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).

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.

Here they are again:

These are the only 5 polyhedra satisfying all of the following wonderful properties:

- Every face is a regular polygon
- Every corner (or vertex) is identical to every other (
*vertex-transitivity)* - Every face is identical to every other (
*face-transitivity*) *Convexity*(roughly speaking, no holes, dents, or spikes)

The main goal of this post is to explore the geometrical landscape that opens up when we drop requirement 4. But first…

We’ve focused on 3-dimensional polyhedra. Last time we also looked at 2-dimensional structures – what about 4-dimensional structures? Just as in 3-dimensions there are 5 regular convex polyhedra (the Platonic solids), so in 4 dimensions, it turns out there are 6 regular convex polychora. You might object that we don’t have much chance of actually building them… but, just as a 3d shape such as a tetrahedron can be unfolded into 2-dimensions…

…so a 4-dimensional shape can be unfolded into 3-dimensions, where their nets form interesting non-convex polyhedra:

I don’t have enough pieces to create nets of the remaining Platonic polychora – in fact you might notice some squares missing in the hypercube. The hyperoctahedron is within range if I had *a few* more triangles but the remaining three – yes six in total! – Platonic polychora have impractically many faces. But do check out Hyrodium’s amazing models of them.

However, we needn’t give up on 4D just yet. There are always the Archimedean polychora, such as:

And exciting new constructions such as the duoprisms:

There are several other avenues to explore up in 4D, but for now let’s come back down to Earth. What other non-convex polyhedra might we try? The mathematician Bonnie Stewart^{[2]} discovered several interesting ones, such as this compact little beauty known only as G3:

We’ll come back to G3 later. Stewart also found a few others, including this one, X:

You might wonder *how* Stewart found these – we will come to the answer in a little while.

Let’s be more systematic. Conditions 1, 2, & 3 above together give the definition of a *regular* polyhedron, the most symmetric type of all. If the Platonic solids are the only convex regular polyhedra, an obvious question arises: are there any *non-convex* regular polyhedra? The answer is yes! There are the four Kepler-Poinsot polyhedra. Great! Let’s grab the magformers and set to work….

But no. The Kepler-Poinsot polyhedra are non-magformable. Here’s why: suppose a solid is non-convex because it has a spike. At the end of that spike will be a vertex. And if one vertex is at the end of a spike, then by vertex-transitivity *every* corner must be at the end of a spike. And that is impossible, unless we relax one other hitherto unspoken condition, and allow the shape’s faces to cut through each other along “false edges”. The great dodecahedron, for instance, is composed of 12 regular pentagons, just like the ordinary dodecahedron, but this time they are configured so that they intersect:

(Image credits: [1])

Meanwhile the small stellated dodecahedron is made from 12 regular *pentagrams *– because if we’re going to allow faces to cut through each other, then why not also permit polygons whose edges cut through each other?

That’s all very well, but there’s no hope of making these without taking a hacksaw to my precious magformers. Still, there is one nice shape we can make along these lines. The polyhedra just discussed share one other property that we have taken for granted: they come in one piece. That is, if you start at any corner, travelling only along the shape’s edges, you can reach any other corner. That sounds obvious, but once we allow faces to slice through each other, it is no longer automatic. There are five regular *polyhedral compounds* which satisfy 1-3 above, have faces which cut through each other, and don’t come in one piece. One of them is the compound of two tetrahedra, also known as…

You can think of this shape in two different ways: as a *stellated octahedron* (which matches how one builds it – first make an octahedron and then construct a tetrahedron on every face); or as a polyhedral compound: two big tetrahedra (the yellowy one and the blueish one) pushed through each other until their centres coincide.

So much for non-convex regular polyhedra and compounds. So much, in fact, for this entire line of thought. There is a beautiful theory of non-convex uniform polyhedra & polyhedral compounds, those satisfying conditions 1 & 2 above. Besides everything we have met so far, there are the non-convex (e.g. pentagrammic) prisms and antiprisms, and analogues of the Archimedean solids: 53 uniform star polyhedra such as the dodecadodecahedron build from 12 intersecting pentagons and 12 non-intersecting pentagrams:

There are also 70 uniform polyhedral compounds, on top of the 5 regular compounds already discussed. But, alas, none of these are of any use to the aspiring magformist.

