A while ago I became interested in `captured lightning‘ Lichtenberg figures, but without access to megavolt-scale physics gear, I wondered if I could simulate them in software instead. I was reminded of this when I had my JMM art exhibition entry printed on glass, as this would give me something a bit closer to the acrylic blocks. Some initially vague google searches eventually lead me to Diffusion-limited aggregation, a process that generates trees somewhere between ferns and lightning bolts. I set about implementing this in processing, and you can play around with a small version of it here:

Controls: To set things in motion (or pause them), press SPACE. To restart, press R. You can cycle through various colour options with G, and toggle rendering of the random walks with W. The screen is redrawn after a fixed number of points have been tested, which can be decreased with Z or increased with X: if your computer is powerful enough, it can cope with updating the screen more often; even if it can’t, you can decrease this to 1 to watch a step-by-step construction.

How does it work? There is an initial core disc of points which are included in the structure, and its horizon – the distance of the furthest point from the centre – is tracked. New points are launched from a ‘birth’ circle with radius a fixed multiple (until the edge of the screen gets in the way) of the horizon; further out, there is a ‘killing’ circle. Once launched, points take steps in random directions of size large enough to move them within the horizon- although of course they may go the wrong way! If they ever cross the killing circle they are abandoned, and a new point launched; if they move within the horizon, they switch to taking steps of unit distance instead. If at any stage they bump into a point already in the structure, they stick to it: they stop moving, become part of the structure (possibly increasing its horizon, and thus pushing out the birth/killing circles) and a new point is launched. There’s a fixed maximum radius for the horizon, and once this is reached, no more points are launched.

I’d recently ordered Ben Fry‘s Visualizing Data and started reading it this weekend; just a few pages in I learnt how to import data to processing and a project was born… Since New Orleans I’ve been increasingly interested in mathematical art, and whether in particular I could create something interactive. Here’s what I’ve come up with after a couple of rainy afternoons:

So what is it? Each point represents a number up to 10,000, arranged on an Archimedean spiral, and coloured depending on its smoothness: a smooth number is one with only small prime factors. More precisely, N is B-smooth if the largest prime dividing N is at most B (so 2-smooth numbers are powers of 2; 3-smooth numbers are multiples of 2 and/or 3 only; any number shown will obviously be at worst 10,000-smooth). You can adjust the smoothness bound with the slider: in ‘gradient’ mode the brighter a point, the smoother it is; whereas in ‘threshold’ mode a point is simply plotted or not depending on whether it passes the smoothness test (the mode can be toggled by pressing space).

The least smooth numbers are the primes, and it was thinking about prime spirals that lead me in this direction: the Ulam spiral is one of the first examples of computer-aided mathematics visualisation, and I’ve taken the circular layout from its close relative, the Sacks spiral. In fact, my original plan was to use the number of prime divisors, rather than smoothness, for deciding when to plot points, with the Sacks spiral as a special case. But the pictures for larger bounds weren’t particularly interesting- 10,000 just isn’t big enough to allow much of a range of behaviour. So I switched to smoothness, and whilst that means you can’t identify the primes directly, sometimes they’re conspicuous by their absence: in the Sacks spiral there are curves with an unusually high concentration of primes, and in the smoothness spiral there are similarly ‘missing’ curves. There seem to be lots of other features too- if you’d like a closer look, here’s an enormous render of the 101-smooth numbers shown above, created using processing’s PDF mode.

Today’s post by Haggis the Sheep demonstrates how crochet can help understand some topologically-interesting surfaces, so I felt I should mention a similar piece of fibre art I encountered this weekend. The object on the left is a Lorenz Manifold made out of over 25,000 stitches (plus three wires), and took Bristol mathematician Hinke Osinga 85 hours to assemble. Osinga (along with Bernd Krauskopf) had been experimenting with computer visualisation of the manifold, and developed an algorithm which ‘grew’ the image from a small disc, adding layers with additional or fewer points at each step to specify the local features of the surface. This approach conveniently works just as well for wool as pixels – each row of a crochet pattern differs from the last by increasing or decreasing the number of stitches to alter the shape.

