Monday, November 06, 2006

Bucking the trend

As the title of my dissertation indicates, I've done research on crystals. Most non-mathematicians think that crystals refer to three-dimensional polytopes with evident symmetries; I've had to tell many a suddenly eager listener that what they call polyhedra are not the topic of my work. There is a path between crystals and polytopes, but it makes a lengthy journey through Lie theory and algebraic geometry (sometimes with a tropical substition), which makes even my head spin.

One reason I suspect that so many educated people feels a certain comfort with 3-polytopes is their diagrammatic use in chemistry. I know I have a fondness for my days spent at a lab bench playing with wooden balls and springs, building crude models for carbon rings and water molecules; I'm sure I'm not the only one with this sentiment.

Geodesic domes are a famous family of polytopes, closely related to buckyballs, named for the man responsible for bringing these structures into the public eye. Not only are they good for housing humans, they can also deliver microscopic payloads when its vertices are atoms and its edges are bonds between them; its facets will be pentagons and hexagons, thanks to the chemical nature of carbon. Generally the pentagons in these nano-scale buckyballs (also called fullerenes) must not be adjacent, as is the case for your garden variety soccer ball. However, a team of chemists (including some at my alma mater) have found an ovoid counterexample.

It seems the motivating application is to get heavy molecules such as triterbium nitride to slip into the human body undetected by encasing them in a carbon cage; we therefore have discreet metals thanks to discrete geometry!


Ryan said...

Okay, since I've been wondering and meaning to ask for a few weeks now, and since your dissertation link seems to be dead, I'll go ahead and bite: what exactly (in point of fact, I don't mean exactly at all. I mean generally; that is, in layman's terms) are crystals in the mathematical sense? The idea of polyhedra didn't actually occur to me; I was thinking more along the lines of matrices and data structures...

P. Sternberg said...

Good question! I'll field your accidental question ("what exactly...") here, and get to the intended question in a forthcoming post.

Crystals are colored directed graphs in which the vertices correspond to a basis of a vector space and the edges correspond to operators on that space; the operators form a quantum group, or rather, a quantized universal enveloping algebra.

For examples, I encourage you to play around with this little web-app I put together.

You can also read all about them in the corrected link to my dissertation.