The nice chaps at Kitki, an educational board game company based in India, have come up with a cool idea for a mathematical board game. They’re funding it through IndieGoGo (which if you haven’t heard of it is a bit like Kickstarter), and they’re looking for your help.

The game, called **Three Sticks**, is based on a square grid of dots. The playing pieces are sticks of lengths 3, 4 and 5 units, and you play them on the board by joining two dots. You score points by using your sticks to form geometric shapes, and refill your pile of sticks through picking up cards and stealing sticks from other players.

It sounds like a fun game, and if it goes ahead, it’ll also be pretty good as an educational tool. Scoring is based on the perimeter and type of shape created, which means you have to do a calculation to work out how many points you get. You can score extra points by being the first person to make a given shape, and you can also overlap different shapes – so you can make more than one shape which share some of the same sticks.

Pledge rewards range from a freshly-minted copy of the game up to a beautiful collector’s edition in a wooden box. They’re looking to raise $10,000 to get the first run of games made, and if it’s something you’d like to see happen, you can donate on their IndieGoGo page.

Three Sticks fundraising page on IndieGoGo

Three Sticks on Twitter

]]>Eugenia Cheng (of nonsense formulas *passim*) has “found” the formula for the perfect doughnut, for Domino’s Pizza. Coincidentally, they’ve recently started selling doughnuts.

Actually, “formula” should be in quotes as well – the “formula” she gives is, *drumroll…*

\[ \frac{(r-2)^2}{4(r-1)} \]

Note that that’s not a formula.

Apparently the best value for this ratio is “3.5 : 1″, which I reckon implies that $r \approx 1.063$. That’s $1.063$ of whatever units $r$ is measured in. This is one case where I’d particularly like to see some working out. Knowing Eugenia, she’s made up some crazy model and done variational methods on it to come up with something that’ll fit nicely in the Domino’s press kit.

Her students have picked up her habit: they made up this set of formulas for the perfect christmas tree, for Debenhams.

… which they also came up with in 2012 (although they were pipped to the post in 2013 by Kingston University’s Gordon Hunter, playing for Team Dobbies Garden Centres).

The “Christmas tree facts” box in 2014’s Christmas tree formula press release has some pretty bold claims about the amounts of baubles, tinsel and lights the tree in Trafalgar Square, London, would need. We tried to contact London City Hall to check if the figures match up, but received no reply.

In conclusion: it’s such a shame Eugenia Cheng keeps doing this. She does plenty of other stuff that’s actually worthwhile, like this introduction to higher-dimensional category theory for non-mathematicians. We invite Eugenia to get in touch, if she’d like to explain how these “formulas” concocted for advertising purposes benefit mankind.

]]>Honeybees collect nectar from flowers and use it to produce honey, which they then store in honeycombs made of beeswax (in turn derived from honey). A question that has puzzled many inquiring minds across the ages is: why are honeycombs made of hexagonal cells?

The Roman scholar Varro, in his 1st century BC book-long poem *De Agri Cultura* (“On Agriculture”), briefly states

“Does not the chamber in the comb have six angles, the same number as the bee has feet? The geometricians prove that this hexagon inscribed in a circular figure encloses the greatest amount of space

^{1}.”

This quote is the earliest known source suggesting a link between the hexagonal shape of the honeycomb and a mathematical property of the hexagon, made more explicit a few centuries later by Pappus of Alexandria (sometimes considered to be the last Ancient Greek mathematician). Writing after the Roman Empire’s glory days, Pappus points out that there are three regular polygons that tile the plane without gaps—triangles, squares and hexagons—and bees, in their wisdom, choose the design that holds the most honey given a set amount of building material^{2}.

The idea that bees economically choose the regular tiling polygon with the greatest area for a given perimeter satisfied the ancients. But why should all the honeycomb cells be identical—could bees do any better than using regular polygons?

The amazingly regular honeycombs we are used to seeing are built by domestic honeybees, using perfectly regular wax foundations provided by their beekeepers as a guide. Wild honeybees, however, don’t have the luxury of nestboxes fitted with wax foundations. Their honeycomb cells, though still very regular, aren’t always hexagonal—sometimes pentagons and heptagons creep in. After choosing an irregularly-shaped tree cavity, many worker bees will start independently building the hive’s hexagons at once, meeting in the middle along seams. It is along these seams where the five and seven-sided irregularities and other defects tend to appear^{3}.

More intriguingly, what happens if you provide honeybees with the wrong size of foundation as a guide? H. Randall Hepburn tried this with variously-sized hexagonal foundations^{4}(with flat bases unlike the standard ridged commercial ones). The largest caused the bees to to build gorgeous rosettes—they filled each large hexagonal foundation with five or six cells (again, typically five or seven-sided) surrounding a central cell. The next size down caused the bees to build in an irregular pattern, while the size smaller than that led to the bees building hexagonal cells on each vertex of the foundational hexagons, and leaving a hexagonal void or ‘false cell’ in the centre.

