“*Whose* theorem?” you may be thinking. That was certainly the question I was asked by several of my colleagues when I mentioned that I was giving a talk on this subject. The slides from that talk are here [pdf], and this post contains some meta-mathematical thoughts I had while planning it. My main conclusion is that Georg’s Pick’s theorem is a truly wondrous thing, deserving of a much higher level of celebrity than it currently enjoys. In fact, in this post I’m going to go further than that, and argue that PT merits a place on the maths A-level syllabus. I should quickly say that I’m only thinking out loud rather than making a considered policy proposal (so I’m not addressing obvious next questions such as what should be cut from the curriculum to make the necessary space). All the same, I’d be interested in any reaction.

Before I go on I had better tell you what the theorem says: the action takes place on a square grid (or “lattice”) comprising those points on the plane whose x & y coordinates are both whole numbers. Against this background we can draw all manner of geometrical objects simply by connecting dots with straight lines. Any non-self-intersecting loop built in this way will carve out a shape (known as a “lattice polygon”). Of course, this figure might be horribly jagged and irregular, with thousands of edges. Nevertheless, Pick’s theorem will tell us its area in a single, simple formula. All you need to do is count the number of grid points which lie on the shape’s boundary (call that B) and the number which lie fully inside the shape (C). Then the area is A=½B+C-1.

Here are some observations which I’d say make this a great piece of maths:

- It is easy to state.
- It is easy to apply: all you have to do is count dots.
- It is very general, valid not just for triangles and quadrilaterals but highly irregular shapes too.

And yet…

- It is by no means obviously true.

Together (1)-(4) add up to…

- It is genuinely useful: it will very quickly tell you the area of shapes which would be horrible to calculate from first principles.

So far this could be an argument for including Pick’s theorem at GCSE or even primary school level…. but I don’t think that would be a good idea. As we all know, mathematicians deal above all in proofs. So if Pick’s theorem is to be on the syllabus, then its proof had better be too. And I think there is a lot to recommend this as well.

So, before I go further, here’s an rough outline of how a typical proof goes (see my slides for a more detailed sketch, or Cut the Knot for an alternative approach). First step: establish that the result holds for triangles. Second: prove (by induction) that every lattice polygon can be constructed by gluing triangles together. Third and final step: show that when you glue two shapes together, if PT holds for each separately, then it holds for the amalgam. Here are some remarks in praise of this proof:

- It is a good level of difficulty. It is certainly not trivial, at the same time there are no major technical obstacles to overcome.
- Taken as a whole, the proof is reasonably lengthy – I’d argue this is a good thing, as there is real satisfaction in proving something meaty, rigorously and from first principles. At the same time, the summary is short, and the overarching strategy fairly easy to grasp.
- What’s more, it comes naturally in three pieces, each of which is of a manageable size, any one of which could make a reasonable bookwork-type exam question.
- It is a good illustration of an important philosophy: to address a complicated problem (an arbitrary irregular shape) we break it down into simpler things we know how to deal with (triangles).

Here are a couple of other miscellaneous things in PT’s favour:

- It is a comparatively recent discovery. With much of school-level geometry dating back to Euclid, Pick’s theorem (proved in 1899) would be the most modern thing on the maths A-level syllabus. (I’m open to correction here!)
- It is always good to place science in its human context, and PT offers several possibilities for worthwhile cross-disciplinary research. Georg Pick was an Austrian Jew who lived most of his life in Prague, and was eventually murdered by the Nazis. He was also a friend of Albert Einstein, and played an interesting indirect role in the development of General Relativity.

Back with the maths, PT naturally opens up several further lines of enquiries – these are outlined in more detail in my slides. I don’t suggest these should be on the syllabus, but their proximity is certainly a bonus, and they would make excellent topics for project-work or extracurricular reading:

- What happens if we make the grid finer? If we make it fine enough, can any shape with straight edges be turned into a lattice polygon? (No! This leads to topics like constructible numbers, squaring the circle, and transcendental numbers.)
- Does PT generalise to shapes with holes in? (Yes! This leads directly into discussion of topics like simple-connectedness and Euler characteristic.)
- Does it generalise to 3-dimensions? (No! Or not immediately, anyway. The basic counterexamples are
*Reeve tetrahedra*, which can be grasped without too much difficulty. It is illuminating how these shapes eliminate the possibility of*any*version of Pick’s theorem in 3d: the basic idea being that two Reeve tetrahedra can have the same number of boundary and internal points, but different volumes.) - Beyond this, the more enthusiastic student can delve as deeply as they fancy into the beautiful theory of
*Ehrhart polynomials*, which will lead them to further elegant theorems and very quickly to open problems. This is great for showing that maths is not all finished, and might perhaps inspire them to have a go at tackling these questions themselves.

There is another very deep connection between Pick’s Theorem and the “grown-up” geometry: Pick provides an Euler characteristic-like function defined and additive on a certain class of polygons. For a physicist, it is obvious that any such function with reasonably good continuity properties (the latter are perhaps irrelevant in the “discrete” setting of Pick’s Theorem) should be proportional to the area of the triangle. There are of course fundamentally important examples of additive functions of triangles in any geometry which has concepts of “straight” lines and oriented angles: angular excess/deficit of a triangle which, in case of elliptic, Euclidean, hyperbolic planes becomes proportional to the area of the triangle with coefficients 1, 0, -1 (after appropriate rescaling of the area function) — these are Girard’s and Lambert’s theorems. For polygons, the additive function is the sum of oriented turns of the closed walk along the perimeter; the angular excess/deficit formula becomes, of course, a special case of the Gauss-Bonnet formula. Pick’s Theorem is, in effect, a discrete version of Gauss-Bonnet, with curvature concentrated in lattice points.

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Thanks Sasha – a very interesting remark. These connections are perhaps hinted at in the proof I linked to at Cut the Knot which proceeds by considering the polygon’s “angle of visibility” at every lattice point.

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Richard, the proof at Cut the Knot is wonderful. For more discrete versions of Gauss-Bonnet, have a look at Anthony the Ant story (with my apologies for bad html — I did not even knew that the file was still on the Internet). I got the idea from Israel Gelfand. And I strongly recommend this Java applet by John M Sullivan which demonstrates the parallel transport of tangent vectors on the sphere.

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Pingback: Richard Elwes – In praise of Pick’s theorem

Perhaps I have to add that I am not proposing to include Pick’s Theorem to the A Level syllabus.

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Just to mention that in the 1970s Pick’s Theorem played a major role in many Open University summer schools – doubtless due to the brilliant John Mason. He may even still have copies of the handouts!

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