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Tuesday, April 12, 2011

Cantor Set



Georg Cantor derived some of the most mind-blowing insights of any mathematician; indeed many of his contemporaries found his ideas so wackalooney (excuse the technical language ;-)) that they simply dismissed his notions out-of-hand. But today, while his ideas are no less mind-boggling, they are widely accepted.
One of the most fascinating discoveries, though probably less famous than many of his conclusions regarding infinity, was the "Cantor Set." It is commonly depicted by taking a unit length and then deleting the middle third, to leave two bookend pieces, which in turn have their middle third removed, and on and on and on, infinitely. Many of the properties of this simple, final Cantor Set are quite remarkable...

From the mathacademy.com site (see: http://www.mathacademy.com/pr/prime/articles/cantset/ ):

"What remains after infinitely many steps is a remarkable subset of the real numbers called the Cantor set, or “Cantor’s Dust.”
At first glance one may reasonably wonder if there is anything left. After all, the lengths of the intervals we removed all add up to 1, exactly the length of the segment we started with:


 

Yet, remarkably, we can show that there are just as many “points” remaining as there were before we began! This startling fact is only one of the many surprising properties exhibited by the Cantor set."

The Cantor Set also exhibits self-similarity or fractal-like properties throughout.

...and another site here to explore the Cantor Set step-by-step:

http://personal.bgsu.edu/~carother/cantor/Cantor1.html

One often wonders, in the case of Cantor (and certain other mathematicians), did some sort of madness drive him to many of his remarkable, penetrating insights... or, did his unique insights drive him to madness???
Occasionally (and luckily, only occasionally) it seems as if there is a fine line between succeeding at the highest levels of math... and insanity. Chapter 4 (entitled "Mathematics as an Addiction") of Reuben Hersh's recent "Loving and Hating Mathematics" interestingly addresses the link (if any) between math and mental illness. (He concludes "No," but does add: "Still, there is something different about mathematicians compared to, say chemists or geologists or even English professors. It is possible to be 'crazy' -- that is conspicuously eccentric, very odd, even antisocial -- and still hold a job as a math professor.")
...It all reminds me a bit of Steven Wright's oft-cited observation that "There's a fine line between fishing... and standing on the shore looking like an idiot." ;-)

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