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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?

Instead of using a straight line use a strange attractor ?

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.

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).

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..

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

@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.

(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.