Actually, there is door hiding here, spotted by Bonnie Stewart^{[2]}, that opens to reveal an Aladdin’s cave of beautiful, magformable, non-convex polyhedra. It comes from deconstructing the meaning of *convex.* Remember that a solid is convex, roughly, if it doesn’t have any holes, dents or spikes. Stewart coined the term *quasiconvex* for polyhedra which (again roughly) have no spikes, but may have holes or dents.

More precisely: given any polyhedron, one can form its *convex hull* by filling in all its holes and the gaps between its spikes. Technically, whenever you draw an imaginary line connecting two points within the shape, then all the points on that line are incorporated into the convex hull. As expected, the convex hull is always convex. Stewart said that initial shape is *quasiconvex* if this procedure doesn’t create any new edges, that is to say if every edge of the convex hull is already present in the original. Looking at the stella octangula above, for example, its convex hull will have a new edge joining the tips of the two nearest spikes. So that shape is not quasiconvex. (In fact, it is not too hard to see that the convex hull of the stella octangula is a cube.)

A method for discovering quasiconvex shapes is to begin with a convex polyhedron, such as an Archimedean solid, and then start gouging out, or *excavating* its faces. Excavation is the opposite of augmentation which we me last time. To augment a truncated cube, for instance, we replace one of its octagonal faces with an outward-facing square cupola. To excavate it, one replaces an octagonal faces with an *inward-facing* square cupola. Doing this with an opposite pair of faces, and then removing a final cube to connect the two cupolas, brings us our first Stewart toroid:

The word *toroid* here indicates that these shapes are like a *torus:* they have a hole (or more than one) in the same sense that a donut has. Another can be formed by removing two triangular cupolas and an octahedron from a truncated octahedron:

You can perform augmentation and excavation on the same face. Starting again with a truncated cube, and augmenting two opposite faces with square cupolas produces a biaugmented cube (one of the Johnson solids). Removing two cuboctahedra and a central cube from this shape gets us:

A couple of further examples. Extracting a triangular cupola and trinagular prism from an augmented triangular cupola produces:

Here’s a good one. Taking a pentagonal cupola and pentagonal antisprism out of a rotunda produces:

A different approach, also investigated by Bonnie Stewart, instead of creating toroids by gouging out faces from uniform solids is to glue them together:

Toroids of this form are not quasiconvex. Still, some amazing things are possible – see for example RobertLovesPi’s arrangement of 92 dodecahedra into a single rhomic triacontahedron.

Back to quasiconvex solids, here’s one of an interesting and rare type, again discovered by Bonnie Stewart:

On first glance it looks unremarkable: a Johnson solid with four hexagonal faces excavated. But which Johnson solid? On closer inspection, there is something unexpected: the excavated faces are *irregular* hexagons. This establishes, surprisingly, that a quasiconvex polyhedron built entirely from regular polygons may have a convex hull with irregular faces. K4′ (pronounced “K4 prime”) is built by starting with the truncated cuboctahedron (also known as K4) and excavating four of its octagonal faces with square cupolas. That gives polyhedron with a a sort of belt of squares around it. Deleting that belt gives K4′, with its slightly squashed appearance.

Not many polyhedra like K4′ are known. But the good news is that K4′ can itself be drilled out with two more square cupolas and a cuboctahedron:

So how many Stewart toroids are there? If we allow shapes like the ring of hexagonal prisms above, then there are infinitely many. What if we limit ourselves to quasiconvex polyhedra whose convex hull (unlike K4′) above is an Archimedean solid? So far as I am aware, the exact number isn’t known, but is at least guaranteed to be finite. Stewart estimated, in February 1980, that there are 6031 ways of drilling through the truncated cuboctahedron alone, and several quadrillion ways of tunneling through the (decagon-requiring and therefore non-magformable) truncated icosidodecahedron. This is how Stewart discovered the unusual non-convex solids G3 and X above: by carving smaller polyhedra out of a bigger one, those were the parts which were left behind.

