Wednesday, August 1, 2012

Apocrypha Topologica II

Dear Mathematicelli,

If I may proceed,

The Cretan king Minos seems to have hit upon the psychological trick of bewildering his prisoners into imagining a topological obstruction where in fact the obstacle was only metric: that of the Labyrinth (with intimidating monster to keep you distracted). The solution that let Theseus escape the Labyrinth has found renewed popularity of late, and so is worth elaborating. Ariadne's reasoning might conceivably have run thus:

• If one enters the labyrinth and comes to its center, then one has got (and can get) from there to here

• Walking the same way backwards, one can get there from here

• What is wanted is some means to remember the path one took in getting here, and then to reverse it

• Since I don't know how long a path one might need to follow, the keeper of the memory had better be long!

In other words, we keep more information than just the fact that two points are connected by a path: we remember the whole path between them. Some recenter mathematicians more inclined to vandalism have suggested painting the walls of the labyrinth to remind yourself of where you have been --- which is sufficient data to escape, if you paratroop into the maze under cover of darkness; but the solution proposed by Ariadne and adopted by Theseus makes it easier to tidy-up afterwards: unroll a string as you walk along, and then follow it in reverse, winding the string again!

A point worth noting, which may have escaped Ariadne and Theseus in their flight, is that the re-winding and the follow-backwards phases of the solution can be performed in either order if only the string itself is slippery enough --- this could have been remarkably handy if the labyrinth had been under water, and had they wanted to get fish out of it without the fish seeing them and becoming suspicious. It also points out something special in the total path followed by Theseus through the Labyrinth: it is contractible!

Dungeons we may pass over, as well as the castle and siege warfare. Chain mail is about as old. But Somewhen between the Visigoths and Polyphony was discovered knitting. These, like most mechanical inventions, rely on metric phenomena to make their topology useful, but it is unquestionably their underlying topology that is used. (As a side-note, there's a lovely film-reference back to Ariadne in The Name of the Rose, where Adso returns his knit sweater to a trivial topology in order to escape a non-planar labyrinth! I'm curious how the Vandal painters would fare here!) If you'll forgive a jump-forward, the topologist Poincaré seems to have observed a woman knitting, and independently invented purling on the spot. I don't know what the full topological significance of that is, but the ubiquity of What Is certainly makes itself plain to those who can see it; for which give thanks to God, I think. About such other oddities as the Borromean rings we have remarked elsewhere.

I have no idea what's coming next; but this isn't a bad lot. We've more-or-less covered ±1800. Prof. Cauchemar


TGWWS said...

As a rather avid knitter and purler who can wrap her fingers (if not always her mind) around the topographical puzzles involved, I humbly submit that Adso's reduction of his sweater amounts to vandalism in itself.

Belfry Bat said...

I'm inclined to agree (it also led to chattering teeth).

Salome Ellen said...

On the other hand, I own a bracelet (and my son owns a belt) strategically topologified from parachute cord so that if we are ever in a dire situation like a labyrinth they will return to the simplest equivalent form with ease!

Belfry Bat said...

Well, that's handy indeed, and I'm glad to know you are both so well-provided. I think I've seen things like that on thinkgeek...

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