So bees don’t necessarily need to use regular polygons at all. Allowing for the possibility of a mixture of shapes leads to two subtly different mathematical problems:

**(P)** Which mixture of shapes that tile the plane, and each have unit perimeter, have the maximum average area?

Fejes Tóth, a Hungarian mathematician, had proved by 1963 that the regular hexagonal tiling was the solution to the isoperimetric problem, (P), but only proved that it solved the iso-areal problem, (A), under the assumption that all the shapes were convex. This convexity condition is more restrictive than it may at first seem: a bulging edge on one shape leads to a concave indentation on a neighbouring shape, so the only convex shapes that can tile the plane are polygons.

It wasn’t until 1999 that Thomas Hales proved that hexagons also divide a plane into shapes of equal area with the least perimeter.

A key step in the proof (though not Hales’ breakthrough) is showing that, on average, such shapes would have six sides. Take a planar graph (ie. with no edges crossing) on a sphere. Using the Euler characteristic of a planar graph (which also holds on a sphere), we know that the number of the vertices, edges and faces of the graph satisfies $V-E+F=2$.Nudging the edges slightly, it turns out to be sufficient to consider only vertices of degree 3. It’s quite easy to convince oneself with a diagram (like the one above) that this is plausible. Moving edges by a tiny amount, you can make sure no more than three lines meet at one vertex. You might change the numbers of edges and vertices, but without changing the perimeter or area by much.

If the $i$th face has $e_i$ edges (and vertices), then adding these up for each face, we would count each edge twice and each vertex three times^{5}. So, by the Euler characteristic of the whole graph, overall we must have $$V-E+F=\sum_i \left ( \frac{e_i}{3}-\frac{e_i}{2} + 1 \right )=\sum_i \left ( 1 – \frac{e_i}{6} \right )=2.$$As the number of faces increases, $1 – \frac{e_i}{6}$ must become very small, and so the average of $e_i$ over all faces $i$ tends to $6$, which is the number of sides we were hoping for.^{6} This step can be used to show a finite version of the theorem on the plane, which is a good start for the infinite version.

So out of all the prisms arranged side-by-side, hexagonal prisms use the least beeswax to build a unit bee-sized volume. But honeycombs are not made up of hexagonal prisms: the hidden end of the honeycomb cell is not flat. There are two layers of cells, back-to-back and offset, and the end-cap between them is a pyramid made of three rhombuses, so the whole shape could also be described as ‘half of an elongated rhombic dodecahedron’. The ridged wax honeycomb foundation pictured near the start mimics these rhombuses. This shape is more efficient than a hexagonal prism with a flat base.

What bees haven’t realised, is that there’s a slightly more efficient way to cap off their hexagonal cells. Keeping the hexagonal footprint of their hive, they could make repeated use of a truncated octahedron, formed by lopping off the vertices of an octahedron. This would lead to a saving of almost 2% in the idealised version with walls of negligible thickness, which isn’t much.

Using a curvy non-polyhedral variant of the truncated octahedron, Lord Kelvin, the 19th century physicist, held the record for ‘least surface area for a fixed volume’ in the three-dimensional version of the honeycomb conjecture for a full century^{7}. This was assumed to be the best possible until 1994, when two physicists, Denis Weaire and Robert Phelan, armed with computers, devised an ever-so-slightly more efficient way of dividing space into cells of equal volume, using two different curvy solids (2.2% more efficient than the rhombic dodecahedra). It was after this discovery that the long-standing two-dimensional honeycomb conjecture seemed less obviously true: a suggestion from Weaire is what nudged Hales to attempt his proof^{8}.

Despite the fact that honeybees do not achieve a theoretical minimum surface area for a set volume, honeycombs are still a sensible way to reduce the need for building materials. However, we should question the assumption that evolution is purely trying to economise on the bees’ behalf, and ask whether this actually is the main reason why honeybees build hexagonal honeycombs. Building a typical wild nest takes approximately a kilogram of wax, produced by consuming about eight times that amount of honey. Since this is also a third of the amount of honey the bees require to survive the winter, there is a strong evolutionary incentive to use as little wax as possible. There is evidence that bees do economise in another sense when building hives: they have some preference for nesting sites where honeycombs are already present^{9}, so they don’t need to build new ones. An alternative explanation to the economic argument is an engineering one: that hexagonal honeycombs are strong structures. It seems pretty hard to design experiments to determine how much each factor contributes to the bees’ motivations^{10}, so for the time being the explanations may well remain as Just-So stories.

One useful way of looking at the causes of a particular animal behaviour is to ask four questions, known as Tinbergen’s questions. The idea is to explain the current and historical reasons for a behaviour’s existence by asking questions about the individual and the species.

(Current & species: Function)Why have honeybees evolved to build combs in this shape?

(Historical & species: Phylogeny)How has honeycomb building evolved over time?

(Current & individual: Mechanism)How do the bees construct the honeycomb?

(Historical & individual: Ontogeny)How does the behaviour develop in an individual bee?

We’ve been looking at one possible answer to the first question, however this still leaves three unanswered. The ontogeny, or biological and neurological development of bees’ building behaviours is well beyond the scope of this article, which is convenient because I know absolutely nothing about it. As for the other two questions, Charles Darwin (among others) had a good crack at them.

