What is a buckyball? Part 3: Fullerenes

In the previous post we saw how we could project polyhedra into the plane, and use some simple properties about planar graphs to classify all the possible Platonic solids. In this post we’ll finally get to the buckyball, by considering a less restrictive class of polyhedra: the fullerenes.

The Platonic solids were extremely regular: every face had to be the same, with all angles and side lengths the same, and the same number of faces meeting at each vertex. In a Fullerene we allow the faces to be either pentagons or hexagons, but we require exactly three faces to meet at each vertex. We can still take a one-point projection and get a planar graph: it’ll be 3-regular from the vertex condition, and every face has degree five or six.

This turns out to force a seemingly stronger condition on our graphs:

A fullerene has exactly twelve faces of degree five.

To see this, suppose there are P faces of degree five (pentagons) and H faces of degree six (hexagons). Then all-in-all there are F=P+H faces, and we know from Euler’s formula that F = 2 + E -V. By 3-regularity we know 2E=3V. So P+H=F=2+3V/2 – V = 2 + V/2. Further, by handshaking for planar graphs we know 2E = 5P + 6H; so

3V=2E=5P+6H=5(P+H)+H = 5(2+V/2) + H.

This tells us that V=2H+20, so H = V/2 -10. As P + H = 2 +V/2, we conclude P = 2 + V/2 – H = 2+ V/2 – (V/2 -10) = 12, as claimed.

So in a certain degenerate sense we’ve already seen a fullerene – if we have 12 pentagons and no hexagons, with three pentagons meeting at every point, then we have one of our Platonic solids – specifically, the dodecahedron. However, the motivation for studying fullerenes comes from molecular chemistry, where they arise as different allotropes of carbon. But the laws of physics get in the way of having adjacent pentagonal faces when building with carbon – the bonds are not stable. To be a viable fullerene in the chemical sense, our fullerene graph has to have isolated pentagons. That means that none of the five vertices of each of the twelve pentagons can be shared, so a fullerene has to have at least sixty vertices. But, remarkably, we can exhibit a 60 vertex planar graph with twelve pentagonal faces, all other faces hexagonal, three faces meeting at every vertex, and no two pentagons touching:

A sixty vertex fullerene

This, as you may have guessed, is our long-awaited Buckyball! Or, more properly, Buckminsterfullerine. This is the simplest possible isolated pentagon fullerene, but it is still much more complicated than the more familiar allotropes of carbon: graphite and diamond. The theoretical existence of the C60 allotrope had been advanced several times in the 60s and 70s, but was not generally accepted as a realistic possibility by the scientific community. That had to change in the 1980s, when it was first synthesised by Kroto, Curl and Smalley. They named it Buckminsterfullerene due to its resemblence to geodesic dome constructions by the architect Richard Buckminster Fuller:

From my travels: The Montreal Biosphere, the largest of Buckminster Fuller’s Geodesic Dome constructions. These are the namesake for Buckyballs, but are not actually fullerenes!

They also produced C70, showing that C60 was just one instance of a general class, the fullerenes: they received the 1996 Nobel prize in Chemistry for opening up this field of study. It has subsequently been shown that C60 is naturally occuring – it can be found in soot, created by lightning, and has even been identified in clouds of cosmic dust!

What is a Buckyball? Part 2: Projection

A cube in two-point perspective.

How can we represent a 3-dimensional object such a cube in only 2-dimensions, such as on a flat piece of paper? This is the problem of projection, and it inevitably introduces inaccuracies. Different choices of perspective will alter what features survive the projection process. For instance, a perfect cube has all faces square, with corner angles of 90 degrees, and opposite sides of each square are parallel. But in the two point perspective shown only the vertical lines remain parallel; the introduction of vanishing points has distorted the horizontal ones and thus the angles.

Instead of thinking of the cube as a solid object, we can describe it in terms of its vertices (the corners or points) and the edges that join them – that is, as a graph! But in the previous post we were interested in planar graphs, and in our 2-point perspective we have edges crossing. This might seem unavoidable – the `front’ blocking our view of the `back’ – but through a different choice of projection we can get a planar graph of the cube, or indeed any suitably well-behaved solid. This is the key to using graph theory to study those solids.

What is a Buckyball? Part 1: Planar Graphs

Earlier this year I was involved with the construction of the `Giant 4D Buckyball‘, as part of the University of Edinburgh’s Innovative Learning Week. The sculpture was actually of something rather more complicated – the Cantitruncated 600-cell – but buckyballs (in various representations) were a fundamental building block. So, what exactly are they?

