Welcome to Fractal Forums

Maybe it would be an interesting experience to try and find a one dimensional representation of the Mandebrot Set?

I tried z[n]=z[n-1]+c but that seems to produce no complexity at all. All points from c=-2 to about c=0.25 are inside the set and all others are outside the set.

Any ideas?

lol, congratulations, you found the 1 d mandelbrot ... :D  O0
what did you expect ?!?!?
you can use another line through the set ... :D

Instead of using a straight line use a strange attractor ?

For the real z[n+1]=z[n]*z[n]+c, if you plot the C values from -2 to 0 int the X axis and the the iteration values in the Y axis, you will get a bifurcation diagram: http://en.wikipedia.org/wiki/Bifurcation_diagram (http://en.wikipedia.org/wiki/Bifurcation_diagram)

I've thought about a 1D Mandelbrot set too. It was in desperation trying to find the 3D version. I thought if I can't succeed go up, what happens when you go down.

The problem seems to be that you can't properly rotate using only 1 dimension. in 2D you can, and in 3D, you can rotate in any of the 3 planes. But 1D lacks this ability.

Interesting concept though - we'd expect to see a dot-dash kind of pattern appear, but with lots of patterns of dots and dashes etc.

>> we'd expect to see a dot-dash kind of pattern appear, but with lots of patterns of dots and dashes etc.
I would expect to see a line, which is what you see if you use 1d numbers (the reals). Mandelbrot is (almost certainly) connected in 2d, so a connected version in 1d could only be a single line segment.

Wouldn't the line have some on/off structure to at least give the appearence of dots and dashes (though obviously some very short/long dots/dashes sometimes).

the reason why it gives a line on the x axis is because it is connected i believe ....  :alien:

With power 1 it is not a fractal anymore...no folding or stretching.
You can go close to 1 though and less than 1.

Less than 1 inverts the fractal..what usually is the inside guts is outside and vice versa..

mrrgu:
I think the question is not for power one but for dimension one which already was correctly answered by Aexion but nobody seemed to pay attention lol.

However I wonder what happens if you represent the Mset in Bifurcation-style... Maybe you get a somewhat-3D sheet-furcation-thing :)

I just looke at the formula and it had power one... but you are right though  ;)

This is the logistic formula, not the Mandelbrot formula, but Mandelbrot would probably look somewhat similar:
    (http://www.hevanet.com/bradc/ComplexLogisticBifurcationDiagram3D.png)

@I think the question is not for power one but for dimension one which already was correctly answered by Aexion but nobody seemed to pay attention lol.
Actually, I don't see how the logistic map is the same as a 1d mandelbrot, since r*x*(1-x) isn't  x^2 + c

But a 1d mandelbrot is surely simply the set of points at i=0 on the mandelbrot. i.e. a line segment.

it's not exactly the same but it's of the same kind.

you could do those bifurcation map for the Mset aswell.

If you coordinate-transform the logistic bifucration, you can get it to exactly match the Mset and its special positions like the biggest Minibrot.
Afaik, that Minibrot is the first bigger lake of stability after the beginning chaos...

Basically do the Mset recursion with x²+a and do the same as you'd do for the logistic map with a is what in the logistic map would be r.
The results are equal.

As you see, the picture of the "complex logistic map" above, it features a lot of mandelbrot-ish bulbs :)

(I'm pretty sure the OP meant to write the formula with a second power - he even said it went from -2 to 0.25)

All these things such as the bifurcation diagrams produce 2D (or 3D) objects. I think the idea of using a strange attractor sounds cool, but not very mathematically "pure". Maybe taking the segment from (0,i) to (0,-1)? In other words, z[n+1]=z[n]*z[n] + i*c? This is reasonable, and can produce interesting segments. I'm sure that in places where it intersects (for instance) a spiral, you might get something interesting; increasingly rapid dashes, maybe something like   --------    ----  -- - - -- ---- -------- ? If you just want one curve (in 2D), maybe something like plotting y = the greatest imaginary component so that a number with the x-coordinate as its real component belongs to the Mandelbrot set. Even more interesting, the absolute value of the difference between two nearby pixels. This would produce something akin to a pin plot, with high pins where represent places where a branch extends over another.