Let’s end with a couple I am rather pleased with. Last time, I said that my favourite Johnson solid was the hebesphenerotunda. The back of this is built from pentagons and triangles, like a patch of icosidodecahedron. So you can use a hebesphenerotunda to excavate an icosidodecahedron, and if you do that twice on opposite sides, happily their hexagonal faces match up exactly:

(Strictly speaking the hexagonal face should be removed from the right hand picture to give a proper “tunnel”, but aesthetically and structurally it’s nicer to to leave it in. This is of course the elephant in the room in building with magformers – faces are only there in skeletal form anyway.)

Likewise, Stewart’s G3 (pictured above) has on one side three pentagons, like a patch of dodecahedron. So you can use G3 to excavate a dodecahedron:

You can then do the same again…

What this gives is a decomposition of the dodecahedron into two G3s, four pentagonal pyramids, and one of… whatever this is:

Having got this far myself, I had wondered whether anyone else had followed the same line of thought, an perhaps even given this shape a name. The answer is that Alex Doskey and a few others have, but the only written description of this shape I can find is Alex’s “another figure with fewer faces”.

Anyway, putting all those back together gives a very pleasing decomposition of the dodecahedron, which is where we leave this post.

[1] All the pictures in this post are either my own photos of magformers, or are created using Robert Webb’s Stella 4d, the best software I know for easily manipulating a wide variety of polyhedra.

[2] The ideas in this post are based on Bonnie Stewart’s wonderful, but sadly largely unavailable, self-published, hand-lettered, and hand-illustrated book *Adventures Among The Toroids (1980, 2nd edition).*

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:

Whenever we think of the Archimedean solids, we mustn’t forget (explanation below) to mention in the same breath the prisms. There are infinitely many of these, of which five are magformable (although one, eagle-eyed readers will notice, we have met already under a different name):

There is also the infinite family of antiprisms of which we can make six (two may look familiar):

What to do next next? In our enthusiasm for polyhedra, let’s not overlook 2-dimensional structures. There are three regular tessellations:

…and eight semi-regular tessellations, of which six are magformable (the other two this time require dodecagons [corrected, see comments]). For example:

Tessellations are not limited to 2-dimensions though. Of the shapes discussed so far, the *space-filling polyhedra* (i.e. ones which can tessellate single-handedly) are: one Platonic solid (the cube), two other prisms (triangular and hexagonal), and one of the Archimedean solids (the truncated octahedron).

Beyond these there are 23 other convex uniform honeycombs, in which combinations of Platonic, Archimedean, and prismatic solids together fill space. In principle, all are magformable! So this a rich seam for further exploration, although you do start to need serious numbers of pieces to get anywhere. A couple of examples:

Before moving on to the topic in this blogpost’s title, let’s step back. All the shapes we have met so far are characterised by two things: *symmetry* and *convexity*. Firstly, all their faces are highly symmetrical: only equilateral triangles, squares, regular pentagons, hexagons, or octagons are used, that is to say regular polygons. (Magformers do offer some less regular pieces – rhombuses, isosceles triangles, etc. – but I shan’t be using any.) But the Platonic solids exhibit global symmetry as well. Every face is identical to every other, in form, but also in how it relates to the rest (*face-transitivity*). More subtly, every corner is identical to every other *(vertex-transitivity)*. Obviously, the Archimedean solids, having faces of more than one shape, cannot be face-transitive, but they are vertex-transitive. Indeed this is their defining property. To coin one more piece of jargon, a solid is *uniform* if it is vertex-transitive and built entirely from regular polygons.

The geometrical theorem we have danced around is that the Platonic and Archimedean solids, the prisms, and the antiprisms are the only polyhedra which are both uniform and convex.

…which brings us to *convexity*. Roughly, this means that the shape doesn’t have any holes or dents, or any bits sticking out too far. More precisely, a shape is convex if whenever you take two points on different faces and connect them with an imaginary straight line, that line lies entirely inside the shape. If you can find two points whose line passes through fresh air on its way between them, the shape is non-convex.

A detail: the requirement that the imaginary line lies *inside* the shape rather than on the surface, separates polyhedra which are *strictly *convex from those which are merely convex, where adjacent faces may lie flat.

