With nonmonotonic irregularity, it’s time for another Follow Friday – a round up of the maths people on Twitter you should be following, or at least some fun links you can look at.
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- Drew @twentythree said: “for all the budding mathematicians in waiting of course”.
- Will Davies @notonlyahatrack said: “my simplistic answer would be when teaching it ;)”
- Ian Robinson @IanRobinson said: “it allows you to work with precise fractional values without rounding errors in calculations. Useful for engineering etc.”
Later, Colin Beveridge @icecolbeveridge suggested something similar: “in computing, roots are expensive — if you can consolidate them, you save computing time.”
This rings true for me but it was a mathematically-inclined structural engineer who asked the original question. Is this really used in engineering? - Christian Perfect @christianp said “anything involving making rectangles” thinking particularly of “carpentry and landscaping“.
Carnival of Mathematics 91
The next issue of the Carnival of Mathematics, rounding up blog posts from the month of September, is now online at Matheminutes.
The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. For more information about the Carnival of Mathematics, click here.
Henry Segerman’s 30-cell puzzle
Henry Segerman is a mathematician at the University of Melbourne with a keen interest in 3d-printing mathematical shapes. He’s just uploaded a video showing off his latest creation, a 30-cell burr puzzle created in collaboration with Saul Schleimer:
[youtube url=https://www.youtube.com/watch?v=FJwqT_sbB_A]
Pretty cool, eh?
As well as providing a PDF describing the puzzle, Henry’s uploaded the design to Shapeways so you can have your very own copy to play with.
Earlier this year, Henry and Saul’s half 120- and 600-cells won the “Best Use of Mathematics” award at the 2012 Bridges Conference.
Proof News
Here’s a little catch-up with the status of the claimed proofs of some big statements that were announced recently.
At the end of August, Shin Mochizuki released what he claims is a proof of the abc conjecture (link goes to a PDF). Barring someone spotting a huge error, it’s going to take a long time to verify. It’s mainly quiet at the moment, apart from a claimed set of counterexamples to one of Mochizuki’s intermediate theorems posted by Vesselin Dimitrov on MathOverflow, which was quickly shut down because the community there didn’t approve of MO being used to debate the validity of the proof. No doubt there are other niggles being worked out in private as well.
At the start of September, Justin Moore uploaded to the arXiv what he claimed was a proof that Thompson’s group F is amenable. Like Mochizuki’s abc proof, experts thought Moore’s proof was highly credible. We were waiting for my chum Nathan to write about it, since his PhD was all about Thompson’s groups F and V, but it turns out we don’t need to: at the start of this week, Justin retracted his paper because of an error which “appears to be both serious and irreparable”. The amenability of Thompson’s group F has been proven and disproven many times, so I still want Nathan to tell me (and you) all about it.
In lighter news, via Richard Green on Google+, recent uploads to the arXiv show that Goldbach’s conjecture and the Riemann hypothesis are true. I’d love to know how it feels to upload a six-page paper which you know proves something like the Riemann hypothesis. It must be a lovely state of mind. Certainly much better than what people like Moore and Mochizuki must go through, waiting for the first email to arrive telling them they’ve made a terrible mistake and their work is not yet complete.
If I’ve inspired you to have a go yourself, look at Wikipedia’s list of unsolved problems in mathematics and take a crack at one this weekend. Can’t hurt ((Disclaimer: depending on levels of ability, perseverance and agreement with consensus reality, attempts to solve these problems may well ruin your life)) to try!
Puzzlebomb – October 2012
Puzzlebomb is a monthly puzzle compendium. Issue 10 of Puzzlebomb, for October 2012, can be found here:
Puzzlebomb – Issue 10 – October 2012
The solutions to Issue 10 can be found here:
Puzzlebomb – Issue 10 – October 2012 – Solutions
Previous issues of Puzzlebomb, and their solutions, can be found here.
Surds: what are they good for?
Here is a question I was asked:
Why is rearranging equations containing square roots on the curriculum for GCSE? What might it be useful for in later life?
This is a two-part question, one part of which is dynamite. When I put the question to Twitter, Paul Taylor @aPaulTaylor was the first to take the bait:
Is usefulness in later life a necessary condition for inclusion on the GCSE curriculum?
