A masterpiece of popularisation of mathematics, by Steven Strogatz in the NYT: fingerprinting, the index theorem and works by L. S. Penrose, “Dermatoglyphic topology,” Nature, Vol. 205 (1965), pp. 544–546, and R. Penrose, “The topology of ridge systems,” Annals of Human Genetics, Vol. 42 (1979), pp. 435–444.
“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.
- 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.
I wanted to get in touch to let you know about a new maths website Everything is mathematical, a site that we’ve built to support a brilliant new book collection that explains how maths shapes the world around us. I can send you a PDF of the first book, ‘The Golden Ratio’ to preview, so please let me know, by emailing at
challenges.hothousedevelopments.com >>>at<<< mail.opal-solutions.com,
whether this is of interest.
Presented by Marcus du Sautoy, the 44-part series is aimed to introduce you to a range of mathematical topics in an approachable style, and is aimed at young and old alike – in fact the only requirement for enjoying these books is a curious mind and a thirst for understanding. The series approaches the subject of mathematics in a completely new, fresh and reader-friendly way, and covers a range of topics such as: the Golden Ratio, Prime Numbers, the Fourth Dimension, Fermat’s Enigma, the Secrets of Pi and Chaos Theory.
We’ve also built a website that goes along with the series that will feature news, videos and puzzles from the world of maths, as well as stories about maths innovators and heroes. We’ll be updating this every week, so please check it out.
We will be setting a weekly video maths challenge, the first of which is presented by Marcus du Sautoy, and will post the solution a week later. Visitors to the website will be able to enter a competition every week.
The first challenge video can be found here.
We were wondering if you would like to review the series for us, and we’d like to offer some copies of some books from the series to give away as a competition prize on your site or blog? We would love it if you wrote about the books and the site, as well as checking out the challenges and solutions on the website.
It would be great to hear your thoughts on the site and series, so please drop me a note and let me know what you think.
Tony Gardiner said:
The best mathematicians – from Poincare to Thurston – are sometimes surprisingly sensitive to, and sensible about, educational issues.
The sad news is that Bill Thurston has just died.
He got his hands dirty with mathematics education from at least 1980, and wrote several articles which are quite inspiring. Here is one that is easily to hand for some more refreshing reading than the stuff I usually highlight!
Bill Thurston died on 21 August 2012. Obituaries: The Mathematical Legacy of William Thurston (1946-2012) and Bill Thurston (October 30, 1946 – August 21, 2012)
Still life is the most philosophical genre of traditional figurative painting. It saw some of its most famous manifestations in the Flemish tradition of the XVII century, but it evolved and survived as a meaningful presence through much of XX century art, adopted by avantarde movements such as cubism and dadaism.
The purpose of this essay is to dig into the philosophical meaning behind the still life painting and show how this genre can be regarded as a sophisticated method to present in a pictorial and immediately accessible visual way, reflections upon the evolving notions of space and time, which played a fundamental role in the parallel cultural developments of Western European mathematical and scientic thought, from the XVII century, up to the
A challenge for the artists of today becomes then how to continue this tradition. Is the theme of still life, as it matured and evolved throughout the dramatic developments of XX century art, still a valuable method to represent and reflect upon the notions of spacetime that our current scientic thinking is producing, from the extra dimensions of string theory to the spin foams and spin networks of loop quantum gravity, to noncommutative spaces, or information based emergent gravity? Some may feel that the notions of spacetime contemporary physics and mathematics are dealing with nowadays are too remote from the familiar everyday objects that form the basic jargon of still life paintings. However, much the same could be said about the notions of relativistic spacetime and the bizarre world of quantum mechanics that were trickling down to the collective imagination
in the early XX century, and yet the artists of the avant-garde movements of the time were ready to jump onto concepts such as non-euclidean geometry, higher dimensions, and the like, and bring them into contact with a drastic revision of what it means to “represent” the everyday objects that surround us, and that come to occupy a profoundly altered concept of space and time. So, I believe, the challenge is a valid one, even in the light of the ever more complex landscape of today’s thinking about the concepts of space and time, and I take the occasion to make an open call here to the practicing artists, to take up the challenge and paint a new chapter of the “still life” genre, suitable for the minds of the current century.
Thurston embraced efforts to make mathematics more accessible and enjoyable for students and the general public, especially in later years. In a 1994 article for the Bulletin of the American Mathematical Society, he wrote that the fundamental question for mathematicians should not be, “How do mathematicians prove theorems?” but, “How do mathematicians advance human understanding of mathematics?” He believed that this human understanding was what gave mathematics not only its utility but its beauty, and that mathematicians needed to improve their ability to communicate mathematical ideas rather than just the details of formal proofs.
He worked on projects to increase public understanding of mathematics and saw the mathematical sides of art and design. He co-developed a course called “Geometry and the Imagination” designed to introduce deep geometric concepts to people who did not necessarily have an advanced background in math.
Bill Thurston, the famous topologist and geometer, died on 21 August 2012.
“Cosmologists have drawn on Dr. Thurston’s discoveries in their search for the shape of the universe.
On a more unlikely note, his musings about the possible shapes of the universe inspired the designer Issey Miyake‘s 2010 ready-to-wear collection, a colorful series of draped and asymmetrical forms. The fashion Web site Style.com reported that after the show, the house’s designer and Dr. Thurston “wrapped themselves for the press in a long stretch of red tubing to make the point that something that looks random is actually (according to Thurston) ‘beautiful geometry.’”
Mathematicians can cite many other examples of surprising applications. Could the 19th-century founders of mathematical logic have imagined where Alan Turing would take their new field a hundred years later? With the computer science that Turing founded, the once-abstract field of number theory became a foundation of cryptography. The mathematics of origami have contributed to designing solar sails and automotive airbags. In the 1980s, the topological subfield of knot theory became a powerful tool in particle physics. Symposia have already been held on applications of topology to the design of industrial robots. I’ve even read the statement — but haven’t been able to find the reference again — that every significant pure math idea has an application. We just haven’t discovered some yet.
All this is timely, because in some quarters of neo-mercantilist, managerial academia, some mathematics is considered too pure for the national economy, especially in the UK.
In a famous paper on the uncanny way that math describes reality, the physicist Eugene Wigner concluded:
“The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.”
Bill Thurston was one of the great bestowers of that gift.