## You're reading: Posts Tagged: maths

### Dani’s OEIS adventures: triangular square numbers

Hi! I’m Dani Poveda. This is my first post here on The Aperiodical. I’m from Spain, and I’m not a mathematician (I’d love to be one, though). I’m currently studying a Spanish equivalent to HNC in Computer Networking. I’d like to share with you some of my inquiries about some numbers. In this case, about triangular square numbers.

I’ll start at the beginning.

I’ve always loved maths, but I wasn’t aware of the number of YouTube maths channels there were. During the months of February and March 2016, I started following some of them (Brady Haran’s Numberphile, James Grime and Matt Parker among others). On July 13th, Matt published the shortest maths video he has ever made:

Maybe it’s a short video, but it got me truly mired in those numbers, as I’ve loved them since I read The Number Devil when I was 8. I only needed some pens, some paper, my calculator (Casio fx-570ES) and if I needed extra help, my laptop to write some code. And I had that quite near me, as I had just got home from tutoring high school students in maths.

I’ll start explaining now how I focused on this puzzle trying to figure out a solution.

### Pale imitations: newcomers in the Math/Maths Podcast hiatus

Since the start of the year, the Math/Maths Podcast has been on hiatus. I’m very much enjoying the extra thesis-writing time but apparently this has left some missing their regular mathematical listen. Not infrequently I get an email from someone wishing me well with my thesis and asking when we’ll be back podcasting. Well, nature abhors a vacuum and here are three offerings that I’m aware are working to fill the void. (Oh, and “pale imitations” – I’m joking, of course!)

My Aperiodical co-conspirators Katie Steckles and Christian Perfect started All Squared, a maths magazine podcast, in February. The description for the first episode (or “number”, as Katie and Christian have it) overtly points out the “unusual paucity of maths podcasts at the moment” and promises “a half-hour podcast featuring maths, guests, puzzles and links from the internet”. The name is designed to be recognisable to mathematicians, who might find themselves reporting that an expression is “all squared”. As someone who named a podcast as overtly as it is possible to be, “Math/Maths”, this obfuscation amuses me. The three episodes so far have been enjoyable with a guest and main topic in each. As far as I’m concerned, this is far more the Aperiodical podcast that should exist than is The Aperiodcast with that third guy.

#### TES Maths Podcast (iTunes)

This one started just before Samuel Hansen and I went on our hiatus, but if you enjoyed the teaching aspects of what we did you can get a lot more on the theme from Craig Barton and his guests on the TES Maths Podcast. Craig promises “to share the latest news, resources and ideas that are relevant to secondary/high-school maths teachers and general number enthusiasts”.

Wrong, but Useful is a new podcast featuring “a mathematical conversation” between Colin Beveridge and Dave Gale that sets out its stall as a response to the lack of Math/Maths episodes. The title is another nod to the mathematically minded without being overt, referring to a quote from George Box and Norman Draper who wrote “essentially, all models are wrong, but some are useful” (Empirical Model-Building and Response Surfaces, 1987). Episode 1 sees Colin and Dave finding their feet in a rambling, wide-ranging mathematically-themed discussion. There were a couple of awkward moments that gave me Math/Maths early episode flashbacks but I’m looking forward to Colin and Dave getting into the swing for the next episode.

Happy listening!

### Podcasting update

I have a job. This is not the podcasting update, but it does affect it! If you have listened to the latest Math/Maths Podcast you will know that I will be lecturing mathematics from January while trying to finish my PhD thesis, and that we will be putting that podcast on hiatus while I do so. This means no more talking to Samuel Hansen for at least six months.

There is something you can do to fill this mathematical podcasting gap, however. Samuel is trying to raise money through a Kickstarter to allow him upgrade his equipment and improve the quality, to pay for the travel to conduct face-to-face interviews and to make this his full-time job so he can concentrate on a regular release schedule, for his work in maths (math) and science communication over at ACMEScience.com.

At Kickstarter, Samuel says:

ACMEScience.com has spent the last four years trying to do something that very few others have ever attempted, create entertaining, insightful, and interesting content about mathematics and science. Started by Samuel Hansen in the beginning of 2009, ACMEScience has produced a pop-culture joke filled mathematical panel show, Combinations and Permutations, a show that interviews everyone from the CEO of a stats driven dating site to a stand up mathematician to Neil deGrasse Tyson, Strongly Connected Components, a show that tells the stories of the fights that behind DNA, dinosaurs, and the shape of the universe, Science Sparring Society, a video interview show that has featured predatory bacteria and crowdsourced questions, ACMEScience News Now, and a series of hour long journeys into the world of competitive AI checkers computers and stories of the most interesting 20th C mathematician and much more, Relatively Prime.