Among the biological investigations recorded in Darwin’s “On the Origin of Species” were experiments and observations relating to honeycomb formation. He proposed that a rough evolutionary progression (the phylogeny) could be inferred from various types of modern bee nests. Bumblebee nests are rough conglomerations of almost spherical cells found underground. The nest of the Mexican stingless bee, *Melipona domestica*, lies somewhere between that of the bumblebee and honeybee, with a hexagonal arrangement but more rounded cells and cylindrical walls.

Looking at these other bee nests, Darwin suggested that honeybees once had nests similar to bumblebees, then more like those of the Mexican stingless bee, before their current honeycombs. Through conversation and correspondence, he came to the same conclusion as some others before him, that if bees built regular arrangements of cylindrical or spherical cells closer and closer together over the generations, then at some point they would crowd together into the famous honeycomb arrangement.

In more formal modern mathematical terms, this is the idea of *Voronoi cells. *Take* *some points in space: the region closer to one point than any other is known as that point’s Voronoi cell. If you take points in two slightly offset layers, all spaced equally to their closest neighbours (ie. joining the points would form equilateral triangles and hexagons), then the three-dimensional Voronoi cells form the honeybees’ hexagonal cells with their rhombic bases (partial rhombic dodecahedra).

Darwin wanted to know whether this theory matched up with the way bees actually build their hives. He watched very closely as his own honeybees constructed their nest and his observations support the Voronoi cell idea of how the nests evolved and how they are built. Darwin conducted some brilliantly simple experiments that involved introducing either a thick wax block or thin coloured wax sheet into a hive. With a wax block introduced, each bee dug a small hemisphere into the block, until it touched or got extremely close to an adjacent bee’s excavation, at which point the bees built the hexagonal walls up, leaving a smooth basin. This suggests that in the usual situation, the rhomboid basins are caused by pressures from bees working on cells on the other side, whereas in this case there were none. With a thin sheet introduced, the bees started excavating but stopped before they broke through to the opposite side, leaving very shallow and flat basins. This is all supports Darwin’s suggested mechanism for the cell construction.

In the classic 1917 book “On Growth and Form” D’Arcy Thompson points out that the beeswax is warm and slightly liquid when being used for building, which may cause tension effects to come into play. The same effects cause bubbles to naturally seek shapes that minimise their surface area. This is attractive as an alternative to the Voronoi cells idea, and so gets frequently repeated, but to me it doesn’t yet seem to be supported by the evidence.

The surface tension explanation is most vocally put forward in the paper *Honeybee combs: construction through a liquid equilibrium process?* (££)^{11}. The authors suggest that—in analogy with experiments where wax is melted around close-packed rubber bungs—the bees first construct cylindrical cells, and then heat them until the wax melts enough for surface tension to take over and form the hexagons.^{12}

A rebuttal, *Hexagonal comb cells of honeybees are not produced via a liquid equilibrium process *(££)^{13}, points out some weaknesses in the paper above. The bees are seen to actively move their mandibles across the cell walls (video, 22mb AVI), mechanically shaping them, as well as feeling them with their antennas. Social wasps build their hexagonal nests with cellulose, a rather different material to wax. Cellulose doesn’t have the same fluid properties under heating either, so it is merely softened by the wasps using saliva. This means that wasps achieve similar results to the honeybees by actively kneading cells into shape without surface tension effects. If in doubt, we should prefer arguments that have more explicative power, and not resort to special cases for bees and wasps.

Central to the argument is the question of whether bees construct intermediate cylinders. This observation only seems to appear once in the literature, in a paper published in 2013 (after the above rebuttal)^{14}. This paper shows a close-up photo of the few circular cells found after the honeybees were smoked out during the construction of their hive. The circular cells were near the edge of the growing comb, a bit shorter than the usual 10mm, and the walls nearest the base of the circular cells were hexagonal. After the bees returned and completed their work, these cells were modified to their normal hexagonal specifications.

Unfortunately, this observation doesn’t clinch the argument either way. While it is necessary for surface tension to play a major role, intermediate cylinders are consistent with both explanations. The fact that bees rework their cells as the comb expands was noticed by Darwin, who observed that some vermillion-coloured wax introduced at the edge would become dispersed throughout much of the comb.

It’s going to take a very well-designed experiment to convince me that bees rely largely on surface tension effects, as opposed to moulding the exquisite hexagons themselves. After all, the worker bees are rarely depicted as lazy.

Charles Darwin’s “On the origin of species”: is freely available on Archive.org and Project Gutenberg, for instance, and cheaply available in book form. Chapter VIII: Instinct, *Cell-making instinct of the Hive-Bee, *is the section referred to above.

After reading about Darwin’s experiments in his own words, I strongly recommend reading an account of Darwin’s honeycomb correspondence which includes some modern recreations of Darwin’s original experiments.

“On Growth and Form” by D’Arcy Thompson not only has a detailed history of the study of honeycomb geometry, but has many interesting facts about biological and geological shapes and their growth, as well as speculation about their root causes.

