Welcome to Fractal Forums

Besides the Cantor dust, i can't think of any....  O0

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 ;)

• 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.

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!

see, that's why I asked

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.

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!

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.

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/ (http://www.fractalforums.com/ifs-iterated-function-systems/dragon-curve-featured-in-jurassic-park-%28book%29-explained-by-numberphile/)

ayup, that's exactly what I'm talking about :)

   Contemplate this:

  .

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

and so on. Try figuring out the rule :)

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.

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..

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.

It is surely not a fractal from a strict point of view, but what about the Feigenbaum graph? You can generate it from the real component (bi=0) of the mandelbrot iteration. The graph itself might not be 1-dimensional but it arises from a 1-dimensional parameter. And there are bifurcations all over the place, so there is some sort of self similarity inside too. Maybe not a fractal and not 1-d in the end. But there is definitely a close relation.

You mean mine, right?

I'm too wondering, which one you mean
do you say the Feigenbaum-graph "maybe not a fractal" or did I mmisunderstand something?

Sorry, I think, I was unclear. From a strict point of view, the Feigenbaum graph is not a fractal. This is what I meant. Until today, I saw it as a chaotic map, not as a fractal.
And after a little bit of google, I see that it is a fractal. With a Hausdorff dimension below one (and below the one of cantor dust) it might also be one dimensional, but I'm not sure.
Does it have a topological dimension?

And to be precise, I was talking about the one below (from Wikipedia). The pure mandlbrot iteration
(https://upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Verhulst-Mandelbrot-Bifurcation.jpg/385px-Verhulst-Mandelbrot-Bifurcation.jpg)

The Feigenbaum graph is really a collection of many 1D point sets. Any point on the X axis of the above depiction represents a single parameter, a real number. Above that point, parallel to the Y Axis, the corresponding 1D point set for that parameter is plotted. That point set is the attractor of a dynamic process, i.e. the "infinite limit" of some iterated computation, parameterized with the value from the X position.

Across the whole parameter range (i.e. the relevant interval of the X axis), the corresponding attractors fall into two broad categories. Either an attractor consists of a finite number of isolated points, or the attractor has an infinite number of points that densely fill (part of) the Y axis.

"Simple" attractors with a finite number of points mean that the underlying iterated computation ends up cycling through a fixed list of values. Such an attractor is not fractal.

"Chaotic" attractors with infinitely many densely spread points indicate unstable, "unpredictable" behaviour of the iterated computation. These attractors are usually fractal point sets.

(There is also a theoretical possibility to get an infinite number of non-chaotic points, but I don't know if that case can happen in the Feigenbaum graph. An example of such a point set would be all points at positions 1/n for n any positive integer. But to be relevant to the Feigenbaum graph, an iterated computation would have to visit each point infinitely often - I am not sure if Feigenbaum's formula can do that when iterated.)

Depending on your definition and representation, the Stern-Brocot tree for rational numbers may be of interest.

https://en.wikipedia.org/wiki/Stern%E2%80%93Brocot_tree