In 1966, Norman Johnson posed a question: forgetting about global symmetry (i.e. face or vertex transitivity), what strictly convex polyhedra can be built from regular polygons? He came up with a list of 92 (on top, of course, of the uniform solids just discussed). In 1969, Victor Zalgaller proved that this list of 92 *Johnson solids* was indeed complete.

By my reckoning, 74 are magformable, with the need for decagons again the impediment to the rest. (I hope my new friends at Magformers will take two things from this post: the benefits of making octagons more widely available, and the urgent need to start creating decagons!) Let’s see some examples. The simplest Johnson solids are the pyramids (notice an old friend sneaking in amongst the newcomers:

These already illustrate a theme, which is that many of the Johnson solids are obtained by fiddling with uniform solids: the triangular pyramid is a tetrahedron (and thus not one of the 92), the square pyramid is half an octahedron, and the pentagonal pyramid is the hat of an icosahedron.

Gluing matching pairs of pyramids together gets you the bipyramids (no surprises to see a familiar figure this time):

The bipyramids (that is to say the triangular and pentagonal ones, not the Platonic interloper) are interesting for another reason: they are face-transitive. Every face of the pentagonal bipyramid (say) is identical not just in shape but in function to every other. But it is *not* vertex transitive: the corner at the top is qualitatively different from those around the equator, in that five faces meet at the top, but only four at each equator corner. (So it’s no injustice that the pentagonal bipyramid is omitted from the list of Platonic solids.)

Several of the Johnson solids are derived by either chopping bits off (*diminishing*) or adding bits on to (*augmenting*) uniform solids. E.g….

As another illustration of the sort of cutting and gluing you can do, the hat of a rhombicuboctahedron (one of the Archimedean solids) is a Johnson solid: the square cupola.

Gluing two of these together gets you a square bicupola. In fact, it gets you two, depending on how you align the halves.

Likewise, the hat of a rhombicosidodecahedron is a pentagonal cupola (one of the Johnson solids I can’t make properly – by now you know why). Connecting two of them with a ring of triangles (setting them out of phase à la antiprism), gives you this:

Let’s have one more example. Chopping in half an icosidodecahedron (another of the Archimedean solids) gets you two (decagon-requiring and therefore not pictured) pentagonal rotundas. Giving one of these halves a twist, and then gluing them back together with a ring of squares in between gives you:

You get the idea! So, are any of the Johnson solids space-fillers? Just one: the gyrobisfastigium. But actually, even this is a bit of a cheat. The gyrobifastigium is built from two triangular prisms joined with a twist, so the gyrated triangular prismatic honeycomb (pictured above) can be re-imagined as constructed from these instead of prisms. Sadly, there are no genuinely new honeycombs built from Johnson solids.

Beauty is often held to be closely related to symmetry. By definition, Johnson solids can’t have the high level of symmetry of the uniform shapes we met earlier. But let no-one say that makes them ugly! For one thing, there is still room for symmetry of the traditional reflectional or rotational variety. Surprisingly (at least to me), *all* of the Johnson solids exhibit some symmetry of this kind. How much varies considerably, but it supports my view that many of these shapes are really quite beautiful. The most intriguing are the *elementary* Johnson solids, those *not* obtained by chopping or gluing uniform solids using the sorts of techniques described above. For instance:

One I like very much:

But my favourite of all is the last of the 92:

This shape is elementary: it cannot be broken down into smaller regular polygonal convex solids. It has attractive three-fold rotational and vertical reflectional symmetry, but also a touch of the spectacular: it’s the only one of any the polyhedra discussed in this post which requires triangles, squares, pentagons, and hexagons.

I hope you enjoyed this quick tour of the Johnson solids! But where does the newly enthusiastic magformist go from here? Answer, they begin building *non-convex* polyhedra – and that’s when the fun really starts! A follow-up blog-post is on its way.

In the meantime – hopefully it’s obvious by now – but let me say this: I like Magformers a lot. What I enjoy most is not just that you can make all these wonderful shapes (there are numerous ways of doing that, from origami to 3d-printing) but that you can do it incredibly quickly and easily. You can knock up a dodecahedron in 30 seconds, an icosidodecahedron in a minute, and a rhombicosidodecahedron in two. So, even in a short amount of time, you can lose yourself in a lovely world of geometry.