But what does it actually represent? Lorenz was one of the founders of chaos theory, discovering the ‘butterfly effect’, the way in which seemingly small changes to a system such as the weather could escalate into major differences in behaviour. The Lorenz oscillator is a set of rules for evolving the position of a point in 3-dimensional space which exhibits this chaotic nature: starting points generally find their way to the Lorentz attractor, a complex pattern that never repeats itself. However, points on the Lorenz manifold manage to avoid this trap, and instead settle at the origin, the ‘central’ point of space.

Some of Hinke and Krauskopf’s computer visualisations, their crochet of the manifold, and a rendition in steel by Benjamin Storch can be viewed for the rest of the month at The Bristol gallery, which can found down by the harbourside. They’re there as part of one of the Changing Perspectives exhibitions, which also includes work from my department’s invaluable Chrystal Cherniwchan: the photographic project Exploring the Valley, and the Mathematical Ethnographies films. As well as maths, there are exhibits inspired by scientific topics from shifting glaciers to high voltage electricity, so if you’re local, why not take a look in person? If not, well, you can get a taste from the links above, or if you’re feeling brave, grab the instructions to crochet your own Lorenz manifold!

AMS Contributed Paper Sessions: Combinatorics and Graph Theory, I
Y. Kim, Cycle-saturated graphs with minimum number of edges.
D. Pragel, Algebraic and Graph-Theoretic Properties of the Box Product of Two Paths.
A. Barghi, Firefighting on Random Geometric Graphs.
J. Ellis-Monaghan, Ribbon Graphs and Twisted Duality.
J. Fierson, Some graph theoretical results for the task mapping problem for parallel computers.
S. Raval, Complex Contagions on Graph Dynamical Systems.

Although I’m officially a number theorist (honest, it says so right there in the sidebar!) much of my thesis topic and subsequent work has been more concerned with graphs, and there was plenty of interest on offer here.

From a research perspective the box product construction particularly caught my attention: in the presented work, products of paths were considered, which yield grids that can be sliced vertically into copies of one factor, and horizontally into the other. This carries over into some nice structural properties of the adjacency matrix, and they were able to come up with a particularly neat characterisation of its determinant based on the length of the paths. The obvious next step would be to try something more complicated than paths, and I wonder if some candidates from my own studies of cyclotomic graphs might be suitable.

On the other hand, the firefighting problem is something I’d have no idea how to solve, but it seems like I could make an undergrad project out of it – or a web game! Given a graph, some vertices are specified as being on fire. Each round, firefighters may be placed at any vertices that aren’t on fire, then the fire spreads to any neighbouring vertices that haven’t been protected in this way. On an infinite graph, the question is whether such a fire can be contained or could burn indefinitely.

AMS Colloquium Lectures
A. Lubotzky, Expander graphs in pure and applied mathematics, I.

More in the graph-theory line: unfortunately I was only able to attend this, the first of a series of three talks by Alex Lubotzky on the subject, but at least I now know what expander graphs are and why I might care! The original motivation was practical: in designing a communications network (be it mobile phones or multicore processors) you want short routes between nodes for speed and reliability, but as few connections between nodes as possible to minimise cost. Expander graphs are those which (remarkably) manage to balance these opposing properties, but they also find application in a surprising range of abstract mathematical topics.

MAA Contributed Paper Sessions: Cryptology for Undergraduates
D. Cabarcas, Algebraic Cryptanalysis as a tool for teaching Cryptology.
D. Spickler, Cryptography Tools: A Teaching Tool for the Investigation of Classical Cryptography and Cryptanalysis. (Cryptography Tools)
C. Beaver, Group Signature Schemes: How to share a secret without telling it.
A. Li, Cryptography, a Great Topic for Undergraduate Mathematics Courses.
T. Feil, A Cryptology Course for the Non-Mathematician.
R. Talbert, A Brief Fly-Through of Cryptology for First-Semester Students using Active Learning and Common Technology.
R. Beezer, A first-year seminar in cryptology. (slides).
S. Boersma, Student Codebooks: An in-depth writing assignment.
K. Smith, Codes in History, the Arts, and Literature.
K. Meyer, Making Cryptography Come Alive.
M. May, Using Cryptography to Show Students that Math is Everywhere.