And on the pure mathematical geometry side, Fejes Tóth’s “What the bees know and what they do not know” gives the best overview of minimisation problems that bees’ honeycombs do solve, and then many more they do not solve.

- Translation by Hooper and Ash in the Loeb. I’ve been told that ‘Hexagonon’ is in its singular form, and the only Greek word (also having Greek grammar) amongst this part of Varro’s Latin text. I would be happier that Varro understood what he was writing about if the text more explicitly described the construction, perhaps ‘Three hexagons encircling a point’, or ‘Six hexagons arranged around a seventh’. In translation, it could be viewed as falsely suggesting that the hexagon is the polygon with the greatest area that fits inside a circle. In his defense though, Varro also earlier suggests that orchards be arranged regularly in quincunxes, the arrangement of spots representing the number five on dice, to take up less room and give better quality produce. The centres of hexagons in a regular hexagonal tiling can be thought of as an elongated quincunx, repeated. As this is essentially the same result used in another context, I’ll give Varro the benefit of the doubt and defer to Varro’s poetic license.
- A translation by Ivor Thomas in the Loeb “Greek Mathematics: Volume II“, of a passage from Pappus’ Synagogue, Book V: “That [bees] have contrived this in accordance with a certain geometrical forethought we may thus infer. They would necessarily think that the figures must all be adjacent one to another and have their sides common, in order that nothing else might fall into the interstices and so defile their work. Now there are only three rectilineal figures which would satisfy the condition, I mean regular figures which are equilateral and equiangular, inasmuch as irregular figures would be displeasing to the bees . [These being] the triangle, the square and the hexagon, the bees in their wisdom chose for their work that which has the most angles, perceiving that it would hold more honey than either of the two others. Bees, then, know just this fact which is useful to them, that the hexagon is greater than the square and the triangle and will hold more honey for the same expenditure of material in constructing each.”
- Hepburn and Whiffler, “Construction defects define pattern and method in comb building by honeybees“, Apidologie (1991) observe that sometimes the pentagons and heptagons appear together, and at other times pentagons appear in clusters. They suggest that the latter can happen when a heptagonal comb is bisected by an extra wall, giving rise to two pentagonal cells.
- H.R. Hepburn “Comb Construction by the African Honeybee, Apis mellifera adansonii“,
*Journal of the Entomological*Society*of Southern Africa,*Vol.46, No.1, pp.87-101 (1983) - Explicitly, $V=\sum_i \frac{e_i}{3}$, $E=\sum_i \frac{e_i}{2}$, and $F=\sum_i 1$.
- This part of the argument was adapted by Hales from Fejes Tóth’s proof, except Hales used a torus instead of a sphere, which has the more computationally elegant Euler characteristic V-E+F=0.
- On the Division of Space with Minimum Partitional Area, Sir William Thompson (1st Baron Kelvin), 1887 – given that the polyhedral truncated octahedron beats the rhombic dodecahedron anyway, I’m confused by what problem Kelvin claims is solved in his Yoda-like comment: “Certainly the rhombic dodecahedron
*is a solution of the minimax, or equilibrium-problem*; and certain it is that no other plane-sided polyhedron can be a solution.” - Weaire to Hales: “Given its celebrated history, it [the honeycomb conjecture] seems worth a try . . . ”
- Seeley and Morse, “Nest site selection by the honey bee, Apis mellifera” £29.95, Insectes Sociaux (1978)
- If the regular hexagonal array is both locally optimal for structural integrity as well as for wax conservation, then separating the two contributions is even more difficult
- Pirk et al (£29.95), Naturwissenschaften (2004)
- The paper also makes an interesting mistake, by remarking that the shape of the honeycomb base is rounded, and not rhomboid as usually described. This is based on moulds they made of the cells. It was later pointed out that they must have used old honeycomb cells, which gradually become rounded over time. The authors ascribed the previous rhomboid observations as either having been taken from cultivated honeycombs with wax foundations, or the straight-edges being an optical artefact from viewing two offset layers of hexagons. While being independent-minded and relying on your own observations is admirable, overturning several centuries of observations shouldn’t be taken lightly.
- Bauer and Bienefeld (£29.95) Naturwissenschaften (2013)
- Karihaloo et al. “Honeybee combs: how the circular cells transform into rounded hexagons“, J.R. Soc. Interface (2013)

*3rd February 2015 (hardcover); Simon & Schuster/TED*

Hannah Fry, who’s a lecturer and public engagement fellow at UCL, has written a book. Following a TEDx talk she gave last spring, Hannah was invited by TED to be one of 12 speakers who got the chance to put their ideas into book form. Her topic was the mathematics of love, and the result is this collection of mathematical stories and techniques for navigating the world of romance, from choosing a partner to keeping hold of one.

It’s not a huge book – Fry herself describes it as ‘[not] exactly War and Peace’ in the acknowledgements – but I still found out plenty of things I didn’t already know. It covers a good range of topics, and goes into a decent amount of depth – enough to pique your interest, and with detailed and well-chosen references for anyone keen to find out more. More complicated mathematical clarifications are tucked away in footnotes, meaning the text can get on with telling the story in an engaging, and often funny, way.

