Knots, fishing and the origin of the universe
Most awful post title ever, I know, but maybe I’m still hooked on prof. Schaefer’s conference from two weeks ago.
I went fishing on Sunday and although my luck was better this time (I caught four fish!) I spent a great deal of time tying hooks, untangling my line from others or even from my own. Whenever the knot became too complicated to solve I just cut the line and tied a new hook or floater. At some point I was wishing there was a tool that could help me to untie those nasty knots and make better ones, I would have settled at least for a recipe! That tool/algorithm exists, of course, and it’s called topology; and within this branch there is a whole area devoted to knots (knots theory.) Of course in topology a knot has no ends, that is, they consist of single loops. This is one of those math areas which found little use during the time of their development but that in time became the framework for complex physical theories such as quantum gravity or string theory, these theories account for the wacky title, of course.
Within topology we come accross graph theory too, which is an everyday chemist’s tool although most of us are unaware of it. 2d representations of chemistry structure are graphs, dots joined by edges. If you look at an old text, the 2D representation of norbornane looks like two fused squares with a methylene in the middle of the common edge. This representation is topologically correct but geometrically incorrect. more complicated molecules were just drawn into texts.
In chemistry, although molecular symmetry is described by group theory (and this in turn connects molecular structure to its quantum properties,) many computational chemistry efforts are conducted on topology and graph theory. For lack of a better example think of SciFinder’s molecule builder tool: in it you can draw a molecule (or a piece of it) disregarding everything you know about structural chemistry, hybridization, the VSEPR model, Bent rules, and so on, and still SciFinder can find related structures to your query because all that it reads are labeled points (atoms) and edges (bonds); it understands the graph, not the symmetry arising from geometry, let alone the molecule. Another example of graphs theory applied to chemoinformatics are those softwares that take a IUPAC name and yield the structure (the graph) or viceversa; what the algorithms do is interpreting or generating graphs once a set of rules were provided.
Among graphs there is a particular kind that is called planar graphs; these can be presented in such a way that no edges overlap each other. There is an online game with which I came across a few years ago and I’m still addicted to it, its name is planarity and it can be found here (NSFW). Molecules are planar graphs but their non-overlaping-edges representation is hardly of any help since their chemical properties rely on their 3D structure.
Now, if I was to set my mind to evil, could we think of people as dots or connectors and their relationships/story-lines as edges and ultimately come up with an algorithm for untangling a lie? It would require a lot of data (the edges) if we were to untangle a lie made by others, but what if we want to weave a life of lies? we know what vertexes are around us and up to some extent the edges between connectors close to us; therefore we could draw bogus edges (lies) provided we could come up with a planar graph in which no two bogus edges overlap. That could be a planar graph plotted on top of a non-necessarily planar one. Definitely unethical but nonetheless feasible from my point of view.
Maybe I should just stick to untie knots in my fishing line next Sunday.