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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

Sure.


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)