Popular maths book aficionados will find some well-worn examples (the prisoner’s dilemma and the secretary problem, to name a couple), but they’re introduced with wit and charm, and Hannah acknowledges the shortcomings of such models in dealing with real-world scenarios – giving an idea of the ways in which they break down, and emphasising that these models are merely something we can use as a rough guide to how things really behave.

“My great hope is that a little bit of insight into the mathematics of love might just inspire you to have a little bit more love for mathematics.”

The book is stuffed with enjoyable examples, and touches on a range of areas of maths – game theory, networks, probability and estimation among others – while sticking always to the theme of love and romance, in a witty and inclusive way.

It’s also a quite nice object in hardback, with a pretty geometric/symmetric patterns of hearts on the cover (two versions exist, both nice), and each chapter is accompanied by a full two-page illustration which playfully summarises the theme, be it online dating or wedding planning.

This would make an ideal gift for lovers of maths, and mathsy lovers – and considering the release date, a spectacular Valentine’s day present. Or, if you’re not currently “studying the two-body problem”, get one for yourself.

Hannah’s TEDx talk on The Mathematics of Love

Hannah is @FryRSquared on Twitter

The Mathematics of Love on the publisher’s website

]]>I regularly review resources written for pupils and teachers that in some way aim to support or extend Science, Technology, Engineering and Mathematics (STEM) education. The most recent campaign in the UK is the *Your Life* campaign and as usual it has a website with short articles designed for teachers and pupils to browse and be inspired.

Imagine my excitement when one of the articles was called “Why Do Penguins Care About Maths?”. Two of my favourite things together in one article, there was even a video. I imagined something about penguins going North, then East then South on their quest for fish and ending up close to where they started. How does the problem change for a beady-eyed Rockhopper over a majestic (but slightly ridiculous) Emperor? How far does a penguin swim anyway? How do you map three-dimensional movement as it glides up and down under the water? So many possibilities for penguins and maths.

You can certainly imagine my disappointment then when the answer was that their keeper needed to work out how many fish to feed 6 penguins and how to calculate the volume of their tank. The video didn’t even have that many penguin close ups. Not many penguins and not much maths.

The Your Life campaign is a collaboration between government and some large employers. It aims to increase the numbers taking STEM subjects, particularly physics and maths, by 50% in three years. The articles on the website are written with explicit links to careers. Understanding just how important maths is to a whole range of careers is poor. This week I have also completed an evaluation of a project on careers advice for careers in the NHS. About 80% of pupils surveyed knew you needed science to be a doctor, around 50% selected maths. There were similar results for other NHS careers. Maths engagement in a campaign like the Your Life campaign does need to show application for careers, and a wide range of careers which use maths at lots of different levels, not only careers for people with degrees in maths. But surely we can do better than some arithmetic?

So why should a penguin (or other favourite animal) care about maths?

]]>There’s a standard format for celebrating a mathematical milestone, perhaps the 80th birthday of some deeply eminent number theorist. His collaborators and graduate students, and their graduate students, and *their* graduate students all gather together in some gorgeous location to regale each other with their latest theorems, while the rest of the world pays no attention. For the London Mathematical Society’s birthday, we had something different. Well, we did have the gorgeous location. The Goldsmiths’ hall in London is a magnificent venue, and the livery hall in particular was evidently designed by someone with a peculiar fondness for Element 79. (See for yourself.) But speaker-wise, a decision had obviously been taken that the party would be an outward-looking affair. The focus was not so much on the LMS, or even on maths per se, but on our subject’s ability to *unlock worlds*, particularly the worlds of TV, film, and computer games.

The event was hosted with great panache by Maggie Philbin, everyone’s favourite *Tomorrow’s World* presenter and, as one of the first journalists to try a mobile phone, probably the first person ever to shout “I’m on the train!” down a comically outsized device.

A recurring theme through the day was that other proud British institution which has recently celebrated a major birthday: Doctor Who. After a greeting from the current LMS president Terry Lyons, the day’s first speaker was Steve Thomson, a former maths teacher and one of the Doctor’s script-writers. (He has also wielded his pen for *Sherlock* and in other honourable causes.) Steve described how mathematical ideas can spice up these stories. These applications are all rather gentle: the non-Euclidean geometry of the corridors of the TARDIS (Series 6, Episode 4, The Doctor’s Wife), naming Time Lord technology with randomly selected mathematical buzzwords (DOCTOR: “The parametric engines are jammed! Orthogonal vector’s gone! I’m almost out of ideas!”, Series 6, Episode 3, Curse of the Black Spot), and so on. Omitted from the talk, as Yemon Choi pointed out on Google plus, were Block transfer computations: mathematical operations so powerful that performing them alters the structure of space-time.

The quaternions provided a second recurring theme. These strange 4-dimensional numbers were invoked by the second speaker, the mathematician Nigel Hitchin, as one of several delightful tales of mathematical Creativity, Discovery, and Curiosity. Their discoverer, William Hamilton, was impressed by the complex numbers as a 2-dimensional number system, but was adamant that he could go one better. As it happened, he went two better, and famously carved the axioms for his system on Broom Bridge in Dublin (a plaque still marks the spot). He became so obsessed with these objects that he came to believe, or so it was said, that they could solve every problem presented by the real world, an affliction his critics dubbed “Irish madness”.

