(Sorry for the very long upcoming text.)

The question:

For an M-set: Can points that escape for a given escape radius and max iteration be classified again as members when radius and max iteration are increased 10-fold?

The background:

I am computing 3D sub-spaces of a 6D M-set for tricomplex numbers using the Lyapunov sequence ABABABAB... in the case of z^16.

The basic formula is f( (x,y,z) ) = (x,y,z)^16 + c where c is one of two tricomplex numbers cA and cB.

(x,y,z)² := (x²-y²-z², 2xy, -2xz) (formula by Eric Baird from

httpr://www.bugman123.com/Hypercomplex/index.html)

If the origin (0,0,0) stays bounded, the 6D number (cA,cB) is marked in a 6D cube. Afterwards I display a 3D slice of that multi-D space by fixing 3 of the 6 coordinates (to random values) and aligning the other three with the axis of a 3D cube to look at.

The yes/no images below used the slice: (_,0.028,_,0.724,_,0.496) where '_' means the viewed-upon-axis x,y,z and the numbers are the fixed values of the other 6D coordinates. Dimming was added for depth effect. Axis values range from -2 to +2.

The problem:

What I don't understand is the following: I've read that, in the 2D Mandelbrot, if an orbit point is outside the 2-disc, it accelerates and gets to infinity ever faster. That means to me, if I increase the escape radius by a factor of 10 and the iteration count as well, a point, that escapes using the smaller values still escapes later (10 iterations more mean 10 quadratic equations, hence the distance is around d^1024 when starting with distance d because the value of the constant can be neglected for large distances.

Below is a series of those fold-increases for the 6D case: 10 max iterations with escape radius of 10 show a cake (upper left), so does 100 (for both values).

But at 1000 it changes. Points formerly escaping now are back in the M-set. (The upper row is the same size, so the cake is smaller than the tube to the right, in the second row I had to reduce the cube size from 200^3 to 100^3 for speed reasons. The blue lines are the coordinate axis later introduced, z-axis is straight upward.)

And from 10^4 to 10^5 the tube wall gets even thicker.

I'm using standard double floating type in C++, maybe it's a problem of precision?

Is my naive transformation of that concept of accelerating speed with the above formula not valid? I do not know where it originated from, maybe it lacks properties of the standard complex number multiplication? Or the value of 10 (and 100) is not what is considered "large enough" and I simply have to use higher values to get more stable and true images?

Or is the concept of accelerating speed true for the 2D Mandelbrot but not for every 10-fold increase. So are there points that escape, then do not escape when increasing both escape radius and max iteration, then escape again and so on, until they finally forever increase their distance with every step?

Linkback: https://fractalforums.org/fractal-mathematics-and-new-theories/28/3d-m-set-for-tricomplex-numbers-escape-time-and-shape-change/2793/