There are many days when I like to focus on something specific but then there are those days when I’m running around developing a bigger picture. To some extent this is mathematical voyeurism. I’m looking but not participating in the process. Doing mathematics is very different than just reading about it.
There are a number of blogs that excite me, for example John Baez’s blog Azimuth (extended from his “This Week’s Finds in Mathematical Physics”). In his posts, Baez usually wanders around several topics connecting them together in a way that excites more investigation. He feels free to skip details if they’re going to muddy the presentation and leaves them to you or to the comments.
I find myself mimicking this process sometimes when I’m ‘looking about’ on the web for connections between interests of mine. I thought I’d share what’s on my mind at the moment (and what’s occupying my tabs).
Kepler-Poinsot polyhedra. This has been on my list for a while. I was reminded of it because I needed star for the top of my holiday tree and I decided on the great stellated dodecahedron. I had remembered hearing that these don’t satisfy Euler’s Formula V-E+F=2. I was confused as to why because the paper models I made did. I was further confounded wondering how they could be regular polyhedra. This revealed an unspoken assumption of mine about polyhedra and reminds me to reread Lakatos.
Penrose Tilings have always interested me and I’d like to write a blog post about them. They’re beautiful, harder to make than you’d think and in some sense were inspired by a problem in logic (the paper of which I’ve just downloaded to my dropbox) by a fellow, Hao Wang, who published it to the Bell System Journal. What a combination! Hao turns out to have been a confident of Godel and I just came across a proof by Professor Juliet Floyd (BU) on Wang and Wittgenstein and their interaction. Neat, I am currently reading the lecture notes from Wittgenstein’s Foundations of Mathematics course.
I came across a comment that there are not just an infinite number of Penrose tilings…there are an uncountably infinite. Oh, why’s that? (open tab!)
I have a side investigation going as to why there is no fullerene with 22 vertices. I discovered this problem on my own while exploring the consequences of the defining two properties of buckyballs (aka fullerenes). Each vertex must have exactly three edges incident upon it (so the resulting planar graph will be cubic) and the faces must be pentagons and hexagons. You can show pretty easily that every buckyball must have twelve pentagonal faces but it’s a bit harder to find out anything about the number of hexagons. Zero hexagons results in dodecahedron. The next one you’re likely to find (I found mine using planar graph doodles) will have two hexagons and this turns out to be one several hydrate crystals that form in gas pipes! After finding examples with 2,3,4,5 etc hexagons I set out to figure out what a buckyball with one hexagon would look like. Try as I might I couldn’t do it. This was wonderful! What a problem! Turns out this was investigated by a fellow named Goldberg about 100 years ago. I still don’t know why it’s the case that you can’t have just one hexagon but I have a tab open in Google books reading “An Atlas of Fullerenes” which may have some nice tools.
Baez has a nice series of articles on platonic solids in three and four dimensions (where there are six, the most that ever occurs in a dimension) which adds to the growing suspicion about n=4 dimensions being the very best (a thought not original to me, but inspired by many others, for example Sir Michael Atiyah).
Baez also wrote a book I’m working through: Gauge Fields, Knots and Gravity. I’ve been sniped lately by a statement that diagonalizable matrices are dense in the general linear group. I can’t explain why, but this seems fascinating. To what degree does this generalize? GL is the space of linear operators on a vector space (I should write something like GL(F,n) where F is the field from which entries in GL are drawn and ‘n’ is the dimension of the matrix: i.e. nxn). So all linear operators on a finite dimensional vector space can be approximating by those that are diagonalizable? Neat! Is there something like a power series representing this? Fun, Fun, Fun!