His discovery received a luke-warm reception at the time, with Lord Kelvin bitchily remarking: “Quaternions came from Hamilton after his really good work had been done; and, though beautifully ingenious, have been an unmixed evil to those who have touched them in any way”. All the same, quaternions have subsequently proved themselves useful, both within mathematics and, as we heard later, in engineering. From this story, Nigel drew the moral that “curiosity driven research is certainly worth pursuing, but you may have to wait 150 years in order to see the fruits of it”.

For me, Rob Pieké from the Moving Picture Company delivered the day’s stand-out presentation, describing how to create fluid special effects in films. Smoke, dust, and flame all count as fluids, so imagine, let’s say, a giant monster made of molten lava smashing a mountain to bits. It is obviously important that the resulting plume of dust be visually realistic, otherwise how can the audience possibly believe what they’re seeing? (See MPC’s film reel for other similar examples.)

First you divide the space into tiny cubic boxes called voxels. Then inject some virtual fluid, and encode in each voxel simplified versions of the Navier-Stokes laws of fluid dynamics. For instance, since fluids are assumed to be incompressible, the amount of fluid flowing into each voxel through its six faces should be equal to the amount flowing out. But once you’ve adjusted the outflow to make this true at one voxel, you may have messed up the neighbouring one. Bring in the heavy duty computers: you simply iterate this correction procedure until it stabilises, or near enough. Robert gave such an exquisitely clear exposition of this process that by the end of his talk I almost felt that I could go and code up a simulation myself. His advice: “Try it, it’s a good weekend project if you already have the right framework in place (i.e., grid/voxel lib, image writer)”. So next time you’re in the cinema watching a two-headed winged dog creature breathing fire, remember to say a little thank you to the mathematicians in your life for that amazingly authentic nasal smoke.

Another terrific talk followed, from Andrew Blake, Laboratory Director at Microsoft Research Cambridge, and one of the people behind Kinect. His central question was: How can you teach a machine to see? Examples of genuine machine sight include facial recognition in cameras, and pedestrian detection in Mercedes. The particular problem he and his colleagues were faced with was getting a computer to see a human body. Needless to say, human bodies come in many shapes and sizes, can adopt a number all manner of contorted poses, and camouflage themselves in all sorts of inconvenient ways. The team used *machine learning* techniques to teach the computer to recognise the different parts of the body, with sophisticated ideas from information theory serving to home in on the right answer with high probability and within a manageable time. (One of their crowning triumphs was to use their device to create a 3D scan of the graph theorist Béla Bollobás.)

Robert Calderbank spoke next and again invoked the quaternions, this time to highlight the difficulty of distinguishing between pure and applied maths. Robert (who has won several awards for his decisive contributions to mobile phone technology and related areas) admitted that he is not clever enough to be able to tell the difference, and, goodness knows, nor am I. GH Hardy, who is not known as an idiot, famously got it completely wrong, writing in his Apology “I have never done anything ‘useful’. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world.”

Nowadays quaternions sit within the broader context of matrix algebra. In this modern incarnation, Robert explained that they provide exactly the right formalism for encoding telephone calls across two channels, giving a much more robust signal than either could achieve alone. So much, then, for “pure” mathematics.

The Doctor regenerated for the next talk by James Reid, Head of Effects at Milk, who talked about mathematical ideas in the artistic side of special effect design. So, imagine you’re a Dalek exterminating the residents of Arcadia (The Day of the Doctor, 50th Anniversary Episode). The question is: when you blow up a Gallifreyan building, how should it fragment, so as to look most realistic? Obviously, splitting it into little cubes would look rubbish. The answer (I’m reconstructing the idea here, since James didn’t go into much detail – any errors are mine) is to use a Voronoi partition. You might start by randomly distributing nodes around the virtual building. Then every location in the building sticks with its nearest node, and when the whole thing comes apart those are the lovely irregular chunks that you see. This simple, attractive idea has numerous applications throughout science (and not only if you’re an insane genocidal pepper-pot).

The final talk came from Frances Kirwan, a former president of the LMS (and, as it happens, my algebra lecturer from about 15 years ago). Lest we forgot what we were all doing there, she finished the day with a short account of the history of the LMS. We learnt that the society is the world’s third oldest surviving national mathematical society (props to the Dutch and Moscow Societies), that it was founded in 1865, and that the inaugural president was Augustus de Morgan. It more or less took the reins from the Spitalfields Mathematical Society which had been running for the previous 150 years or so, where maths and merriment cheerfully mixed, and “every member had his pipe, his pot, and his problem”. You can read the lyrics to the Spitalfields society drinking song online, at Songs from the History of Science. It’s all about how alcohol makes you a better scientist. Here’s a taster:

“When Ptolemy, now long ago,

Believed the earth stood still, Sir,

He never would have blundered so,

Had he but drunk his fill, Sir”.