This session was one of my reasons for making the long trip, and was definitely worth it. Based on the enthuiasm of the speakers, the feedback they’ve received from their students, and the sheer number of people who turned up for this session, I think it’s safe to say that cryptography is definitely worth offering in the undergraduate syllabus. The American undergraduate experience is rather different to the English one I had, or the Scottish one I tutored for, and in particular there’s a need for mathematics courses for non-mathematics students. Several speakers were able to provide a cryptology course for such an audience, as the mathematical prerequisites can be made fairly modest and supplemented by the history of the subject, or its relevance today to topics like privacy and security online. One even managed to assess it through written projects, despite the protests of the more mathematically inclined students! The consensus seems to be that if you’re going to teach such a course, your starting point should be Cryptography by Trappe and Washington, and -despite my love of the discrete log problem – it’s probably best to stick to symmetric crypto and a bit of RSA. Various speakers had developed software to remove some of the computational grind (such as crypto tools, linked above), but the coolest contribution was probably instructions (PDF) on how to make an Enigma machine out of a pringles can!

AMS-SIAM Special Session on Mathematics of Computation: Algebra and Number Theory, I & II
M. O’Sullivan, The sum-product algorithm for binary codes having check nodes of degree two.
D. Harm, Complexity of the Graph Isomorphism Problem.
N. Boston, Combining Group Theory and Number Theory Computations.
M. Jacobson, Class Group and Regulator Computation in Quadratic Fields.
A. Sutherland, Genus 1 point counting in quadratic space and essentially quartic time.
A. Silverberg, Finding the rational points on a certain genus 12 curve.
R. Scheidler, Efficient Divisor Reduction on Hyperelliptic Curves.
D. Moulton, Finding small sets whose subset sums include a given set.
J. Silverman, Lehmer’s Conjecture and points on elliptic curves that are congruent to torsion points.
C. Smyth, Minimal polynomials of algebraic numbers with rational parameters.
K. Hare, Pisot and Salem polynomials dividing Newman polynomials.

This session was the other reason for my attendance – Mahler measure is quite a niche topic, so with two talks on the agenda here I felt I should turn up, but they weren’t the only draw. If you dig deep enough in this blog, you’ll find that I spent the start of my PhD thinking about point counting problems and hyperelliptic curve arithmetic, which both featured here. A particular highlight was Andrew Sutherland’s talk, which presented improvements to SEA which have led to a substantially larger record for point counting on elliptic curves.

MAA Session on New and Continuing Connections between Math and the Arts, I

Fractal Tree No. 3 by R. Fathauer

`Mathematical Art’ usually conjures up images of fractals, but there’s a lot more to it than that and several themes emerged from this session and the attached exhibition.

The Alhambra in Spain gets another bump up my list of potential mathematical tourism sites: although it seems that debate continues over whether all seventeen wallpaper tilings can be found there, it seems to have the best (and best known) collection. But other talks mentioned their appearance in everything from Tibetan sand mandalas to Norwegian rosemaling. I discovered that there’s such a thing as ethnomathematics, which aims to go beyond cataloguing such connections between mathematics and culture and attempt to explain them.

Also finding its way to the travel list is the Museum of Mathematics, although I’ll have to wait a bit as it doesn’t exist yet… hopefully it’ll open in 2012. Rather than focus on dry historical exhibits, their vision is for installation pieces like a race circuit for square-wheel tricycles, large geometric sculptures, and interactive digital art. The target audience might be schoolkids, but I suspect I’d walk around with a big smile on my face too!

Another exciting project I was oblivious to is the Bridges series of conferences on connections between maths and art: these combine invited talks and papers (with peer-reviewed proceedings) with hands-on activities, an art exhibition, film screenings, all in a location chosen to inspire! The next one is at the University of Coimbra, Portugal, in July.