*[Update: the sequel to this post is now online – Magforming the Stewart Toroids]*

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’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.

]]>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:

**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.**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.**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.**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.**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.**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.**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.**“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.**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.

** 10. Mochizuki’s claimed proof of the abc conjecture.** The countdown kicks off on an awkward note. If Shinichi Mochizuki’s 2012 claimed proof of the abc conjecture had gained widespread acceptance, it would definitely top this list. As it is, it remains in limbo, to the enormous frustration of everyone involved.

**9. The weak Goldbach conjecture.** “From 7 onwards, every odd number is the sum of three primes.” We have known since 1937 that this holds for all large enough odd numbers, but in 2013 Harald Helfgott brought the threshold down to 10^{30}, and separately with David Platt checked odd numbers up to that limit by computer.

**8. Ngô Bảo Châu’s proof of the Fundamental Lemma.** Bending the rules to scrape in (time-wise) is this 2009 proof of a terrifyingly technical but highly important plank of the Langlands Program.

**7. Seventeen Sudoku Clues.** In 2012, McGuire, Tugemann, and Civario proved that the smallest number of clues which uniquely determine a Sudoku puzzle is 17. (Although not every collection of 17 clues yields a unique solution, their theorem establishes that there can never be a valid Sudoku puzzle with only 16 clues.)

**6. The Growth of Univalent Foundations/ Homotopy Type Theory.** This new approach to the foundations of mathematics, led by Vladimir Voevodsky, is attracting huge attention. Apart from its inherent mathematical appeal, it promises to recast swathes of higher mathematics in a language more accessible to computerised proof-assistants.

**5. Untriangulatable spaces.** In sixth position is the stunning discovery, by Ciprian Manolescu, of untriangilatable manifolds in all dimensions from 5 upwards.

**4. The Socolar–Taylor tile.** Penrose tiles, famously, are sets of tiles which can tile the plane, but only aperiodically. It was an open question, for many years, whether it is possible to achieve the same effect with just one tile. Then Joan Taylor and Joshua Socolar found one (pictured above).

**3. Completion of the Flyspeck project.** In 1998, Thomas Hales announced a proof of the classic Kepler conjecture on the most efficient way to stack cannon-balls. Unfortunately, his proof was so long and computationally involved that the referees assigned to verify it couldn’t complete the task. So Hales and his team set about it themselves, using the Isabelle and HOL Light computational proof assistants. The result is not only a milestone in discrete geometry, but also in automated reasoning.

**2. Partition numbers.** In how many ways can a positive integer be written as a sum of smaller integers? In 2011, Ken Ono and Jan Bruinier provided the long-sought answer.

**1. Bounded gaps between primes.** It’s no real surprise to find that the top spot is taken by Yitang Zhang’s wonderful 2013 result that there is some number *n*, below 70 million, such that there are infinitely many pairs of consecutive primes exactly *n* apart. The subsequent flurry of activity saw James Maynard, and a Polymath Project organised by Terence Tao, bring the bound down to 246.

Where’s Hairer’s work on the KPZ equation? What about Friedman’s new examples of concrete incompleteness?! What can I say? It’s just for fun, folks. If you think I’ve got it horribly wrong, then feel free to compile your own lists. (The real answer for such things being left out is that I couldn’t easily update my book to include them.) And now…

In no particular order:

- The simple continued fraction of π has now been computed to the first 15 billion terms by Eric Weisstein, up from 100 million.

- The decimal expansion of π has been computed to 13.3 trillion digits, up from 2.69999999 trillion.

- The search for the perfect cuboid: if one exists, one of its sides must be at least 3 trillion units long, up from 9 billion.

- Goldbach’s conjecture has been verified for even numbers up to 4 × 10
^{18}by Oliveira e Silva, up from 10^{18}.

- The largest known twin primes are the pair either side of 3756801695685 × 2
^{666669}, up from 2003663613 × 2^{195000}.

- The largest known prime, and the 48th known Mersenne prime (up from 47), is 2
^{57885161}-1, up from 2^{43112609}-1.

