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Author Topic: Any interesting 1D fractals around?  (Read 4336 times)
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matsoljare
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« on: February 05, 2013, 04:33:30 PM »

Besides the Cantor dust, i can't think of any....  afro
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cKleinhuis
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« Reply #1 on: February 05, 2013, 05:19:45 PM »

what about the logistic function !?
http://en.wikipedia.org/wiki/Logistic_map

it is always visualised using the bifurcation diagrams, but this should be a one dimensional function...

and beside of that, the mandelbrot set is as well one dimensional when only reals used for calculation wink
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kram1032
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« Reply #2 on: February 06, 2013, 12:02:00 AM »

  • What do you count as 1D
  • What do you count as interesting?

Technically, the simplest cellular automata are 1D. It's just the typical visualization, where you put time dimension (e.g. iteration count) in a second spacial dimension so you can see how the interesting patterns unfold.

The same thing is true for how Cantor dust typically is depicted, as well as for the Logistic map.

Just as well, however, you could think of the generic cellular automaton as a high-dimensional fractal of which each individually valued automaton (e.g. rules 30 or 110) are mere 1+1D slices of that.

Plotting the Mandelbrot set's real axis as a typical f(x) plot reveals some rarely explored patterns as well. They might not be all that interesting but they are there.

If you fold a long piece of paper alway in the same direction and then unfold it all the way, you get a fractal pattern of valley and, uh, hilltops? - if you map those to 0 and 1, what you have is essentially a discrete 1D fractal pattern. If you interpret those 0s and 1s as left and right turns of 90░, you'll end up with the 2D dragon-curve which, however, is only a line.

Similarly, all space-filling curves at least on some level can be seen as 1D objects. Of course, those monster-curves were traditionally what first really challenged simple, discrete dimensionalities, so their case isn't so simple.

That being said, if you strictly only allow 1D-representations where you can't clarify what's going on by a "fake" (purely representational) second dimension, only so much can be done and most of it might not be all that interesting to look at.

Many things you'll try to keep in 2 or 3 dimensions, simply because those are the cases we can deal with most easily and efficiently.
1D structures tend to be way too wide and thin for our eyes to be read easily.
With 3D, things already start to obscure and only because we learned to live in a 3D world can we make a good amount of sense from such representations.
If you really shoot for representational clarity, you usually want to go 2D. It's as easily overviewable to us as 1D while putting information into a much more compact form without yet occluding anything like 3D would.
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taurus
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« Reply #3 on: February 06, 2013, 11:02:30 AM »

Besides the Cantor dust, i can't think of any....  afro
who said, that cantor dust is 1d? its topological dimensuion is 0 and the hausdorff dimension is about 0,631.

topologically the menger sponge is 1-dimensional!
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kram1032
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« Reply #4 on: February 06, 2013, 02:01:42 PM »

see, that's why I asked
  • What do you count as 1D
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matsoljare
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« Reply #5 on: February 06, 2013, 11:07:29 PM »

I mean any that can be represented as a 1D line of pixels, while having fractal properties even within that mode of display, which most 1D slices of 2D fractals does not. Especially looking for interesting one that have both solid and "hollow" parts, unlike the Cantor dust which has zero density.
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taurus
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« Reply #6 on: February 06, 2013, 11:25:28 PM »

Quote from: wikipedia
In mathematics, the Menger sponge is a fractal curve. It is a universal curve, in that it has topological dimension one, and any other curve (more precisely: any compact metric space of topological dimension 1) is homeomorphic to some subset of it.
don't know if this matches your definition of interresting, but although one might not await, it is a 1d fractal curve. Menger himself prooved that!
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kram1032
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« Reply #7 on: February 07, 2013, 12:20:51 AM »

in that case I guess, my mentioned folding fractal is in the race.
Fold a long strip of paper always in the same direction
assign emptiness to valleys and fullness to pe/\ks and after an infinitude of folds you should have a pretty fractal string of pixels.
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taurus
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« Reply #8 on: February 07, 2013, 12:36:15 AM »

in that case I guess, my mentioned folding fractal is in the race.
Fold a long strip of paper always in the same direction

Absolutely!
just for the heads up:
http://www.fractalforums.com/ifs-iterated-function-systems/dragon-curve-featured-in-jurassic-park-%28book%29-explained-by-numberphile/
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kram1032
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« Reply #9 on: February 09, 2013, 12:13:43 AM »

ayup, that's exactly what I'm talking about smiley
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M Benesi
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« Reply #10 on: April 29, 2013, 07:47:07 AM »

   Contemplate this:

  .
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kram1032
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« Reply #11 on: April 29, 2013, 03:26:19 PM »

 . ..  ..  .. .  . .. .  . .. .  ..  . ..  .. .  ..  . .

and so on. Try figuring out the rule smiley
« Last Edit: April 29, 2013, 03:28:37 PM by kram1032 » Logged
MateFizyChem
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« Reply #12 on: October 24, 2016, 08:11:43 PM »

How about a number line?

0 1 2 3 4 5 6 7 8 9 10 ...
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 ...

The idea is:

The number has got x digits after its point/comma.
The marked numbers follow this rule:
after removing the point/comma, it is a product of x+1 prime numbers.

The only problem is, it might not have a repeating visual pattern.
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Chillheimer
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« Reply #13 on: October 24, 2016, 08:32:49 PM »

while we're at number-sequences, I started wondering if the fibonacci sequence + golden mean might be called a fractal.

0,1,1,2,3,5,8,13,21,34,55,89...
scale invariance (no matter how high the pair of number is that you use to calculate the ratio)
and self similarity (you always get a close (self similar) approximation of Phi)
also, it is an iterative process that you can continue to infinity.

for my personal definition it has all the key features of a fractal..
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claude
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« Reply #14 on: October 24, 2016, 08:42:46 PM »

while we're at number-sequences, I started wondering if the fibonacci sequence + golden mean might be called a fractal.

What about p^n for any real p > 0, p != 1, n integer...   that doesn't have the awkward problem of imperfection...

However these are only "self-similar in one direction", unlike say the Cantor dust which is "self-similar in two directions", so I think it's less interesting.
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