At the early LMS meetings, in contrast, De Morgan commanded that not a drop of liquor should be seen.

As well as banning the booze, another LMS innovation was to admit women from the start (unlike other learned societies I shall resist naming and shaming), with Ada Lovelace being an early member. (Ada has multiple claims to fame: the poet Lord Byron’s daughter, she was the Countess of Lovelace, Augustus De Morgan’s maths student, and as Charles Babbage’s collaborator, was arguably the world’s first computer programmer. What is more it’s her 200th birthday this year. Happy birthday Ada!).

Some other LMS factoids: Mary Cartwright was the first female president; GH Hardy is the only person to have been president twice; another president, Henry Whitehead, claimed to do his best mathematical work while talking to his pigs. Furthermore, and in the spirit of the day, it has always been a firmly outward-looking society, with many of the greatest mathematicians from beyond these shores honorary members: Cantor, Hilbert, Poincaré,…

And on that note, and with Misery De Morgan’s stricture consigned to the dustbin of history, the assembled company wandered out for birthday cake in another room apparently constructed from solid gold. The reception was opened by Pavel Exner, president of the European Mathematical Society, which is also celebrating a major milestone this year: its 25th birthday. (Disclaimer: I’ve recently become the EMS’s publicity officer, so can’t not mention that!)

The final thing to say (apart from the fact that we ate beetroot flavoured meringues at some point, which were strangely delicious) is that this event was only the *launch* of the 150th anniversary celebrations. Events are continuing throughout the year, and throughout the country. You can find more details on the LMS’s events listing page.

Lastly then: Happy Birthday to the LMS! (And to the EMS, to Ada Lovelace, and, belatedly, to Doctor Who.)

]]>The film, which was funded by the Simons Foundation, has contributions from a large number of mathematicians, including Daniel Goldston, Kannan Soundararajan, Andrew Granville, Peter Sarnak, Enrico Bombieri, James Maynard (based at Oxford, who did further work to reduce the prime gap following on from Zhang’s), Nicholas Katz, David Eisenbud, Ken Ribet, and Aperiodihero Terry Tao, as well as Zhang himself.

*Counting From Infinity *features interviews, conversations between groups of mathematicians, and footage of Zhang’s life and workplace. It also includes interviews with his wife, who was as surprised as anyone by her husband’s sudden rise to fame, and his friends and family. Erica Klarreich, mathematician and science writer, narrates the story, and animator Andrea Hale produced 28 animated sequences which are used to support mathematical explanations in the interviews.

George Csicsery, the film’s producer/director, came up with the idea after talking to the director of the MRSI, David Eisenbud. They felt that much of the coverage of the story had focused on the mathematical result, while nobody really knew much about the mysterious character of Zhang himself. The story of his mathematical achievement, along with a flavour of his character and personality, make up the film.

The film’s world premiere took place at the Joint Mathematics Meeting, in San Antonio, Texas on 10th January, and it is hoped that it will be broadcast in the US on PBS. An order form is available on the film’s website to purchase individual copies of the DVD, and licenses for public performance rights. A review of the film has been posted on the Joint Mathematics Meetings blog.

You can watch the trailer here:

The brief drive to the cinema was not Hawking-free time – no, I had time to enjoy my brief Hawking playlist put together for the occasion: Keep talking (Pink Floyd), repeated. Lots. Armed with cinema purchased consumables, your selfless reporter stepped back in time to a young Hawking’s early days at Cambridge.

As I sincerely hope you’ll go and see the film, I’ll refrain from giving away the storyline. However, like all dramatic re-enactments, the film-makers are faced with a genuine problem: how do you sustain drama when the audience knows ~~the boat sinks at the end~~ he beats the famous short prognosis of his life expectancy? With this in mind, I don’t think the following count as spoilers so here we go…

Based on Jane Hawking’s book *Travelling to infinity: My life with Stephen*, The Theory of Everything tells the story of a young Cambridge postgraduate researcher who falls in love with a fellow student, is diagnosed with a life threatening disease and coincidentally changes our understanding of the universe in which we live. The Stephen-Jane relationship is at the centre of this film and we get to see a young Stephen – who had clearly watched *A Beautiful Mind* and improved on Nash’s technique – woo the fair maiden using only the power of physics. The two leads, Eddie Redmayne and Felicity Jones, give impressive turns as they portray a young couple dealing with the difficulties of living with motor neurone disease. Some of the moments of the couple’s domestic life really highlight how much we shouldn’t take the ready availability of modern supportive/assistive technologies for granted. Also, the dalek impression is great.

Hawking’s supervisor, Sciama, features prominently throughout the film and we are treated to repeated scenes with Sciama’s future all-star group of Rees, Ellis, and Carter in addition to our hero. I confess to feeling rather sorry for Ellis in the film because, while the release of 1988’s *A Brief History of Time* is given special mention, the 1973 release of *The Large Scale Structure of Space-Time* is ignored.