AMS Special Session on Self-Organization in Human, Biological, and Artificial Systems, II
R. Niemeyer, Graphs, Dynamical Systems, Fractals: A Heuristic Framework for Modeling the Structure and Dynamics of Complex Interactions Across Multiple levels of Analysis.
L. Smith, An Agent-Based Approach to Modeling Gang Rivalries.

Although it’s a long way from my research activities, emergent systems is one of the topics that first steered me towards mathematics and computer science. So with a spare hour to fill, I decided to indulge an old interest by sampling a couple of talks from this session. Laura Smith’s was particularly intriguing: based partly on geographic constraints, her team of mathematicians and criminologists was able to build a model of the (violent) interactions of LA’s numerous gangs. The hope is that such a model would be accurate enough to predict where best to focus police efforts to reduce conflict, although because I’ve been watching too much Castle lately I found myself dreaming up scenarios of mathematically-savvy gang bosses using optimization theory to maximise their territory instead…

MAA Invited Addresses
M. Matchett Wood, Binary quadratic forms: From Gauss to algebraic geometry
R. Bell, Lessons from the Netflix Prize

Melanie Matchett Wood’s talk was in the rare category of those from which I felt I’d gained some insight into abstract algebra. Whilst modern terminology is probably the best working language, I think there’s a lot to be said for tracing the historical roots of a topic, rather than simply overwriting it with what can be opaque notation. Gauss may have essentially being doing group theory, but he didn’t know that, and the motivation and inspiration is perhaps easier to understand without that abstraction.

The Netflix prize offered US$1million for a 10% improvement to their film recommendation algorithm. That might seem a lot easier than other million dollar prize problems, compared to the ferociously difficult millenium problems, for instance. But it also meant a lot more viable competition, especially as when Robert Bell’s team hit the required 10%, they didn’t simply win but triggered a 30 day endgame which saw alliances form and the leadership change hands repeatedly: in the end, “BellKor’s Pragmatic Chaos” triumphed by just a fraction of a percent and a twenty minute earlier submission time than their closest rivals. His talk captured this drama, entertained with some of the sub-problems encountered (Why is it so hard to tell who’ll like Napoleon Dynamite? What happens if a user gets a girlfriend? and just who has the time to rate 99% of the netflix database?), and also described plenty of the mathematics behind their algorithm. There’s a documentary film in there somewhere…

AMS-MAA-SIAM Gerald and Judith Porter Public Lecture and Special Film Presentation
R. Lang, From flapping birds to space telescopes: The mathematics of origami.
Film: Between the Folds.

…which leads me neatly to the final events. Robert Lang seems to have been central to the revolution in Origami caused by the mathematisation of the discipline. The ability to algorithmically create folding patterns of stick-figure skeletons has pushed forward the level of detail that can be achieved with a single sheet; but as with other media, the possibility of greater realism has led also to a reaction in the form of abstract works, from mathematically-inspired patterns to ‘single crease’ sculptures. But it’s not just about art: origami folding lends itself to the design of airbags and heart stents, or to the problem of getting large structures into space.

All of which appears in the film Between the Folds, that I’m going to recommend regardless of the contents of your netflix queue. Here’s the trailer:

So all in all I had an excellent time at the JMM; I’m certainly planning to attend the next one, which it seems will be held in Boston even earlier in January. Hopefully I’ll be able to give a talk too- the question is, in which session?

The default is as Vi illustrated, where each dot has one of three colours. At no point do I bother to calculate values in Pascal’s triangle to work out a colour: instead, I implemented the ‘group law’ – the rule that combines pairs of dots into a new dot – and used that to find each new dot based on its parents above it. This underlying group structure is the same as Vi’s drawing, except the colours are different: you could write down a rule for changing my colours into hers and applying it consistently would recover her sketch. That probably doesn’t seem too surprising: what’s more remarkable is that any group of three elements is, in the most abstract sense, the same as these: which is why the original approach, of taking remainders modulo 3, also corresponds perfectly to manipulating coloured dots directly instead of numbers.

