Another way to make a "3D Mandelbrot"

[Tater]

This is not a method to make "the grail" but rather a different way of making a 3D object from a standard Mandelbrot set.

Calculate many points in the boundary of the M set. These are the points c for z_n+1 = z_n^2 + c that neither go to zero or infinity under iteration. These points map to other points in the boundary of the M set when iterated, so for each c value stored, calculate c^2+c, and make a continuous interpolation between c and c^2+c. This object shows, in some sense, what happens to the boundary of the M set if there was a continuous smooth transformation of M under the suspension of the map.

If the continuous interpolation is linear, (1-t)*c + t*(c^2+c) you will get a Mandelbrot set at the t=0 end of a cylinder and the Mandelbrot set at the t=1 end of the cylinder, and a continuous transformation of the Mandelbrot set into itself in between. One could also map the cylinder to a torus, or animate the process. I suspect the animation would be fascinating.

As with other 3D chaotic sets, the suspended M set should be organized by its periodic orbits, so one could compute important skeletal points by just suspending the low order periodic points

http://www3.amherst.edu/~rlbenedetto/talks/mhc_ug14.pdf

[bkercso]

Interesting idea!

(If you calculate these interpolated points after every iteration and do the further iterations with them, you get the coupled Mandelbrot set (two sets with linear coupling), which also can be considered as a 4D object:
http://www.fractalforums.com/mandelbrot-and-julia-set/coupled-mandelbrot-sets-%28cms%29/

There are some 3D pictures in the articles I linked in Reply #11 and #12, but not high quality ones.)

[claude]

Calculate many points in the boundary of the M set. These are the points c for z_n+1 = z_n^2 + c that neither go to zero or infinity under iteration.

Sure.

These points map to other points in the boundary of the M set when iterated

I don't think this is true at all, the dynamical plane z is different to the parameter plane c, and numerical testing seems to show no matching.

Examples:
c = -2, z = 0, -2, 2, 2, 2, 2, .... --> 2 (not in M or its boundary)
c = i, z = 0, i, i - 1, -i, i - 1, -i, ... --> cycle of length 2 (and i - 1 is not in M or its boundary)
c = 0.25, z = 0, 0.25, 0.3125, 0.34765625, 0.3708648681640625, ... --> 0.5 (not in M or its boundary)
c = -0.75, z = 0, -0.75,-0.1875,-0.71484375,-0.2389984130859375,-0.6928797585424036, ... --> -0.5 (in M but not on the boundary)

[Tater]

"I don't think this is true at all, the dynamical plane z is different to the parameter plane c, and numerical testing seems to show no matching.

Examples:
c = -2, z = 0, -2, 2, 2, 2, 2, .... --> 2 (not in M or its boundary)
c = i, z = 0, i, i - 1, -i, i - 1, -i, ... --> cycle of length 2 (and i - 1 is not in M or its boundary)
c = 0.25, z = 0, 0.25, 0.3125, 0.34765625, 0.3708648681640625, ... --> 0.5 (not in M or its boundary)
c = -0.75, z = 0, -0.75,-0.1875,-0.71484375,-0.2389984130859375,-0.6928797585424036, ... --> -0.5 (in M but not on the boundary)"

Yes, you are right. I need to rethink this, but it is certainly true that the Mandelbrot set interpolated to its first iterate would be interesting to see. Thanks for straightening me out.