Julia’s description gives a good overview of the geometric route to the buckyball (and onwards to the full sculpture). But one of the wonderful things about mathematics is that there are often multiple routes to the same place, with seemingly disparate sub-disciplines actually being the same idea in disguise. So my first exposure to the buckyball came not from thinking about physical geometry, but the rather more abstract world of graph theory. In this post I’ll describe the first step along this alternative route, planar graphs.

2014 Joint Mathematics Meetings Art Exhibition

Some of my work will once again be included in the art exhibition at the Joint Mathematics Meetings– a selection of stills from my video project x<–>t, which I described on my main site here. The image above is a more recent rendering using the same ‘strip photography’ technique: it captures the changing behaviour through time of the chaotic pendulum I wrote about a few years ago!

The benign dictatorship of the London Underground

Earlier today I spotted this video, featuring the stand-up mathematician Matt Parker and all-round interesting person Tom Scott exploring some oddities of the tube:

Matt’s ability to beat Tom around the network depended on local knowledge of hidden shortcuts. You might wonder why these quicker options aren’t indicated by signs, but as they explain in the video, for some the problem is that they’re only quicker because they’re secret. Were the full force of the travelling public to try and use them, the resulting congestion would lead to travel times even worse than the officially sanctioned route.

This is a real-world manifestation of the price of anarchy, an area of maths I find fascinating, as it collides game theory with networks. An obvious starting point is `Pigou’s Example’, which I’ll simplify here slightly.

Suppose there are two choices for a journey across a valley: a long but reliable bypass that loops around it, taking ten minutes to complete the journey; and a direct bridge that suffers badly from congestion, so that journey times grow with the number of users. To be precise, let’s imagine that if x people cross the bridge, they each take x minutes to do so. Now imagine that ten people want to travel across the valley. Which route should they take?

With a bit of thought, we can see that they should all try for the bridge: at worst it only takes ten minutes anyway, but for each person that opts for the bypass you’ll save a minute by sticking with the bridge. Put another way, if you’ve opted for the bridge, you gain nothing by switching to the bypass, regardless of the choices of the other travellers. In game theory terms, everyone using the bridge is a Nash equilibrium – a stable situation in the sense that no-one can improve their journey by changing strategy (from bridge to bypass).

However, it’s not a social optimum: with ten people on the bridge, they each have a journey time of ten minutes each; but if we could force five to take the bypass instead (requiring the same ten minutes) then those on the bridge now have a journey time of five minutes. This means that the population as a whole spends 75 minutes travelling, rather than a hundred. This is a genuine improvement for all concerned, with no-one being asked to take a longer journey than before, and some experiencing a shorter one. (One can devise scenarios where forcing a longer journey on a few makes things better on average, but that seems less fair.)

Unfortunately, allowing selfish travellers to self-organise, this arrangement is not stable: those on the bypass will look enviously at the speedsters on the bridge, and opt for it next time. The problem is choice, or rather that we can’t trust others not to make a choice that gives them an advantage. If our valley crossers could enter into a gentleman’s agreement that half of them would use the bridge on the opposite day to the rest, all would be well. But the non-cooperative approach to game theory takes the pessimistic view that people will agree to anything – particularly if that puts you on the bypass! – then proceed to selfishly use the bridge every day anyway. Under such assumptions, we then need an external authority – be it a government, the mafia code of honor, or Transport for London’s crafty signwriters – to force us to take action that’s for the greater good: even if it’s good for us too.

This sheds some light on a related problem that seems at first counter-intuitive, and has thus become known as Braess’ Paradox: adding more capacity can make everyone’s journey slower! This will occur if the network is already experiencing congestion, and once again is due to the perils of choice. If the new route becomes the obvious way to go, it can create an even greater bottleneck than before as everyone tries to save some time by using it.

