56:42 An argument can be made that something like the fractal canopy tree has countably infinite "sub-objects" as its "cardinal number". The primary root being 1, two secondaries being 2 and 3, four tertiaries being 4,5,6, and 7, and so on. But in general, fractals are geometric shapes and it's hard to define "cardinality" for geometric objects (unless you construct some sort of a set out of them). Cantor set is special here because it can be seen as a construction on the Reals, so we are able to inherit a notion of cardinality.
DR. Strogatz, thank you once again for solid introduction to Fractals and the geometry of strange attractors. Canter Sets is also important in fractal theory.
Dr strogatz, I appreciate and i'm very thankful to your great courses. i hope that you give to us those homework's too as a PDF version ,we want to try work on them too. Big salute from Algeria.
A great lecture, I enjoyed every minute! Thanks. ~ A thought, imagine I am a computer, binary notation is all I understand, Base 3,4,5...10...Etc. I have to translate into binary before I comprehend the value. Pi, an irrational number, is 11 + 1/1000 + 1/1000000 + 1/100000000000 ...etc.. I read Cantor's 'Diagonal Argument' of uncountable irrational numbers, and my fuse blew! because the only transformation I could perform in each line the diagonal intersected was to turn a 1>0, & 0 >1. So although there were now two lists:- list "0" & list "1", they were still countable, though my silicon neurons did recognise that no number on list "0", matched any number on list "1". A paradox for me, and I hate it when I blow a fuse! all my registers default to zeros! ~ 00000000...etc.
57:10 (countable) and in general. Consider the sequence n modulo 4, which is a cycle of 0, 1, 2, 3, 0, 1, 2, 3, ... Now consider this sequence of S[i ] as a cycle where each step spawn TWO new cycles. Each cycle is of period 0 (instead of a period 4 for the modulo 4) since, indeed, S[1] can be seen as two S[0] scaled and offset (normalized, would you say). Such structure of cycle of cycles, first, does not have a convergent value when n tends to infinity (neither has n modulo 4), so S[infinity] makes no sense, and, second, has a countable period (of 0, as of 4 in the case of residue modulo) of the "same"(once normalized) element. Thus, a cyclic sequential representation rather than a sequential linear representation seems more appropriate. Sure, using a linear approach rather than a cyclic approach can be elaborate, but it does not seems the best that we can do.
I don't totally understand your illustration, but from what I understood, I would guess it is uncountable. A simple way to see this would be as follows. Each S[i] doubles from S[i-1]. Thus even though the "i's" are countable, the total number of cycles (as you term them) would be Lim_{i-> infinity}2^i. But this has the same cardinality as R which is uncountable. i.e. Lim_{i-> infinity}2^i = 2^|N| = |R|
Have you tried using statistical inference, and extrapolation in relation to fractal patterns and interactions? Mainly the Mandelbrot set? If not I would love to share formulas!!! and see if we can create something new!
How Scaling Exponent, Holder Exponent And Hausdorff dimensions are related? With what logic (Mathematical and intuitive) we can navigate between these concepts.Please comment your expertise on this.
Just deleting the second half of the interval, therein including the right but not the left end point, could qualify as a fractal leading to a countable instead of uncountable set (namely to only the first, left-hand point)?
If two layers actually merged into one, then all you would end up with is a straight line rather than a 2D image resembling the rings of Saturn. Actual mergers would look something like the figure 8 or the sign for the infinity loop. So I don't think having infinite planes with no set starting or stopping point is a cohesive explanation of what is happening at the point of merging.
Although I think he did say when addressing this paradox that perhaps they merely *appear* to merge, which I think is a better explanation. The pastry examples are great because they give you a sense of how something can consist of very thin layers, yet appear to the naked eye as one layer.