You may remember that Samuel raised money through a Kickstarter before, for the extremely well-received documentary series Relatively Prime. So you might judge this as evidence that he is capable of delivering this project. However, you may also remember that if he doesn’t raise the whole amount he needs then he gets nothing.

There are various pledge levels, with various rewards. Some of these are aimed at the individual who wants to own a piece of the project. Others are aimed at people who want to sponsor/advertise via the shows and get their message out there. Looking at the level of pledges so far, Samuel could really do with a few companies or individuals who want to get a message out to a mathematics or science audience coming forward and pledging some money. Relatively Prime was very well listened to, and you could get your message to a large, focused, engaged set of listeners.

There is not long to go (only four days at time of writing) and it doesn’t look good. So please pitch in and also tell everyone you know via your own blog/podcast/social networks/etc. so that others will support his effort.

Here is the video in which Samuel makes his case. It’s six minutes so at least watch that! The Kickstarter page is ACMEScience.com by Samuel Hansen. Donating is easy through Amazon payments.

### Martin Gardner celebration week in Nottingham

Last Sunday saw the anniversary of the birth of Martin Gardner, and as a celebration, the Gathering for Gardner people planned a world-wide party ‘G4G Celebration of Mind‘. It happened to be Maths Jam night on Tuesday, so we put the Nottingham Maths Jam on the G4G-COM map. Then on Friday three of us had agreed to take a puzzles stall to the Nottingham STEM Pop Up Shop, so I added this to the map as well.

A Celebration of Mind party is supposed to “celebrate the legacy of Martin Gardner on or around Sunday, October 21, 2012 through the enjoyment of [one or more of] Puzzles, Magic, Recreational Math, Lewis Carroll, Skepticism and Rationality”.

At Maths Jam I printed a bunch of flexagon material from the Flexagon Party page. I also had a plan: having finished two jobs in recent years with piles of business cards outstanding, I brought these to try some business card origami. In fact, we decided to make a business card Menger sponge. So we started folding.

Meanwhile, John Read had come equipped with some colourful designs to make hexahexaflexagons.

 John Read’s first hexahexaflexagon of the evening. Designs from flexagon.net.

 John Read’s second hexahexaflexagon. Designs from flexagon.net.

At the same time, Jon made a trihexagon.

 Trying to make a triflexagon

Finally, after much business card folding,we had a Menger sponge* (*not a real one, it being a fractal after all!). Here it is, in Maths Jam-style, balanced on a pint of beer.

 The completed business card Menger sponge

And here’s a shot through the Menger sponge, where a geometry puzzle is being attempted.

 Through the completed business card Menger sponge, some geometry is happening

Here are a couple of the other puzzles that we tried, some from the #MathsJam tag on Twitter:

You have 100 coins, 10 of which are showing heads and 90 of which are showing tails (though these are indistinguishable by touch). Blindfolded, you must divide the coins into an even number of heads and tails.

Which is bigger, 3^(21!) or 2^(31!)?

Then on Friday we made our way to Broadmarsh shopping centre for our afternoon at Nottingham’s STEM Pop Up Shop.

 Nottingham STEM Pop Up Shop welcome notice

Here’s a picture of our stall, with Kathryn Taylor presiding, and in the foreground the posters about Martin Gardner, mathematical games and mathematicians that I had printed. Someone did ask me who Martin was and I explained a little; I think I also convinced him to come to Robin Wilson’s talk on Lewis Carroll next month in Derby.

 Martin Gardner posters on our stall

Here’s the detail of our stall, which we called ‘Solving it like a mathematician’. You can get details of the set of puzzles on my website.

 The STEM Pop Up Shop ‘Solving it like a mathematician’ stall

Looking around the shop, I requisitioned the Alan Turing postcards from the ‘My Favourite Scientist’ set for our stall!

 Alan Turing postcards

Here’s a wider view of the stall, with Kathryn entertaining a customer.

 Kathryn Taylor at our stall

And finally, here’s John Read bewitching a crowd with the loop on a chain trick.