As an example, consider our same ten travellers, now trying to get from A to B via two alternatives C and D as follows:

As before, we have a social optimum by sending half of the traffic via C, and the rest via D. In this way, everyone has a journey time of 17 minutes. But as both routes involve a road that is subject to congestion, this is also a Nash equilibrium: if C and D are equally used, then you make your journey worse by swapping to the other. However, a well-meaning city planner can ruin everything by adding a new, fast road between those two:

Now a Nash Equilibrium arises in which everyone takes the route A-C-D-B: for a total travel time of 21 minutes, four worse than before! We see that this is an equilibrium because if you switch to A-D-B your journey time becomes 22 minutes (because everyone is still on D-B); A-C-B also takes 22 minutes since then everyone uses A-C. Of course, with a benign dictator we can ignore the C-D road entirely and have 17-minute journeys for all. But that can’t arise from selfish decisions, as anyone on the A-C-B route will try to trade the 12 minutes of C-B for the 1+x of C-D-B, and similarly A-D-B travellers will spot that A-C-D is more appealing than A-D. So everyone defects to A-C-D-B in the hopes of a 12 minute journey- and they all end up worse off than before…

Such phenomena are not just mathematical curiosities, as they’ve been encountered in real travel networks too. A New York Times article, What if They Closed 42d Street and Nobody Noticed?, discusses Braess’ paradox in reverse- closing a road in a notoriously busy city sounds like a recipe for travel disaster, but by removing an option the flow of traffic actually improved. Similarly travel times in Seoul were reduced after a 6-lane highway was demolished in favour of a park: a double-win for the environment!

Still, as a fairly-frequent user of the Underground, I can’t help but fear I’ll be selfishly adopting some of Matt and Tom’s hacks- at least until everyone else catches on…

Lehmer’s Conjecture for Hermitian Matrices over the Eisenstein and Gaussian Integers

My third paper on the Mahler measure problem has been accepted by the Electronic Journal of Combinatorics, and is available here freely under the E-JC’s open access policy.

This is joint work with Gary Greaves, and completes the proof of Lehmer’s conjecture for matrices with entries from rings of integers of quadratic extensions: a project that has spanned five papers and two PhDs between us!

‘Introduction to Graph Theory’ Lecture Materials

I have now completed a series of introductory lectures on Graph Theory, for 3rd/4th year students at the University of Bristol. A set of written notes plus (less usefully) my slides are mirrored here. One student declared this to be “[the] most interesting course I have taken at Bristol Uni”, so take a look to see how graphs can be applied to everything from organic chemistry to cold war politics!

JMM 2013 Highlights

The San Diego Convention Center, home to the 2013 Joint Mathematics Meetings.

Earlier this month I attended the 2013 Joint Mathematics Meetings, trading freezing Britain for sunny San Diego! This was the 119th annual meeting of the American Mathematical Society, the 96th of the Mathematical Association of America, my third trip to the US for a JMM, and the first at which I’ve given a talk. There are also sessions organised by the Association of Women in Mathematics, the National Association of Mathematicians, the Association of Symbolic Logic, and the Society for Industrial and Applied Mathematics, and with over 6500 people in attendance, no shortage of topics to hear about! One of the things I love about these meetings is that in addition to the sessions that you’re involved in, you have an opportunity to attend events that catch your interest, but you couldn’t normally justify travelling to on their own. Two years ago, I gave a complete rundown of my schedule for my first JMM; this year I thought I’d pull out some highlights.

As a photographer I have a particular love of architectural subjects. But even without a camera in hand, I can spend many a happy hour wandering an unfamiliar city absorbing its particular style (conversely, few things will put me off a place faster than ugly buildings – 60’s brutalists, I’m looking at you!) However, I don’t generally know much about what I’m looking at! So I was pleased to find a trio of consecutive talks in the MAA Session on Mathematics and the Arts of an architectural bent. One of the speakers turned out to be the author of a book – Mathematical Excursions to the World’s Greatest Buildings – which I received for christmas and am very much looking forward to diving into; and another also presented a series of locations used an examples in a course on multicultural mathematics. Perhaps it’s time for me to revive my mathematical tourism efforts! Most intriguing, though, was Frode Rønning’s talk on the Dutch Benedictine monk Dom Hans van der Laan, who sought to find theoretical underpinnings to architecture. His study of the limits in our ability to perceive differences in proportion in the context of 3-dimensional space led him to the plastic number: the real solution to P+1=P3. Much has been made of the alleged aesthetic qualities of its more famous cousin, the golden ratio; that is a solution to &Phi+1=Φ2. Another characterisation of the golden ratio is that it satisfies Φ-1=1/Φ the plastic number is similar in that P-1=1/P4. A number is called morphic if there are positive integers m and n such that x+1 = xm and x-1=x-n; the golden ratio and plastic number are both morphic, and in fact, they are the only morphic numbers! But my own connection to this number is through Mahler measure; in my work, it has only been necessary to study non-reciprocal polynomials, because it is known that the Mahler measure of reciprocal polynomials is bounded below. What by? The plastic number!

