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.