The OP was looking for the 1d equivalent rather than an arbitrary slice, and since complex numbers just extend real numbers, R^2 + C seems to be the rather obvious 1d analog.
-Mandelbrot is connected, the line segment -2 to 0.25 is connected
-Mandelbrot inside has dimension 2, this line has dimension 1
The Mandelbrot has border points at the connection with the first great circle, and more further to the head in ever increasing numbers. So R^2 + C  has an increasing number of border points (as R goes towards -2), in fact I would conjecture that it has so many border points that the border has fractal dimension 1. So in fact the line is rather a set of lines with border points (but no space) between them. So..
-Mandelbrot border has dimension 2, this line border has dimension 1

It is actually a pretty interesting shape, and I'm in no doubt that this is the 1d equivalent.
The lengths of the connected line segments don't follow a simple pattern, as can be seen by plotting the bifurcations horizontally on the logistic map.

It's a line. When you say, "border points", this is equivalent to just saying "where the period changes". You'll just get the bifurcation diagram of the logistic map; boring. Just as the Mandelbulb was taken to the 8th power for aesthetic reasons, why not take a slice of the Mandelbrot perpendicular to the real axis? It's way more interesting...

Well you can if you like, and it might be more interesting. I'm just commenting on what the direct 1d analog is.
Besides, this line segment is interesting. Imagine a line going diagonally along a chess board like a bishop on black squares. At the exact point it crosses to the next square, is the colour of that point black or white? It can be neither. It is fair to define this diagonal line as single segments of length root 2 attached by border points to each other. Each being the single 0d point where the horizontal borders cross.
So R^2 + C looks like this -------+----+--+------+----+--+-+--+-+  The line segments are connected in the same way that the black squares on the chessboard are connected and the same way that the mandelbrot insides are connected. But the imaginary axis of the mandelbrot is not connected, so is not an equivalent fractal in my opinion.

The R^2 + C also shows the decimals of PI when calculating the number of iterations to escape as shown in this table:
"here's a table for points of the form (.25 + X, 0)

X               # of iterations
1.0               2
0.1                8
0.01              30
0.001               97
0.0001       312
0.00001       991
0.000001       3140
0.0000001       9933
0.00000001       31414
0.000000001   99344
0.0000000001 314157"
This is just like the 2d mandelbrot does.

What seems more interesting than the similarities are the differences. The border of the 2d mandelbrot is completely non-smooth fractal, the _border_ of the 1d mandelbrot isn't connected (so is dust) and, looking to 4 dimensions, the quaternion mandelbrot border is overly smooth.
This is interesting and could be explained by saying that the ^2 operation does not have enough degrees of freedom in 1d, has just the right amount in 2d and it doesn't take up all the degrees of freedom in 4d (hence smoothness). This would explain why the mandelbulb requires two operations (on longitude and latitude) and why a properly fractal 4d mandelbrot would require rotating in 3 directions.

Of course this all kind of undermines my point because it means R^2 + C isn't the true 1d analog... the true equivalent would have a connected border of fractal dimension 1 and the line segments should also be connected. I'm not sure such a shape exists.

Hello guys, I was reading this old post, and I invite you to read my recent post:
http://www.fractalforums.com/mandelbrot-and-julia-set/mandelbrot-on-real-numbers-t5375/msg26926/#msg26926
Perhaps you can tell if I'm really on something interesting or just loosing time, because I'm not a fractal expert, just curious about them and I try to research all that I can with my limited knowledge. I'll really apreciate some comment. Thanks.

I think 1d mandelbrot would look like the 2d mandelbrot at the line i0. And i0 so happens to contain an unbroken line. (If you do not know what I mean by i0, y0.)