There’s nothing special about having three colours, so I’ve included the option to have anything from two to eight: no reason to stop there, except I was struggling to come up with non-garish palettes! These are all examples of cyclic groups: if you keep applying the group law you cycle through each colour before coming back to the first one and repeating. But not all groups are cyclic, so I wondered what would happen if I picked a group that wasn’t. For instance, with four colours you can write down two equally valid yet inequivalent group laws: the cycle on four colours, or the Klein four-group¹ which combines two copies of the cycle on two elements. But when I tried to plot that, I just got a two-coloured diagram, not four: I was trapped in the smaller subgroup. Similarly, you can make a group of six elements by considering symmetries of an equilateral triangle: from that I could recover either the two or three colour plots.

In fact, this limitation will always apply when the triangle is grown from a single seed, and all the surrounding dots are assumed to be the ‘zero’ or identity colour. Each seed will generate a cycle of some length, and the picture you get will only include colours from that cycle, not the entire palette: except for the happy cases where you picked a cyclic group and a generator as the seed². I tried lopping off the top dot, so that the ‘triangle’ could be grown from the interaction of two elements, but the results tended to be ugly: you lose symmetry, which is always a disappointment when playing with groups!

¹ Which uncoincidentally happens to be the name of this group of mathemusicians!
² Exercise for undergrads: prove this

The pendulum could be considered the symbol of predictability and regularity, serving as the essential component of timekeeping devices for hundreds of years. Yet simple modifications yield devices that instead exhibit chaotic behaviour, the typical example being the double pendulum, where one pendulum is attached to the end of another. The chaotic pendulum at St Mary’s is believed to be unique in design: water is continually pumped into the crossbeam, causing it to tip one way or the other to let it flow out again; but it’s not generally possible to predict which way it’ll go next!

It’s important to note that this isn’t because the motion is random: there are rules governing its behaviour, and if you could return the chaotic pendulum to exactly the same state as it was at the start of the video, then it’d do exactly the same again. However, in chaotic systems tiny changes can be amplified into disproportionate effects, whilst large changes might have no impact. This leads to what is known as sensitive dependence on initial conditions. Think of catching a train, where if you’re a second early then you arrive at your destination on time, but if you’re a second late you’ll have to wait for the next one, so your two second delay grows into half an hour. On the other hand, whether you arrive at the station ten minutes or ten seconds early makes no difference to when you arrive at the other end. Thus your arrival time at the destination depends on your arrival time at the station in a highly non-linear way.

So if your attempt to reset the chaotic pendulum were even slightly flawed, then the video would be useless for making predictions. Similarly, even if you could build a completely accurate mathematical model, any readings taken from the real pendulum would contain errors, and so after a while the predictions of the model would diverge from the real behaviour. Fortunately for a pure mathematician like myself it’s more interesting to build that model, capturing the essence of all chaotic pendulums, than to be able to predict the activities of a single example!

The Clifton scientific trust has this to say about the lessons that can be drawn from the chaotic pendulum:

Some people look to science for certainties on which to base their lives. Increasingly we realise our knowledge can never provide certainty, even for this simple machine. The world is a more wonderful and a more surprising place than we could have imagined.

I’d agree with their conclusion, but not for their reasons… A truly random world, for example, would be constantly surprising. What’s wonderful about ours is that seemingly complicated phenomena can be described by simple rules – complexity can emerge from the interaction of otherwise easy-to-understand parts, rather than requiring some irreducibly complex explanation (such as a creator). Whilst the existence of chaos might restrict our ability to predict, it shouldn’t discourage us from trying to explain.

