Friday, December 01, 2006

Advent calendar

Cool pics and movies.
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8 comments:

My Frontier Thesis said...

The Alexander sphere looks like some high-end spark plug. Very cool.

It's also wild (in the good sense) to see how today's scholars are using contemporary technology to illuminate the past. Very neat. Very, very neat.

mft

12/01/2006 1:48 PM
Tecumseh said...

The Alexander horned sphere is a "wild embedding". Not smooth at all -- JJ could not touch it with a 10-ft pole! Here is another view of it.

BTW, James Alexander was one of the founding members of the IAS in Princeton, in 1933. Einstein was another.

12/01/2006 3:32 PM
My Frontier Thesis said...

I like it. Mathematically, I don't know what it means. Philosophically, it serves to symbolize some kind of inward eternal recurrance. The poles extend out but circle around to project in towards one-another — I'm guessing the offshoots go on infinitely? Why not, eh?

12/01/2006 4:26 PM
Tecumseh said...

Yes, it's an infinite process. The horned sphere bound in its interior a topological disk (as does the usual sphere). But what's amazing about it is that the exterior of the sphere is not simply connected: loops around the horns cannot be contracted without cutting across the sphere. This contradicts a well-known conjecture, which is true in all dimensions >3, but not in dimension 3. Neat, huh?

12/01/2006 5:38 PM
My Frontier Thesis said...

Yeah, that is real neat. And it's nice when you hard scientists deploy art to elucidate what otherwise looks like a bunch of gibberish to us humanities types.

This also smacks of that other more-trendy infinite spiral process — I can't think of the technical term offhand, but I can see it in my brain.

Is that all you guys do: come up with formulas to express the infinite (kidding)?

12/01/2006 7:07 PM
My Frontier Thesis said...

There's certain formula that explains whether or not a line is in or outside of a certain spatial area, at least in raster files within Geographic Information Systems. I'm thinking this has something to do with what you just said, but I could be way off (I used to be able to understand statistics quite a bit better when AA and I sat down at some pub and he began drawing on bar napkins; beers helped).

AI, explain this a little further for me when you have a moment:

what's amazing about it is that the exterior of the sphere is not simply connected: loops around the horns cannot be contracted without cutting across the sphere. This contradicts a well-known conjecture, which is true in all dimensions >3, but not in dimension 3.

This last part, what do you mean by dimensions greater-than three?

12/01/2006 7:21 PM
Arelcao Akleos said...

Next thing we know, MFT will be enrolling at a certain university in Boston to better learn from AI ?

12/01/2006 7:25 PM
Tecumseh said...

MFT: What's magical about 3-D? It so happens we live in a 3-D Universe, but why not think about arbitrary dimensions, too? Or, forget about the Universe analogy -- for example, think of the space of all polynomials of degree 17 (or less). Clearly, that has more than 3 dimensions, right? Can you count how many?

12/01/2006 8:15 PM

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