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Are there non-dense fractal curves with Housdoff dimension 2 besides the Mandelbrot set (and derivatives) and a select few julia sets?

By not dense, I mean that for every point on the curve, there are finite disks points arbitrarily close that con't contain the curve.
Curves like the Hilbert curve are dense and would eventually turn "solid" after infinite iterations.

There are probably plenty in 3 or more topological dimensions... but in a 2d plane...
All Julia sets that lie on the mandelbrot set boundary have dimension 2 I think, but I don't think that is proven.
Derivatives of mandelbrot set are huge though, any function az^n + bz^(n-1) + ... + kz + c would have dimension 2 as it is just composed of multibrots.
z = a(e^z - 1) apparently has haussdorf dimension 2 for all a.

sounds like a reasonable thesis at first. but when i look at the few points i can exactly locate on msets boundary line, i have my doubts. do you really think for example the jset at 0.25 / 0 has a two dimensional boundary line? my math skills are insufficient, to make any attempts of a proof, but visual judgement says: no.

the initial question is an interresting one (at least for me), but i am still stuck with the question: what is the second dimension of a non dense line/curve (i like this term)? can't be the second dimension of the plane, i would assume... :hmh:

I just found the paper showing that julia sets on mandelbrot boundary have dimension 2: http://arxiv.org/abs/math/9201282

Yes I find it a bit confusing too, apparently the mandelbrot boundary is dimension 2 but has no area... a confusing combination.

ok. visual judgement fails here. probably because boundary points are undecidable, means the iteration reaches them after an infinite number of iterations.
btw. (off topic, i know) is there any and if so, how many known rational coordinates are located on mset boundary besides 0.25/0 and -2/0 (the only ones, that come to my mind)


nice - a bit confusing  rofl2
for me, as someone coming from a rather philosophical approach, this is totally confusing, and that is actually a too weak expression...

just a little bit of visual judgement: made two julia set shots. one @ 0.25/0 and one @ 0.2501/0 (sorry for quality) both with 2000 iterations (thx to asimes for his html5 renderer)

(http://bilder.toplist100.org/show-image_350-12082821415524.jpg)(http://bilder.toplist100.org/show-image_350-12082821279609.jpg)

so now we still don't know how the julia set @ 0.25/0 looks like. visual judgement has to fail here.

ps: (off topic again) i really have to thank you tglad, for the discovery of the mandelbox. a big part of my significant artisic expressions of the past two years base on this nifty math object. pragmatists like you deliver the food for the minds of dreamers like me - thanks for that!

just to highlight this, funny thing is when modifying parameters of it, you think you understand - ah, yeah its spiral, ah yeah its the distance, a yeah this is the modifier i put on it - BUT it is just beautiful!

Can someone explain how something can have a dimension of 2 but no area? This makes no sense to me either

making sense? who said, that math have to make sense? in opposite, goedel prooved about 80 years ago, that a complete math allways needs parts, that don't make sense. in form of undecidable statements. http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems (http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems)

that's why modern set theory is consistent, but not complete - it is based on axioms.

the mandelbrot set - or better it's boundary - actually is undecidable, so i assume, it doesn't need to make sense.
the mandelbrot set only shows, that math systems of that complexity can create an entire dimension, independent from the plane it can be drawn on (the math plane, not a piece of paper).

(0.25,+/-0.5) is on the main cardioid and is the tangent point to the 1/4 (and 3/4) disk.  Of course, (0,+/-1) is on the boundary, but on a dendrite, so it may not be what you had in mind.  The period 2 disk is centered at -1 and has radius of 1/4, so (-1,+/-0.25) are there as well.  Using Pythagorean triples, one should be able to find an infinity of rational coordinates on the boundary of the period 2 disk.

On the contrary, I think you know exactly what the (0.25/0) Julia set looks like; it looks just like the left image.  It's not clear to me on visual inspection that the boundary has a dimension of 2; it looks quite a bit smaller in dimension to me.  Also, even if it were dimension 2, that shouldn't be confused with the (standard) topological dimensions--that is, a 2D (in the regular, planar sense) fractal that has a fractal dimension of 2 shouldn't be expected to sprout a third topological dimension.  I think fractal dimension is more a statement about the roughness of the shape than how many numbers are needed to specify a location.

great information, thanks - good to be on a forum with so many experts!


i still don't think so
at first tglads paper seems to proove, that its dimension is definitely 2. 2nd when you approach 0.25/0 from the right one it will stay like the right one (with more and more spirals), as long as you increase the iteration count suitable. you can play this game further and further and the image will stay like the right one, but even arbitrary precision can't simulate an infinite iteration count, that is needed to decide wether the jset is coherent or not (that's why it's called undecidable).
so if we'd talk about 0,24(period)9/0, i would agree, but not for 0.25/0.

absolutely agree! the fact, that msets boundary has no area shows, that it can't be topological, 'cause one of two topological dimensions is empty. even the fractal part of the koch curve is not topological, as the koch curve has no area too.
but an entire dimension of factality is still hard to understand, at least for me. but after following this thread, the confusion is not total anymore...