02 April, 2009

Something to show for a day's work

Click the image for a bigger view

One of the problems with mathematics is that it's difficult to find something to show for one's work. I'm not saying that nothing gets done during a day, (though that too can be a problem), but that it's rare to be able to hold up anything which another person, even another mathematician from a different field, could distinguish from gibberish (with a large percentage of Greek letters). Today though, I made some stripy tubes --- I thought they looked pretty good. (The stripes are just for decoration.)

For those who care, and I'm sure you all do, the picture shows two orbits from the Henon-Heiles Hamiltonian system, projected from 4-D to 3-D. The fatter tube making a loop is a periodic orbit, the thinner tube winding around it is a quasi-periodic orbit which is following the surface of an invariant torus. If it kept going it would eventually make a solid looking shape like a slightly bent doughnut. Both orbits give important information ( just don't ask me what) about what is happening in the model. Depending on the sort of computer algorithm that is used to study the model, these two orbits will either look (quasi-) periodic like they do here, or vanish off into the distance.