Let’s set that aside for now. Whether usefulness is necessary or not, asking what a topic might be useful for in later life is a perfectly valid question for a fourteen year old who is being asked to study that topic.
Surds is one of those confusing areas that I vaguely remember but have to look at a definition to recall properly. The BBC GCSE Bitesize website has “a square root which cannot be reduced to a whole number” and says “you need to be able to simplify expressions involving surds”. Rearranging surds, then, is the business of noticing that the square root of 12 multiplied by the square root of 3 can be combined to give the square root of 36, which is 6.
Surds, then, are a part of general algebraic fluency. I expected, therefore, that one answer would be that this is the kind of manipulation that helps generally with higher mathematics; though I wonder when such neat numbers arise in reality. I also expected to hear that surds were useful in very efficient computation. I remember once speaking to someone who was programming computers to go on board aeroplanes. These had very limited computing power and needed to work in real time; the programming involved all sorts of mental arithmetic tricks to minimise the complexity of calculations.
For the latter, I am not sure how relevant this is to modern engineering or programming. For the former, it might be that we are including this for every student at GCSE simply as part of the algebraic fluency that we hope of from incoming mathematics students at university. When I put the question to Twitter, two responses reflected my cynicism on this point. When are surds useful in later life?
Other, less cynical responses, were available. Early responses:
I put these suggestions – rounding errors and rectangles – to Twitter.
John Read @johndavidread said (tweet 1; tweet 2):
I think it’s unlikely anyone doing practical work would need the accuracy. Feels more pure Maths than Applied. But is it used? For engineers, landscape, carpentry etc expansion to a few decimal places so you can measure to reasonable accuracy is fine.
Carol Randall @Caro_lann said: “engineering isn’t just measurement! There’s lots of heavy maths involved in getting a B.Eng (and beyond).”
John Read @johndavidread asked: “where in Maths do equations with square roots come up that you’d want to simplify without calculating numerical value?”
To this, Daniel Colquitt @danielcolquitt wrote what on Twitter must be considered an essay, a four tweet message (1, 2, 3, 4):
Very simple examples: Computing the eigenfrequencies of beams, or reciprocal lattice vectors & hence in various Fourier transforms. In this case, exact form is required, decimal expansion will not do. For the beam example, a numerical value can be computed for a given set of parameters, but if you want to know that frequency for *any* set of parameters, you need to know how to hand surds.
On algebraic fluency, Christine Corbett @corbett_inc suggested “the umbrella of ‘simplifying equations'”.
To this, John Read @johndavidread asked: “but then why not teach it as ‘simplifying equations’? No kid had heard of a surd in the 1980’s”.
Daniel Colquitt @danielcolquitt replied: “For GCSE & roots of reals >0, I would tend to agree with you. Complex roots are somewhat different”.
But we’ve swayed back rather close to the dynamite, haven’t we? I’ll stop there.
My sense is that I haven’t had a satisfactory answer really. This sort of rearrangement is good for building up the background knowledge of the undergraduate mathematics student or perhaps engineering student, but no one seems to be claiming they are an engineer who uses this outside of the classroom. No one seems to have claimed this topic develops mathematical thinking in an interesting way, or that engineers who don’t think they are using it really are relying on it in the black box of software, or that the topic somehow contributes to an appreciation of the beauty of mathematics in the teenagers who are learning it. (This may be due to my experiences and the experiences of those who have replied, or the way I have misinterpreted their words.) It may be that there’s a bunch of stuff on the GCSE syllabus just for those who go onto A-level or degree-level mathematics, and perhaps that’s fine, but it would be nice to have a more satisfying answer to give. So, dear reader, are you satisfied with these answers? Do you have a better answer?
Follow Friday
I’m hijacking Katie’s newly-instituted series of posts about who to follow on Twitter with a post about who to follow on Google+.
Google+ famously has almost nobody on it. If anyone knows the potential for really interesting exceptions to the word “almost”, it’s mathematicians, so by that mad logic there should be some really interesting mathematicians on Google+. As luck has it, there are! It seems that the unconstrained nature of Google+ posts gives mathematicians the space they need to express themselves usefully.
Here are a few mathsy people you might like to encircle on Google+.