Illustrating surgery

Recently, at the conference in honour of Sir Michael Atiyah’s 80th birthday in April, a solution to a long -standing mathematical problem was announced.  The Kervaire invariant problem was a conjecture that stated (roughly!) that in high dimensions there are spaces which cannot be turned into spheres by a process called surgery.  Just like in the medical world, mathematical surgery is about cutting bits out of shapes and then gluing different things back in.  To everyone’s surprise, the conjecture turned out to be (mostly) wrong: in all dimensions apart from 2, 6, 14, 30, 62 and maybe 126, all spaces can be turned into spheres by surgery.  Those spheres might be very bumpy and weird, but they are spheres nevertheless.

Now, this little explanation of mine is very brief and doesn’t contain near enough detail for anyone mildly interested.  Thankfully, the journalist Erica Klarreich (of Nature and New Scientist fame) is about to write an article that will put everyone’s curiosity to rest.  The article will appear on the web, rather than in a paper publication, and will be aimed at telling the story of the Kervaire invariant problem to a general audience.

“What has this got to do with little Haggis?”, I hear you ask.  Well, Erica needs pictures for her article.  She asked one of the mathematicians who worked on the problem, Doug Ravenel…who asked his friend Andrew Ranicki…who asked his graduate student Julia…who asked her sheepy housemate…me!  Seeing as I had learnt a little about the program POV-Ray in order to make my Möbius strip pictures last week, they thought that I would be up to the task of drawing a picture of surgery.

I’ve been playing around with POV-Ray a bit this weekend and it’s not taken me long to come up with some ideas.  To illustrate surgery, I need to take a shape (I chose a torus, or doughnut), cut a bit out of it and then glue a different thing back in.  Here’s a quick picture I made of a torus with a bit cut out of it:

Torus_surgeryNext I need to figure out how to glue circles onto the exposed edges of the torus, and I’m done!  Honestly, this really didn’t take me long and the final version will probably look quite different to this, but I’m very proud of myself so far.  POV-Ray is such a powerful program and my skills are barely scratching the surface.

I shall keep you posted on my future sketches and when/where the article will appear!

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2 responses to this post.

  1. Posted by myhbar on June 30, 2009 at 8:53 pm

    This looks great! I’m looking forward to seeing more surgery illustrations 🙂

    Reply

  2. Posted by ranicki on July 10, 2009 at 3:45 pm

    I look forward to more surgery pictures! Technically, the Kervaire invariant conjecture applies to spaces which are “framed manifolds” of dimension with remainder 2 when divided by 4. For more information see http://en.wikipedia.org/wiki/Kervaire_invariant

    Reply

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