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Magforming the Stewart Toroids

Posted by richardelwes on June 1, 2018
Posted in: Uncategorized. 1 Comment

[This is a sequel to my previous post Magforming the Johnson Solids. Please see that for a disclaimer and (if you want to understand the words in this post) the geometrical background. If you only want to see some pretty pictures, then ignore all this stuff and just scroll down…] 

Last time, we investigated the shapes that can be built out of regular polygons, focusing on convex shapes – roughly speaking those with no holes, dents, or spikes. There are exactly 98 theoretically magformable convex polyhedra: all 5 of the Platonic solids, 11 Archimedean solids (out of 13 in total), 4 prisms (of an infinite family), 4 antiprisms (ditto), and 74 Johnson solids (out of 92).

Penguin

Once you drop the requirement for convexity, the only limits are your imagination and the size of your magformer collection. In principle there are inifinitely many magformable polyhedra, because you can always stick more bits on. See robopenguin here, for example.

So what to do? Let’s return to the starting point of every discussion of polyhedra: the Platonic Solids.

 

 

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Magforming the Johnson Solids

Posted by richardelwes on May 18, 2018
Posted in: Geometry. 13 Comments

[Disclaimer: this is not a sponsored post or advert – it is a product purely of my own enthusiasm. But in the interests of full transparency, let me say one thing: for reasons you may come to understand, I developed a deep desire for some Magformers octagons. Although these exist, they are like gold-dust. So I wrote to Magformers UK and asked whether they might sell me six octagons as a special deal, and they very generously replied that they would give me six octagons, which indeed they did,  for which I am eternally grateful, and which you can spot in some of the photos below. They feature more prominently in the follow-up post which is [update] here.]

Platonic Solids

The Platonic solids: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron.

When some kind soul gave my children a set of Magformers – a magnetic construction toy mainly comprising regular polygons – needless to say the first thing I did was steal them for myself and set about building up the collection until I could create the five Platonic solids.

The next shapes to move on to are the Archimedean solids. There are 13, of which 10 are realistically buildable (the truncated dodecahedron and truncated icosidodecahedron require decagons which Magformers don’t (yet?) make, and the snub dodecahedron requires unfeasibly many triangles (80) and would in any case collapse under its own weight). Here’s a sample of three:

3 Archimedean Solids

A cuboctahedron, rhombicuboctahedron, and truncated icosahedron.

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Books, Blog, & Song(s)

Posted by richardelwes on August 31, 2017
Posted in: Uncategorized. Leave a comment

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Writing & Europe

Posted by richardelwes on January 7, 2017
Posted in: Elwes Elsewhere, Politics, Uncategorized. Leave a comment

It’s been a long time since I posted anything here… and that doesn’t change now, except in a technical sense. My writing activities are currently split between my work for the European Mathematical Society (see here for why you should join) and writing about the current political situation, with which I am greviously displeased, on social media. If you are interested in the latter, see my Twitter account on your right, and I have also started a blog on Medium. The first post is: Remaining Angry.

Book Review: Red Plenty by Francis Spufford

Posted by richardelwes on May 8, 2016
Posted in: Bookery, Complexity. Tagged: Francis Spufford, Linear programming, Red Plenty. Leave a comment

RedPlenty

Imagine the benefits that could reaped if economic activity could be organised in a rational and scientific way, instead of abandoned to the chaos the marketplace! Imagine the efficiency gains there would be, with workers, managers, farms, and factories all pulling together instead of wastefully competing against each other!

For a period, in the Soviet Union of the 1950s and 60s, there was a genuine and exhilarating belief not just that communism was morally preferable to capitalism, but that it could actually beat capitalism at its own game. There was even a moment, at least for those with the eyes to see it, when it looked as if that might just be beginning to happen.

It is this era which is so brilliantly captured in Francis Spufford’s fictionalised account, Red Plenty. I was recommended the book by the estimable Miranda Mowbray, when we were both speakers at a maths outreach day in London. Her talk was on “Drinking from the fire hose – data science”. Mine was on Linear Programming, and afterwards Miranda remarked that she’d read a book in which Linear Programming was the main character. And so it is.

For the question arises: in the absence of a market to balance supply and demand, how should the central planners set about their work? How much viscose should they instruct a particular factory to produce, given the number and locations of other factories, the availability of sulphur, salt and coal, and the requirements of the fabric, cellophane, and tyre manufacturers?

Astonishingly, the mathematician Leonid Vitalevich Kantorovich was able to devise a tool to answer to this sort of conundrum, in his seminal 1939 work on optimal resource allocation. (It would earn him a Nobel Memorial Prize in Economics in 1975.) The consequence of this breakthrough was spectacular: the political apparatus of central planning could be armed with linear programming, the technical means to accomplish that task, and thus would usher in a new era of Soviet abundance.