- The Encyclopedia of Triangle Centres contains 7719 entries (up from 3587).

- The longest known Optimal Golomb Ruler is now 27 notches long (up from 26):

(0, 3, 15, 41, 66, 95, 97, 106, 142, 152, 220, 221, 225, 242, 295, 330, 338, 354, 382, 388, 402, 415, 486, 504, 523, 546, 553)

- The most impressive feat of integer-factorisation using classical computers, is that of the 232-digit number RSA-768:
1230186684530117755130494958384962720772853569595334792197322452 1517264005072636575187452021997864693899564749427740638459251925 5732630345373154826850791702612214291346167042921431160222124047 9274737794080665351419597459856902143413

into two 116 digit primes

3347807169895689878604416984821269081770479498371376856891243138 8982883793878002287614711652531743087737814467999489

and

3674604366679959042824463379962795263227915816434308764267603228 3815739666511279233373417143396810270092798736308917.(The previous record was the 200 digit semiprime RSA-200.)

- The most impressive feat of integer-factorisation using a quantum computer is that of 56,153=233 × 241. The previous record was 15.

- The Collatz Conjecture has been verified for numbers beyond 2 × 10
^{21}. The previous record was 5.76 × 10^{18}. (However, this has happened via a patchwork of distributed computing projects, and I have not been able to establish with any certainty that*every*number up to the new higher limit has been checked. I encourage someone in this community to organise all the results in a single location.)

I’ve recently been revisiting Euler’s Theorem. Not that one. No, not the one on this commemorative stamp either. No, no, not the totient function one. (Good guess though.)

I mean the one about *partitions*.

What is a partition? It’s simply a way of splitting up a number (a positive integer) into smaller pieces. For instance 2+2+1 is a partition of 5. The question is: how many ways of doing this are there?

Well there’s just one partition of one, namely 1.

There are 2 partitions of two: 2 and 1+1.

There are the 3 partitions of three: 3, 2+1, 1+1+1. (Notice that for these purposes, we count 2+1 and 1+2 as the same partition.)

So far the pattern looks rather easy. But there are the 5 partitions of four:

4, 3+1, 2+2, 2+1+1, 1+1+1+1

And then 7 partitions of five:

5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+, 1+1+1+1+1

Continuing the sequence, we find 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010,…

What’s the pattern now? Finding an explicit formula for the number of partitions of is *seriously hard*. Even Euler couldn’t manage it. But he was able to make some progress: he found a *generating function*.

Next question: what is a generating function? To answer with an example, the generating function for the sequence is the power series .

More generally, the generating function for the sequence is a series: .

Next question: what’s the point? Often the resulting series can be radically compressed. For instance, a bit of mathematical magic tells us that the series can be rewritten^{[1]} as . Now the whole infinite sequence has been encapsulated by a short, simple algebraic expression. So, if you want to know the th element of the sequence, you can extract it (if you know how) fairly easily from the generating function.

Well, there is not a lot of mystery in the sequence – certainly there are no prizes for guessing its th term. But what of the partition sequence: ? Euler’s great insight was that there is a generating function for this too:

You can see that this is not as nice as the function above – and it’s certainly not nice enough to close the book on the whole problem of computing partition numbers. All the same, this function is hugely easier to work with than scrabbling about with partitions with bare hands. Euler’s generating function can also be written as

On first sight, this looks like an algebraic nightmare: you have to open up an infinite number of brackets, each of which contains an infinite series! However, if you want to compute the th partition number, you only care about the first brackets, and only the first few terms in each: from the first bracket you care about the first terms, from the second bracket approximately the first , from the third approximately the first , and so on. In fact the total number of terms you need to worry about is approximately , which is a manageable number when is not too big.

So although Euler didn’t find a formula for partition numbers, his generating function does provide a manageable procedure for computing these numbers. But what of an actual formula?

Further progress came courtesy of a pair of mathematical superstars to rival Euler: Hardy and Ramanujan. The th one is given approximately by the formula

As grows bigger and bigger, this number gets proportionally closer to the right answer. But what about an exact formula? That’s something there has been huge progress on in recent years… but perhaps that’s a story for another day.

[1] There are issues of convergence here, which I will ignore for now.

]]>