Kevin Houston

I teach a course on the History of Mathematics but today’s post concerns a TED talk on the Mathematics of History. This very short talk by Jean-Baptiste Michel serves as a follow up to the one given with Erez Lieberman Aiden on What we learned from 5 million books.

The idea is that we can use mathematics to understand history. Unfortunately, the talk is too short to develop a coherent argument and the examples given are not exactly new so I’m not yet convinced that they have something. It should be interesting to see whether this develops though.

Cover of College Mathematics Journal

The issue is, as one might expect, a mixed bag. Of interest is a reprint of Gardener’s original article on Hexaflexagons (these were a childhood favourite for me); in the provocatively titled Martin Gardner’s Mistake Tanya Khovanova discusses the errors in solutions to Tuesday’s child problem (a problem which surfaced at Gathering for Gardner before ricocheting around the internet); Arthur Benjamin has a very simple (I mean that in a good way) article on Squaring, Cubing, and Cube Rooting. There’s also an article by Ian Stewart, who (I suppose it goes without saying) has a new book out.

Have you seen Maths Busking? I reported my experience of being a maths busker last year.
Well, the concept has been entered for a pubic engagement award and, should you wish, you can vote for it here:
http://www.engageawards.org/entry/18

Willmore surface sculpture

For a surface we can define the notion of curvature in a number of ways. For Willmore surfaces we use mean curvature so let’s define that. At a point of a surface take a normal vector. Any non-zero vector in the line perpendicular to the tangent plane to the surface will do. Next take any plane that contains this normal. The intersection of this plane with the surface is a curve. Take the curvature of this curve at the point , call this curvature . (The curvature of a curve at a point is where is the radius of the circle that best approximates the curve at that point.) Next take the plane containing the normal that is at right angles to the first one. We get another curve and hence another curvature, call it . Take the mean of these two curvatures, . This is the mean curvature at ; we will denote it by . We can do this at every point and so we get a function given by . An alternative viewpoint is given on the Wikipedia page.

Now we integrate the square of this function over the whole of the surface. The Willmore energy of a surface , denoted , is the number . That is we integrate over the whole surface the square of the mean curvature. So for every surface we have a number.

If I deform a surface slightly, then the Willmore energy is likely to change, maybe up, maybe down. A surface is a Willmore surface if any small deformation of the surface cause the Willmore energy to increase. I.e., out of all nearby surfaces this surface has minimal Willmore energy. If we take surfaces with the topology of a sphere, for example a perfectly round sphere, an ellipsoid or an egg, then . The minimum occurs for the perfectly round sphere.

What if we move from spheres to tori? The Willmore Conjecture is that for surfaces with the topology of the torus we have . This conjecture has resisted proof for many years and the recently announced proof looks good to the experts (sometimes people make dubious claims for proofs of famous problems, see NP versus P and the Riemman Hypothesis for examples of this). The proof is rather technical and can be found on the arXiv: Willmore Conjecture paper

For pure mathematicians the main interest of Willmore surfaces is that you get rather good looking surfaces. However, Willmore energy is of use in applied mathematics as it can be used to describe bending energy, for example in biological problems on the elasticity of membranes.

The sculpture is inspired by a four-lobed Willmore torus and as such is not a Willmore surface, one can see this clearly due to the flat looking parts of the sculpture. A picture of a four-lobed torus can be found at GeometrieWerkstatt. The plaque near the sculpture makes it clear that the shape is only inspired by a Willmore surface as can bee seen below.

Plaque

A video was made of the public lecture by Franz Pedit. I’ll link to that when it appears on the web.

Tau vs pi (taken from MIT admissions)

As someone associated with the Tauists (a Times editorial described me as a leader of a maverick band of mathematicians promoting tau) I shouldn’t be celebrating Pi Day today. But Tauists and Piists can live in peace as the following link shows (and from which the above picture is taken):

http://mitadmissions.org/blogs/entry/i-have-smashing-news

Now, if you’ll excuse me I need to rush off to give a talk about card cheating in a local school and then go to Durham for a Yorkshire Durham Geometry Day celebrating the life of Tom Willmore.