### In praise of Pick’s theorem

8th September, 2012

Should Pick’s theorem be on the A-level maths syllabus? In a blogpost at the De Morgan Journal, I argue that it should.

### Linear Programming in New Scientist

10th August, 2012

I’ve an article in the current edition of the New Scientist, about linear programming, convex polytopes, and Santos’ recent refutation of the Hirsch conjecture.

It’s available online here (£) or in a newsagent near you, presuming you live somewhere where the New Scientist is sold…

### Pick’s Theorem & Ehrhart Polynomials

1st February, 2012

Pick’s theorem is a simple, beautiful, and usful fact of elementary geometry. It should be much better known than it is! In fact, I have half a mind that it should be on the A-level (high school) syllabus.

Less famous – but equally wonderful – are Ehrhart polynomials, which are what you get when you try to lift Pick’s theorem into higher dimensions. Though geometrically intuitive, they quickly lead into deep mathematical waters. They’re also valued as tools in optimisation problems and in other areas of computer science (I’m told).

This afternoon I gave a – hopefully fairly accessible – talk on these topics. The slides are available here.

(Update: PDF of slides here).

### Webinar playback: some families of polyhedra

16th May, 2011

On Saturday, I gave my first ever webinar, on the topic of “Some families of polyhedra”. And if you don’t know your tetrahemihexahedron from your tridiminished rhombicosidodecahedron, the good news is that the whole thing is available to see and hear here. It’s just over an hour long, but of course one advantage the recorded version has over the live one is the ability to fast forward, pause, and rewind.

It was hosted over at Mathfuture, by Maria Droujkova. My aim in the talk was to give a very brisk overview of how several different families of wonderful, complex shapes all arise from juggling a very small number of simple criteria. I’m separately uploading the slides for my presentation here [pdf]. They are quite rough and ready, without any detailed explanations, or even any pictures – I used Stella for those. But it does sketch the central story (which I also covered in this blogpost). I may spruce them up one day, if I give the same talk again.

I found the whole thing a thoroughly enjoyable experience, and the Elluminate technology worked extremely smoothly. The format allowed me to talk while sharing my whole desktop with the audience, with the optimal result of people being able to hear my voice and watch everything I was doing, without having to endure looking at my face. And we could all do it from the comfort of our living rooms! This is sort of thing the internet was intended for, isn’t it?

28th October, 2010

Circle packing is a classical topic in discrete geometry. As Axel Thue and László Fejes Tóth showed, if you want to fit as many identical circular coins on a table as possible (all sitting side by side, no piling up or overlapping), the best you can achieve is for around 90.7% of the table to be covered. This is done by arranging the coins along a hexagonal lattice.

That was an interesting result, and can be lifted into higher dimensions in the even subtler science of sphere and hypersphere-packing.

That’s fine, but we can pose the same problem, using coins which are not circular. Now here is an interesting question: which shape is the worst packer?

The question is only sensible for convex shapes, and we further assume that the shape is centrally symmetric.

Then the answer is conjectured to be the smoothed octagon, with a maximum packing density of around 90.2%.

The smoothing is done by rounding off each corner with a hyperpola which is tangent to the two meeting sides, and which asymptotically approaches the two sides beyond those.

[Image from Wikipedia]

### Benoît Mandelbrot

18th October, 2010

The internet is currently full of beautiful fractals, a firework display in honour of Benoît Mandelbrot who died yesterday. I was pleased to see that he even made the front page of the BBC News website.

Contrary to what is written there, Mandelbrot did not discover fractal geometry, as he acknowledged. Earlier thinkers such as Lewis Fry Richardson, Pierre Fatou and Gaston Julia have a greater claim (and much earlier thinkers such as Albrecht Dürer might have something to say about it too).

Mandelbrot did however coin the term ‘fractal’ and he helped bring these earlier ideas together into a coherent theory, recognising the potential for powerful applications in many areas of science. He launched these fantastic shapes into popular consciousness through his books the fractal geometry of nature and the (mis)behaviour of markets. His critics might have dismissed fractals as pretty but useless; today, no-one doubts the central role that chaos theory and dynamical systems have in explaining why the universe looks as it does.

[Update: Tom Leinster at the n-Category Café has a very nice technical post about the Mandelbrot set.]

Well, there is only one picture with which to finish off this post: