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in the previous video we created this
pattern by...
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...bisecting each side but...
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...what if we, say, trisect each side? We can
get a very similar pattern.
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So we trisect the line by cutting it into
three equal parts
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and then cutting each of those into three equal parts...
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...et cetera. We see that...
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...each level is made up...
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...3 1/3 size copies of the previous level
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We can trisect the square and for each
level we get...
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...9 1/9 size copies of the previous level
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And we can trisect the cube, that is by
cutting each side...
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...into three equal parts and, for the cube,
each level's made up of 27...
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...1/27 size copies of the previous level.
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I think you can see the general pattern,
so in general...
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...instead of bisecting or trisecting...
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...we can "m-sec' we can divide...
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...each side into M equal size parts so now
at dimension one...
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...each level is made up of M
1/Msize copies
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...at dimension 2 we get each level
being made...
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...of M squared 1/M-squared size
copies...
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...and similarly at dimension three we get
M-cubed...
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...1/M-cubed size size copies and we
can...
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...keep going and finally we can write down
a general statement...
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...that looks like this: for dimension d...
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...each level is made up of M raised to
the d-th...
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...power number of 1/M raised to
the d-th power sized copies of the previous...
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...level.
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That's our definition...
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...of dimension: that's a perfectly
good definition of dimension
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OK. Well I hope this isn't getting too
abstract for you...
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...but now we can use all this to create a
mathematical definition of dimension...
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...but one warning: the next part
contains a little bit of math.
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In particular it contains some
logarithms so if you're not as...
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...comfortable with logarithms as maybe
you once were...
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...or if you want to review you can watch
the next video...
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...which is a short review of logarithms
otherwise...
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...if you feel very comfortable with
logarithms you can skip to the video...
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after that.