge of classical music and cricket (both of which have substantial vocabularies crammed with specialised and quite odd words), but I can testify that it’s perfectly possible to get by with very little knowledge of, say, association football or popular music recorded since about 1995. In a sense — and I don’t want to push the analogy too far — the odd areas of knowledge that help disproportionately with crosswords are a bit like the collection of mathematical techniques and tricks that each of us learns are disproportionately helpful solving problems in our own area of mathematics. (An example of such a trick from my own field, fluid dynamics, is that of looking for self-similar solutions; I’m sure my colleagues in other disciplines have their own favourites.)

Of course, the danger is that, without an injection of the unexpected, crosswords can become rather stereotyped, and I think the same applies to mathematics. A great compiler has what I think is best described as *wit*: the ability to startle the solver with a surprisingly topical reference or linguistic oddity (the famous anagram of BRITNEY SPEARS, though it’s a bit dated now, is a case in point), or simply to approach the task of clueing from an entirely unexpected angle — again, Araucaria provides some perfect examples. Similarly, the difference between a journeyman piece of mathematics and the work of a real master is that the former will tackle a standard problem very competently in a standard way, whereas the latter will deploy an idea or a technique from an unexpected source, or address a problem that nobody beforehand would have thought was natural or feasible to tackle.

That is the lasting joy of both crosswords and mathematics: not bypassing the heuristics and the conventional (or even algorithmic) elements, but mastering them enough to appreciate the wit that operates within that framework and occasionally dances beyond it. It’s the joy that one also finds in poetry and in music and, I suspect, in every other sphere where an apparently restrictive system of rules becomes the essential support for creativity.

(DP)

PS: ORCHESTRA; TO BE OR NOT TO BE; PRESBYTERIANS. There: you can stop worrying now…

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First, have a look at this 36-second Youtube video from the excellent Steve Mould:

Now try to work out what on earth is going on… Bear in mind that this behaviour doesn’t rely crucially on the beads along the chain, and you can even get the same effect with rope:

If you’re puzzled, you’re not the only one. Steve Mould drew attention to the phenomenon in a blog post back in the summer (it also contains a link to a really neat slow-motion video), which in turn produced some discussion on Reddit (and some follow-up experiments).

Most recently, an analysis by two professional physicists, Prof. Mark Warner and Dr John Biggins, has appeared in the journal *Proceedings of the Royal Society A*. [You should be able to read the full article if you’re on a Strathclyde computer; if not then you might be interested in the podcast version.] If their analysis is correct, then all you need to know in order to understand the chain fountain is some school-level physics (tension, gravity, and a little bit of moments) — and a willingness to think carefully in some slightly non-intuitive directions. (Spoiler: it turns out that the really crucial detail is the reaction force that the jar exerts on the chain as it’s picked up. But you’d already guessed that… right?)

(DP)

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The event will take place from 1500-1600 in LT908, Livingstone Tower, on Friday 31 January, and will be followed by a wine reception. It’s free and open to all, and non-members are very welcome!

(DP)

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In the mathematics I can report no deficience, except it be that men do not sufficiently understand this excellent use of the pure mathematics, in that they do remedy and cure many defects in the wit and faculties intellectual. For if the wit be too dull, they sharpen it; if too wandering, they fix it; if too inherent in the sense, they abstract it. So that as tennis is a game of no use in itself, but of great use in respect it maketh a quick eye and a body ready to put itself into all postures, so in the mathematics that use which is collateral and intervenient is no less worthy than that which is principal and intended.

[The Advancement of Learning, 1605, section VIII(2)]

Happy New Year! Have fun, but I hope your “wit and faculties intellectual” are thoroughly remedied by the time you return to campus…

(DP)

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*Amazingly mathematical music* is an online article from Washington University in St Louis, which explores some of the mathematical patterns to be found in both musical harmonies and rhythms. It’s well illustrated (if that’s the word) with sound clips including the infamous perpetually ascending stairs illusion, barbershop choruses and Tuvan throat singing — well worth checking out! It’s accompanied by a podcast interview with David Wright, who is both the chair of the Mathematics Department at Washington University and a very successful *a capella* choir director.

Hat tip (and jazz hand) to the member of Edinburgh’s Rolling Hills Chorus who supplied the link!

(DP)

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