Sadly, the science is lacking in this film. Other than a few tropes of scribbling equations on a chalkboard, Penrose lecturing in a weird corridor cum cul-de-sac broom cupboard and what must surely be a candidate for the world’s shortest viva, the film is sadly lacking in moments to satisfy an audience who want to see a dramatic re-enactment of the proof that black-holes radiate. Credit must be given for trying this last item using only a mug of beer! As must endeavouring to explain the difficulty of reconciling quantum mechanics with relativity using only vegetables.

Hawking’s work on singularity theorems lurks in the background for the first half of the film while his famous black hole result occupies this space in the latter half when the writers felt they should again allude to his scientific career. For readers unfamiliar with the singularity theorems of general relativity (shame on you) here’s a potted version: the mathematical framework for this result is differential geometry. More specifically we assume that space and time come together in a manifold structure with a certain way of measuring separation of points using a rank two tensor called a metric $g_{ab}$.

A key object here is the Einstein equation

\[R_{ab} – \frac{1}{2} R g_{ab} = 8\pi T_{ab}\]

where $R_{ab}$ and $R$ are curvature terms known as the *Ricci tensor* and *Ricci scalar* respectively and $T_{ab}$ is the stress energy tensor of the matter component of your spacetime universe. For unit buffs, we’re working in *natural units* where Newton’s constant and the speed of light are set to one. Roughly speaking, Einstein’s equation says “curvature = matter” (i.e. matter bends spacetime, or spacetime bends telling matter how to move).

This metric encodes all the information about the curvature of spacetime and frees up limitless pots of cash for demonstrators to spend on rubber sheets and bowling balls.

A stripped down version of the result Hawking and Penrose proved in 1970 shows that if your spacetime satisfies the Einstein equation and the strong energy condition (a condition which requires the matter in your universe to behave nicely) then your spacetime must contain an incomplete lightlike or causal geodesic curve. Roughly speaking this means that there is a path in spacetime which you can follow which stops abruptly due to it meeting a missing or edge point of the manifold (the spacetime singularity). These singularities have enormous significance: the singularity in the Schwarzschild solution is a key feature of this black hole, and that of the Friedmann-Robertson-Walker solution of a homogeneous and isotropic universe is the “big bang”. For readers seeking a more precise statement of this singularity theorem, and its proof, in the full and glorious language of general relativity then please do check out Hawking and Ellis’ book. However, for a free overview you can read Sean Carroll’s lecture notes (PDF).

However, if the film runs the risk of not sating the alpha-nerd in you then be of good cheer for there is definitely something to celebrate: Redmayne’s portrayal of the developing motor neurone disease is utterly compelling as he successfully transforms himself from gawky postgrad to the Hawking we now recognise.

We’re really being spoiled this year with the Turing and Hawking biopics, not to mention that the filming of Ramanujan’s has already started, so get out there and support your local multiplex.

The scene in which Hawking runs over my foot didn’t make the final reel. I’m hoping for a deleted scene on the DVD.

]]>These awards recognise achievement in the field of scientific and technological advancements related to film-making, and have in the past been awarded to a variety of different advancements, including Dolby Surround Sound, the Xenon Arc lamp, IMAX and even Jim Henson’s animatronic muppet technology.

This year though, finally seeing sense, the Academy’s Technical Achievement award goes to a mathematician. Robert Bridson, who’s worked on CGI-heavy films including Gravity, The Hobbit: The Desolation of Smaug and The Adventures of Tin Tin, has been recognised for his work on “early conceptualization of sparse-tiled voxel data structures and their application to modelling and simulation.”

His work involves simulating complex natural structures, like shifting sand dunes, fire (and how it would behave in zero gravity) and dwarves/hobbits riding down a river in barrels. His software, which uses mathematical models, based on physical equations, to realistically simulate the behaviour of objects in these situations, provides data the animators can use to create beautiful and convincing 3D images.

Unlike the actual Oscars, the Technical Awards don’t have a nomination process – you submit yourself for an interview, and the decision is made ahead of the ceremony – so he already knows he’s definitely won. Although, being a mathematician, he’s basically won at life anyway.

Newfoundland’s first Oscar winner recognized for technical achievements, on CBC News

via London Maths Society on Twitter

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If you enjoyed the magnificent ridiculousness of Matt Parker’s MegaMenger international fractal building project, but would prefer something slightly lower-dimensional, we’ve found the collaborative international fractal-building project for you!

A team led by José L. Rodríguez at the University of Almería, in Spain (who also built a Menger Sponge for MegaMenger) are attempting to build a giant Sierpiński carpet, using green and purple stickers, and an army of ~~unwitting~~ excited school children.

The Carpet, which is made up of square stickers attached to paper in a fractal shape, is being built from contributions sent in by schools and organisations all over the world. For a registration fee of €10, you’ll be sent a pack of stickers and a set of instructions for 64 children (aged between 3 and 99) to participate, and your finished fractal can be posted back to become part of the main fractal in Spain.

The project has just about reached its half-way point – involving more than 16,000 students and 1000 teachers at 256 schools in 30 countries – and they’re looking for more to help out and make the second half. If you think your school class, community group or other conglomeration of enthusiastic humans can help, there’s information below about how to get involved. The project will continue running until June 2015, so there’s plenty of time to get stuck in.