The city of retroactive mathematics

[Image: Nested salt shakers spotted on Dezeen... Wait a minute – it's Cameron Slayden's diagram of how the Poincaré Conjecture was proved; via Science].

Science used the above diagram about a year ago to illustrate how mathematician Grigori Perelman had come to prove the infamous Poincaré Conjecture.
I won't get into specifics – after all, I don't understand them (Perelman used "Ricci flow," or "a procedure for transforming irregular spaces into uniform ones," in order to prove that something or other will always be a hypersphere...).
Nonetheless, in the above image we see how "negatively curved regions (blue) must expand while positively curved regions (red) contract. Over time, the original dumbbell-shaped surface evolves into a sphere."
This proves something.
But the above image could just as easily be an architectural diagram.
And so I imagined that a mathematician might show up in a distant city someday, perhaps in the irradiated marshlands of Belarus, only to realize that all the buildings around her are actually 3D illustrations of unsolved geometrical conjectures – only people are living inside them, raising kids and doing laundry. Eating bagels and writing blogs, surrounded by zeta landscapes in glass and brick variations on the Riemann Hypothesis.
That's not a corridor at all but a glimpse of elliptic curve cryptography – it's non-commutative geometry in concrete.
The city is built algebra.
An odd mix of ornamental numerologies in the city's high street adds up to nothing less than a new way to predict prime number sequences.
Our mathematician thus rushes back to her hotel room, frantically writing numbers and making sketches of buildings on the bedside stationery, looking outside as cars pass by, the sun going down, exaggerated shadows of pedestrians looming up and down the facades of cinemas, and she senses proximity to something wonderful... Something hidden in mathematics that she might soon solve.

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5 Comments:

Blogger Murnen said...

Sort of like Ron Eglash hunting for fractals in African buildings [video].

December 16, 2007 2:30 PM  
Blogger HomerTheBrave said...

I'd like to show you around some of the architecture of Second Life.... :-)

December 16, 2007 3:01 PM  
Blogger HomerTheBrave said...

Yikes... I should have linked here.

December 16, 2007 5:05 PM  
Blogger Geoff Manaugh said...

Thanks, Homer - I actually link to the site already in my "Architecture" list. Some great stuff there, even if I do think that Second Life is insanely over-rated.

And thanks for the Ron Eglash video!

December 16, 2007 9:44 PM  
Blogger mb said...

man, write a novel. your imagination is too active and your style too readable to NOT do it.

December 17, 2007 10:04 PM  

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