Fertility in Scotland

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A month in, I am starting to produce my own d3 visualisations essentially from scratch. This example is an interactive version of Figure 2.4 from The Registrar General’s Annual Review of Demographic Trends (158th Edition). The user can select from many more years, but as only one is shown at a time clutter is reduced (whilst animation helps to reveal the changing patterns, and tooltips provide clarification and precise data values). Moreover, there is a cohort effect within this data: it is not entirely accurate that fertility of 25 year olds fell from 1973 to 1974, as these are different groups of women. The transition animations therefore instead show the changing experiences of each of these groups as they age, identified by colour coding. For an alternative slice through the data along these lines, this version instead shows fertility at each age for a selected cohort; I am still considering if there is an effective way to combine the two.

  • Source: Vital Events Reference Tables 2012 Table 3.6: Age-specific birth rates, per 1,000 female population, Scotland, 1951 to 2012.
  • Live births only. Excludes births where mother’s age is not stated.
  • Rate for age 15 includes births at younger ages and for age 44 includes births at older ages.
  • The average age is calculated by adding 0.5 years to the mother’s age at her last birthday (e.g. it is assumed that 30-year-old mothers were, on average, aged 30 years and 6 months when they gave birth).
  • The age-specific birth rates for 2002 to 2010 are the revised figures calculated using the rebased population estimates which were published on 17th December 2013.

The Giant 4D Buckyball

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Earlier in the year I participated in the building of a `Giant 4D Buckyball’ sculpture; the first of its kind in the UK, and assembled by a team of twenty during the opening day of the University of Edinburgh’s Innovative Learning Week. I then represented the project at the ASCUS Art and Science Salon as part of TEDxUniversityofEdinburgh at the end of the week. The build was one of several ILW events organised by Julia Collins from the School of Mathematics, and you can read her account here. There was a lot of coverage of this event, from student blogs to Scottish Television – although of varying standards of mathematical literacy! So I’ve put together a series of posts describing the fundamental building block, the `buckyball’:

Whilst the sculpture definite counts as mathematical artwork, it also gave me a chance to indulge some of my other creative interests. As well as the images above, during the construction (and more recent deconstruction) I was able to capture the action through a pair of time-lapse videos (as always, setting to HD is recommended!):

What is a buckyball? Part 3: Fullerenes

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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

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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.

Continue reading ‘What is a Buckyball? Part 2: Projection’ »

What is a Buckyball? Part 1: Planar Graphs

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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.

Continue reading ‘What is a Buckyball? Part 1: Planar Graphs’ »

2014 Joint Mathematics Meetings Art Exhibition

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Chaotic Pendulum

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

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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

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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!