 John Read enthuses a crowd with a ring on a chain

 Thank you for visiting

### Nobel week – a place for mathematics?

In a blog post last week, Alex Bellos said:

It is often said that the reason Alfred Nobel did not endow one of his prizes in mathematics was because his wife was having an affair with a mathematician.
While this story has been debunked it is nevertheless frustrating to mathematicians, especially during Nobel week, that the noblest of the sciences is ignored by the Royal Swedish Academy of Sciences.

As an alternative, Alex offers the Mental Calculation World Cup.

A while ago, I wrote a flippant little piece in which I claimed that “Most of the Nobel Prizes are for Mathematics“. While perfectly valid and interesting mathematics takes place within mathematics itself, it is an interesting aspect of mathematics that its applications take place on the boundaries with, or even within, other disciplines. This creates some issues for those championing mathematics. Some people would like to assess the economic impact of mathematics but this is a difficult task. At what point does an application of mathematics belong to science, engineering or technology?

The point of the IMA Mathematics Matters series of articles, as I understand it, is to show where modern mathematics research has had its impact, even though that impact may be “perfectly hidden in its physical manifestation”. Some people would take the result of a piece of research to be “not mathematics” as soon as it finds an application. Unfortunately, defining a piece of work as ‘not mathematics’ as soon as it is applied is a way to ensure that all measures of economic value of the impact of mathematics are zero; yet it is clearly the case that much of science, engineering and technology would be naught without the mathematics that underpins it.

This is a balance we try to strike when finding stories for the Math/Maths Podcast; many interesting stories, and certainly those more likely to be written up by university press offices or the media, are those which apply mathematics in some other area. How far do we follow a story before declaring it “not mathematics” and turning our attention elsewhere?

It is with this mindset that I view the Nobel Prizes. Much of the work for which the prizes are awarded is underpinned by mathematics. I see ‘Nobel week’ as an opportunity for mathematicians to go in search of the mathematics behind each prize, rather than to retreat and complain about the lack of a prize specifically for mathematics.

### 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:

• 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“.

I put these suggestions – rounding errors and rectangles – to Twitter.

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?

### "I’m not a mathematician, the maths I’m doing is really just basic modelling"

Last week I attended the first Institute of Mathematics and its Applications Employers’ Forum. The theme was ‘Employability of Mathematics Graduates’. This was an interesting event with many useful views and viewpoints on display.

One speaker, talking about how mathematics student applicants to the graduate training scheme fare, mentioned that during the technical interview some such applicants seem to expect that they will be asked detailed questions about their final year modules. In fact, the questions asked are more like A-level mechanics and this trips up many students. This chimes with a problem I’ve thought about previously about attitudes to mathematics from mathematicians.

I have noticed that many graduate mathematicians who work in mathematical jobs will tell me “I’m not a mathematician, the maths I’m doing is really just basic modelling”. Students and graduates (including, if I think back, me when I graduated) seem to think that if the mathematics they are doing after graduation isn’t at least as hard as final year undergraduate mathematics, then it can’t be ‘real mathematics’ and they can’t be a ‘real mathematician’. As they haven’t moved onto a higher degree to do more advanced mathematics, they must have failed as mathematicians.

I came across this problem somewhat when I worked for the IMA because someone who doesn’t consider themselves a mathematician might ask: since I’m no longer a mathematician, why would I join the mathematicians’ professional body?

I think it is terribly sad when graduates think this. I must be careful here: of course there is more advanced applied mathematics but many graduates find themselves applying fairly basic mathematics to problems and therefore think that they have regressed to an earlier stage of their mathematical development. This rigorously hierarchical view of mathematics – particularly from people who are using mathematics to make a substantial contribution – seems to me to be a real shame. In fact, final year undergraduate mathematics is pretty far up the tree – so far, if we continue the analogy, that it can’t support very many people – but it’s hard to appreciate this when, to overuse the analogy, you’re only looking at the few academic researchers balancing on higher branches.

“If I apply for a job using mathematics, they must want to quiz me about what I learned at the culmination of my degree. And since they’re asking me questions about forces and moments using techniques from A-level, then this can’t be real mathematics and I can’t be a real mathematician.” It’s a real problem.

This is part of where I think the value lies in the IMA series of conferences for the ‘Early Career Mathematician’. Since many mathematicians in industry think of themselves as ‘someone who used to do mathematics’ and may well be the only mathematics graduate in their team/department/company, it can be a very powerful experience to come together and meet others in similar positions. If you’ll excuse a small plug, I am chairing the next of these conferences, the IMA Early Career Mathematicians’ Autumn Conference 2012, in Greenwich in November. Registration is now open. Come along!