The Mathematical Bridge

As I wrote last time about a Mathematical River, it seems sensible to start with this ‘Mathematical Bridge’! Popular myth asserts that the original design was so clever that it was held together simply by gravity, a feat attributed to anyone from undergraduate students to Isaac Newton, depending on who’s telling the tale. But (the story goes) when perplexed Fellows dismantled the bridge in an attempt to understand its construction, they were unable to put it back together again without resorting to using the nuts and bolts which can be seen today. Sadly, the bolts have always been needed, although as first built (in 1749) they were invisible to those crossing the bridge – and it is at least true that it didn’t require any nails!

Trinity College, Cambridge

It’s unlikely, then, that Newton had a hand in the mathematical bridge, since he died twenty years before it’s construction, but that won’t dent his mathematical legacy too much. Trinity College has always had a strong mathematical reputation, admitting almost a fifth of undergraduates in the subject (of 31 colleges in total), with Newton easily their most famous predecessor. He measured the speed of sound in one of the courtyards; the library holds his annotated first edition of Principia Mathematica (and a lock of his hair!); and the tree outside the main gate is claimed to be a descendant of his famous apple tree.

Centre for Mathematical Sciences

Until the mid-20th century, there was no central mathematics department, and instead work was done in the colleges. Today, however, research activity is concentrated at the Centre for Mathematical Sciences. It’s often joked that to find the mathematics department at a university you should look for the ugliest building, but the CMS, which opened in 2003, is a world away from 60s brutalist concrete. The ‘low energy’ design exploits natural ventilation to control the internal environment, and it’s green in a more literal sense too, with the main core having a grass roof. But it’s clearly a building meant for mathematicians, as the first things the architects were asked to design were the coffee rooms. Like the city it resides in, it’s a wonderful place to contemplate mathematics, and I hope I’ll be back again next year.

(First published on the SoSauce travel blog.)

The Union Canal, near Falkirk

In 1822 the Union Canal opened, providing (with the Forth and Clyde Canal) a link between Scotland’s two major cities, Edinburgh and Glasgow. It became known locally as ‘the mathematical river’- by following a natural contour line, the Union Canal maintained a fixed height for its 31 mile course from Falkirk to Edinburgh, removing the need for time-consuming locks. Nor is this its only mathematical claim to fame- in 1834, the scientist John Scott Russell discovered what are now known as soliton waves whilst experimenting on the canal:

“I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped—not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation.”

As Scott Russell described, such waves are unusual in that they can travel long distances whilst preserving their shape, rather than toppling over or simply flattening out with time. Named in his honour in 1995, The Scott Russell Aqueduct carries the Union Canal over the Edinburgh city bypass, yet the thousands of people who drive underneath it every day have probably never heard of his work- many have probably not even heard of the canal! Yet as well as having added to our understanding of physics, electronics and biology, soliton waves are of great practical importance today for their role in long distance communication with fibre-optics.

It seems that a waterside stroll is often of benefit to the advance of mathematics. Nine years after Scott Russell’s discovery – and several hundred miles away, in Dublin – the Irish mathematician Sir William Rowan Hamilton had a ‘flash of genius’ whilst walking along the Royal Canal. He had realized the equations for the quaternion group and, fearful that he might forget them just as suddenly, promptly carved them into the nearby Broom bridge. The original carving did not survive, but there is now a stone plaque in its place, which has been described as “the least visited tourist attraction in Dublin.”

Despite its clever design, the Union Canal’s importance would be short-lived: within twenty years, trains had overtaken barges as the fastest way to travel. The banks became overgrown and the canal filled with rubbish, and the decline continued after its eventual closure in 1965, as the construction of housing and the M8 motorway caused sections to be cut or filled in. Fortunately, an £85-million project – the millennium link – came to the rescue. The two canals had originally been joined by a series of 11 locks in Falkirk, but as these had not survived, a more spectacular solution was found- the Falkirk Wheel.

This engineering marvel is the world’s only rotating boat lift, capable of transferring boats between the two waterways in minutes – and, thanks to physics, using only as much energy to do so as boiling 8 kettles! The wheel opened in 2002, providing the final piece to restore the link between the two cities, providing ideal opportunities for walking, cycling or boating. I can’t wait to explore it further in the spring!

(First published on the SoSauce travel blog.)