Well, it’s hardly a spoiler to say that it didn’t work out quite like that. Red Plenty recounts the rise and fall of that tide: from the elation of discovery and the hope of a better world, to frustration, cynicism, and the ultimate tragedy of failure.

Now, a book about a doomed political philosophy and a technical mathematical procedure may be admirable, but is it entertaining? Reader, it is rip-roaringly so. The story is told episodically, each chapter built around one character, sometimes real, sometimes fictional, each passage invested with the significance that its inhabitants feel. Some are hilarious, some horrifying.

There is Kantorovich, of course, the prodigy and professor. There is the ambitious but sincere (fictional) young economist Emil Shaidullin, trudging through fields in his best city suit, determined to improve the lot of the rural poor. Sasha Galich is a (real) flamboyant song-writer and playwright, becoming uneasy with the ends to which his art is put. Zoya Vaynshteyn is a (fictional) scientist enjoying a mad midsummer’s night, but quietly pitied by her colleagues for the unsayable truth: that her subject, genetics, is afflicted with the plague of Lysenkoism. Sergei Lebedev is a (real) computer pioneer, toiling away in his Institute’s basement to build the machines that will perform the enormous economic calculations far faster than any capitalist market. We meet Mr Chairman, Nikita Sergeyevich Khrushchev himself, travelling to the USA to strike a deal and issue oafish challenges. A (fictional) central planner Maksim Maksimovich Mokhov juggles the balances for 373 commodities in the chemical and rubber goods sector.

What’s so compelling is the colour and humanity of all these people as they live their lives entangled in the Soviet system. Some embrace the socialist dream, some resist, many simply try to organise their affairs around it. There are a few striking characters we meet only once, such as the (fictional) wheeler-dealer Chekuskin, frantically digging his clients (and himself) out of political holes in the Urals. But several we revisit at later stages of their careers, when dreams have died (or been revised downwards), consciences have been pricked, or lines have finally been crossed. Whilst an idea, that of Linear Programming, may indeed be the story’s main character, it is the human supporting cast that makes it so engrossing.

As a postscript, it is worth stressing that Linear Programming really did change the world, and in an altogether more desirable fashion than can be said for the command economy. As so often during the Cold War, very similar work was carried out independently and in parallel on opposite sides of the Atlantic. Linear Programming arrived in the USA with George Dantzig’s 1947 discovery of the Simplex Algorithm. Nowadays, these techniques are employed daily by countless organisations around the world to solve otherwise intractable optimisation problems.

Barry Cooper, 1943-2015

Posted by richardelwes on October 28, 2015
Posted in: Logic. 12 Comments

Barry Cooper, who very sadly died on Monday, was a central member of the Leeds logic group since the 1960s. I joined that group as a graduate student in 2001, and since then have had the pleasure to get to know him. He always took an active interest in his younger colleagues, myselBarryCooperf included, and was enthusiastic about mathematical outreach. Of all the senior mathematicians at Leeds, I would say Barry was the most vocally supportive of my early efforts in that area, and I remain grateful for his support.

Barry’s research interests were in the field of computability (or more accurately incomputability) and in particular the structure the Turing degrees. Roughly speaking, a set of whole numbers X has a higher Turing degree than another (Y) if a computer with access to X has the power to tell which numbers are and are not in Y. Thus, in a very natural sense, X contains all the information that Y does (and possibly more). It may be that Y can do the same for X, in which case the two sets have the same Turing degree.

This simple idea produces a fascinating and fundamental structure, known to the experts as the upper-semi-lattice of Turing degrees. There are all kinds of weird and wonderful configurations hiding within it: two degrees where neither is higher than the other, individual degrees which are minimal (in that there is nothing below them besides the zero degree of computable sets), two degrees which have no greatest lower bound (this is what makes it a semi-lattice rather than a full lattice), and a great deal else besides. This structure (and assorted close relatives) has been the subject of a huge amount of investigation. Barry has played a leading role in this programme over many years.

Outside research mathematics, Barry was popular, active, and successful in a frankly alarming number of different arenas. He was an excellent and well-liked teacher, and will surely be missed by Leeds undergraduate mathematicians as well as by his colleagues and numerous current and former graduate students.

In sport, he was a keen long-distance runner, with a personal best marathon time of 2hrs 48mins. His most recent outing was the 2010 London marathon. One common interest he and I shared was jazz, with Barry having a particular taste for its wilder and more avante garde varieties. He was a founder of Leeds Jazz, and helped attract numerous top artists to the city, including Art Blakey, Courtney Pine, Paul Motian, and Loose Tubes (to pick 4 examples just from 1986). One regret I have is not going to more gigs with him.

A committed political activist and unapologetic left-winger, Barry was involved in various political campaigns, including the the Chile Solidarity Campaign that was set up following the military coup of 1973.

