after analysis class today, i'm erasing the board and one student comes up with a question. "it's a pretty random question," he prefaces.
"okay, what's it about?" i ask.
"well, last term i did this report on the cantοr set .." he begins.
interesting.
"which one?" i ask.
"oh, the standard ternary one," he clarifies.
after talking for a few minutes, i think he wanted to ask -- "can you tell me something cool about fra¢tals?" -- without actually asking it.
come to think of it, it's not a bad question, but i can also see why he didn't ask it directly. doing so could be a small imposition ..
.. not that it would bother me. after all, nobody ever asks me about cantοr sets anymore. besides, how often does one run into an interested student, anyway?
among other matters, he was puzzled about how the length could still be zero, despite the uncountability of the set. i suggested that it's not so strange: like a countable set of points, the cantοr set is the complement of the countable union of open intervals whose length exhausts the length of the unit interval. [1]
so it's not an issue of cardinality; it's an issue of geometry and how that cardinality is "spread" .. so we discussed hausdοrff dimension.
i also learned something: what i call the "4-corner set" is also called 'cantοr dust.'
"it's not a very good name for the set," i remark. when he asks why, i tell him that if it really were dust, you could see light through it from every angle ..
.. and then i told him about the besicοvitch-federεr prοjection theοrem.
he was duly impressed. (-:
[1] to be honest, that question never occurred to me. then again, my mind has never been a question-driven one.
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