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