`Superfrabjous’, by George Hart – from the art exhibition at the 2013 JMM.

Continuing with the artistic theme, I as usual enjoyed the art exhbition (although I didn’t find the time to contribute a piece of my own). Sculptural work, such as shown to the left, always has immediate appeal for me, but there were some strong printed works this year too. Particularly striking were Daniel Gries’ Fractal Cylinder renders: there are lots more of these here, and the code to generate new examples is also available. As beautiful as they are on-screen, they’re even more impressive as a decently-sized and properly mounted print. I was also amazed to find that Charles Redmond’s Starry Pines was a purely generative work, using something called the Context Free Art language, which I’ll have to look into. Finally, I found myself repeatedly returning to Etienne Saint-Amant’s WABE: Broken Perspective.

I attended a couple of talks of a game-theoretic flavour in the AMS Special Session on Mathematics and Social Interactions. Angela Vierling-Classen examined the issue of gendered division of parenting, demonstrating models in which, even if both parents value equality of effort and it forms a valid Nash equilibrium, most starting states converge to equilibria with unequal assignments. Thus it’s not enough to want equality – you have to commit to it from day one! Her presentation also touched on some of the difficulties in publishing highly mathematical work in sociology journals. The other looked at ‘sacred values’, and how inviolable ties between players in adversarial evolutionary games could lead to new outcomes compared to models in which nothing is sacred. In particular, criminal gangs were able to persist in a game which otherwise tends to a crime-free utopia, as players may be compelled to act against self-interest in support of the gang, or alternatively may benefit from allegiance to such a group by way of protection from other criminals!

Loosely on the subject of sacred values, this session also hosted a talk that has surely the strangest abstract I’ve ever seen at a conference; tragically I couldn’t make it to the event, so feel I shouldn’t dismiss it outright as nonsense, but, well, take a look for yourself. Is anyone brave enough to buy the book?

Tony DeRose discussed some of the challenges of animating Merida – or rather, her hair!

Instead, I opted for the MAA Invited Address, How mathematics has changed Hollywood by Tony DeRose from Pixar. Obviously this was a visual treat, with behind-the-scenes (as it were) footage from Monsters Inc, Finding Nemo and Brave. Originally, handling physical constraints like the effects of gravity or momentum, or preservation of volume, were simply down to the animators; trying to implement real physics into the rendering systems introduces all kinds of difficulties when reality clashes with artistic vision! This causes both entertaining mathematical out-takes, and the need for creative solutions. For instance, Merida’s hair – which Tony describes as practically a character in itself – could not be implemented as individual strands, or it would whip around impossibly violently. To get the right mix of softness under low acceleration, whilst holding shape under high acceleration, a system was devised where curls are loosely coupled to a stiff core. Whilst no-one has hair like this, it paradoxically looks more realistic! Similarly, the robot Wall-E performs manoeuvres that would tear apart a real machine of that size; and no human is built like Mr Incredible! For Pixar, then, realistic physics is just the beginning – not an end. More technical detail was given in the context of a problem known as surface subdivision: a seemingly simple process of splitting and averaging raises interesting questions about its behaviour in the limit, with limit curves ranging from smooth, differentiable shapes to nowhere-smooth fractals: which can nonetheless be easily animated! Impressively, Pixar have made all this work freely available as the open source library opensubdiv, and indeed have a research arm that produces academic publications and technical memos.

DeRose is also clearly passionate about enthusing young people about STEM topics, and spoke a bit about his involvement with Young Makers, which supports Maker Faire-style projects. There has been a long-running Art of Pixar travelling exhibition; plans are apparently well underway for a corresponding Science of Pixar in collaboration with the Museum of Science in Boston. I’ll be looking out for that.

The invited addresses were linked to extended sessions on the same theme, with additional speakers- for De Rose’s talk, this was the MAA Invited Paper Session on Mathematics, Computer Graphics, and Film Production. I only had time for one of the talks, Tim Chartier’s Animating Class with Computer Graphics. But if I ever find myself lecturing linear algebra, I’d be highly tempted to use some of his ideas on illustrating matrix operations through their effects on graphics. Working in Matlab, linear transitions between two source images can be used to create animated tricks like invisibility or teleportation. Splitting images into separate matrices for each colour channel then manipulating them can give Warholesque artworks, or apply effects akin to Instagram or Photobooth.