In recent years, Barry devoted a huge amount of energy to the Alan Turing centenary events of 2012. An utter triumph, this anniversary had an astonishing global impact (and overflowed enormously beyond its allotted twelve months), and made great progress in bringing Turing the long overdue recognition he deserves. One outcome of the year was the book Alan Turing: His Work and Impact, edited by Barry and Jan van Leeuwen, a hefty and definitive volume which scooped several prizes, including the Association of American Publishers’ R.R. Hawkins Award. Another result of the increased publicity was a Royal pardon for Turing in 2013; another was the Imitation Game film of 2014. The success of the whole project was in large part due to Barry’s leadership, and the mathematical and computer science communities surely owe him a large debt of gratitude.

Barry announced just two weeks ago that he had been diagnosed with untreatable cancer, a development he met with a characteristic selflessness and equanimity. He died on Monday, surrounded by his family. Over the course of his life, Barry touched many people in many ways, and just as many will now miss him.

Learning to Learn from Babies

Posted by richardelwes on July 16, 2015
Posted in: Brain Science, Education. 1 Comment

cakes

Yesterday, my twin sons turned one. I have spent an amazing number of hours over the last year watching them. I wondered if this experience might teach me something too, about how to learn. After all, babies are the grandmasters on this subject. In the same time that it has taken me to incrementally advance my knowledge of some tiny corner of mathematics, my children have moved from a total inability to do anything besides scream, crap themselves, and scream again, to being able to feed themselves (messily, so messily, but still), crawl, clap, grab, wave, recognise people, stand, and so on. And these are just the most visible manifestations of a deep mental transformation during which their brains have learnt huge amounts about processing sensory data and coordinating muscle movement.

If that rate of learning was to continue through their lives, they would grow into geniuses far surpassing anything humanity has seen so far. So how do they do it? Of course a large part of the answer is physiological: babies’ brains are a lot more plastic than adults’, highly efficient sponges for the absorption of new skills. There’s not a great deal we can do about that (although there may be something). All the same, I think we might be able to see some other, useful principles in action too:

  1. Play. We don’t usually talk about babies “working” – but they are, just as assuredly as a student revising or a scientist researching. The difference is that babies are also undeniably playing – and we wouldn’t usually describe either of the other two in that way. Babies are not motivated by exam grades or pressure to publish. The more fun you find your work, the better you do it.
  2. Be curious. The babies immediately home in on any new item which appears in their playing area, and start investigating. They are always exploring the room’s boundaries, and grabbing at anything unfamiliar or interesting (my laptop, mugs of hot tea, etc..). But they are not searching for anything in particular. Set aside some time for open-ended exploration and experimentation.
  3. First, learn one thing well. The boys learned to clap quite early on. With this under their belt, other manual skills such as waving and pointing were comparatively easy to pick up. Likewise, an experienced mathematician will find it easier than a novice to master an unfamiliar mathematical topic. Even if neither has any directly relevant knowledge, the fact that one is practised in the art of learning mathematics should carry them a long way. Building skills can be worthwhile, even when the skills themselves are not.
  4. A change is as good as a rest. In the opposite direction, the twins do not spend hours at a stretch practising one thing, such as walking. Instead they do it for a little bit, then get distracted by a toy, move onto another toy, have a crawling race, try holding a conversation with their mother, then they do some more walking, and so it goes on. Have more than one project on the go.
  5. Don’t be scared. Most of the time, my sons appear completely fearless. They happily crawl into perilous situations, pull over heavy objects, and invite disaster in any number of imaginative ways. This is despite the fact that they regularly do fall down and otherwise upset themselves. Take risks. Even if they don’t immediately pay off, continue to take risks.
  6. Don’t be embarrassed. Babies are not only unworried about taking a tumble, they’re also unafraid of looking like fools. The more I ponder this, the more important I think it is. In my efforts to learn Japanese, for example, I am hindered (perhaps more than I have realised) by the fear of making embarrassing mistakes in conversation with my in-laws. Likewise, mathematicians do not enjoy admitting errors, or gaps in their knowledge, in front of their colleagues (let alone their students). I think this is a bad habit. In order to learn from your mistakes, you must first allow yourself to make some.
  7. Accept help. The boys are utterly dependent my wife, me, and the other generous people who help us look after them (thank you!). Obviously, adults shouldn’t be that reliant on others, except in extremis. Nevertheless, there may be people in your life who would like to help you succeed. Let them.
  8. “Good enough” isn’t good enough. My children have reached the point where crawling is a highly efficient form of travel – they can zoom around the house to wherever they want to be. Walking, meanwhile, is a faltering, risky business. It would be a perfectly rational short term decision if they opted not to bother with it. Of course, babies don’t reason that way, which is just as well. Invest in your long-term skills, even at short term cost.
  9. Don’t focus on the scale of the challenge. My children’s vocabulary currently consists of little more than “dadadada”, “mamamama”, “aaaaghh”, and “pmmpphh”. It will be quite a journey from these noises to mastery of the language of Shakespeare (and indeed that of Chikamatsu). Of course they have no idea about that. The journey matters more then the destination.

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