23 June 2008

Cloose to the Edge

I had a hypnagogic math moment last night, as we all do from time to time. I drifted off and imagined a geometric sequence s in which each term was exponentially larger than the last. For example, suppose s = n3 for n = 1, 2, 3, .... The 4th term is 81 and the 5th is 243. The numbers x = 90 and y = 700 are "close" in a sense because they're just one term of s apart, yet you might argue they are not close in absolute terms because they differ by an order of magnitude. My foggy brain decided the numbers were not close but cloose (that is, "loosely close," or close only with respect to s). I had the wherewithal to recognize cloose as neologism worthy of Hofstadter, whatever it was I thought it meant, and filed it away for the morning. Now the s I gave above is a simple example, but you can see that with a more extreme definition (say if it relied on factorials or the Ackermann function) cloose will cover far more territory on the number line than close. I started to wonder whether it made sense to define clooseness differently for things other than sequences, like functions or series, but then fell asleep and dreamt about appearing on Iron Chef America (presumably for Battle Toast, Battle Hot Water, or Battle Beer).

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