Saturday, October 30, 2010

Apocrypha Topologica I

Dear Mathematicelli,

I should like today to introduce an apocryphal history of Topology, both as a phenomenon and as a field of mathematical study. It will necessarily be abbreviated, full of fictions, and other more innocent errors --- hence apocryphal. But it should be ordered to the truth so far as illuminating the modern study of topology itself.

Topology was imposed on the visible Creation by God at least as early as the Second Day, when He said
6 ... "Let there be a firmament made amidst the waters: and let it divide the waters from the waters." 7 And God made a firmament, and divided the waters that were under the firmament, from those that were above the firmament, and it was so.
This highlights the first introduction of a disconnected set, a "you can't get there from here" in the world: to reach the waters above from those below you must cross this firmament, whatever it may be. And this notion of separation, whether in the absolute sense of being mutually inaccessible, or the relative sense of inhabiting disjoint neighborhoods has been a puzzle and inspiration for topologists since before we even had the name topology to describe the field.

Some might argue that the temporal ordering of days already introduced an order topology (or the causal partial-order topology we learn from the Special and General Theories of Relativity). I reply that this is missing something of the point, but if you want to write your own paper on the history of Topology that's quite alright.

Other Biblical features of topological interest in denoting separation: the cherubim posted to keep Adam and Eve out of Eden; the Red Sea; the River Jordan; the Rivers Tigris and Euphrates, that separate the Land Amidst the Rivers from surrounding territories (I mean, you can get in and out of that without crossing either, but then you'd have to cross the ~40km line segment between Palu and Hantepe... a narrow road, as geography goes!)

The next great topological discoveries were knots and chains. Windows are a nifty invention too, I suppose, but knots and chains are much more tangible. The way these work, as I'm sure you've experienced, is that various strands of rope or metal, being extended in one direction, have new ways of becoming separated from conditions they might have liked to achieve: while it would take a wall to stop you, and you can walk around a rope quite happily, a string can get stuck by a rope in one perpetual wandering-around! By the by, I don't mean mechanical knots; these are fascinating, but the mechanical distinctions between various rolling hitches are much fuzzier than the topological fact that there are more ways to get tangled in a net than in a string tied in a single loop. Think of the "Gordian Knot" versus an undoubled slipknot bow as you might use on your shoes.

As another aside, there seems some discrepancy between my predecessors in mathematical apocrypha on the one hand vs. closer accounts of Alexander's encounter with Gordy's knot --- what the knot tied up, whether Alexander really sliced it in two or what, etc., are disputed points. One way or another, there were knots and mechanical facts emphasizing topological separations, and it became a point of proverbs through the march of time.

The Gordian Knot brings us to the relatively recent period and setting known as Greek Mythology, and there, for now we will interrupt our History. Next time: strings graduate from topological features to topological tools!

The Math Prof of Your Nightmares


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