M-Set: Example how to calculate the zoom depth of doubling events?

[Chillheimer]

Dinkydau generously shared this secret:

The golden rule behind most of my images is the following. Its aspects can be used to do pretty much everything that you find in my gallery:
Given a location with a zoom level n, moving away from the center to a different center has the following effect:
The shape at zoom level n is doubled at zoom level 1,5n in such a way that the rotational symmetry becomes 2-fold.
At 1,75n the symmetry becomes 4-fold.
At 1,875n the symmetry becomes 8-fold.

Now, I'm not sure, how do I calculate this?

Lets say, I leave the center at 3.84e258
When will the doubling occur? And how do I actually calculating it?
Just 258*1.5--> e387 for the first doubling? that easy?

[PieMan597]

I think that's how it works. However, where you zoom in inside of the point where you go off-center changes the depth slightly too.

[Chillheimer]

confirmed!
thx to pertubation method one can simply test it within some minutes.
as you said pieman, the depth of the doubling changes depending on when you actually start zooming into the "new middle"
So instead of e387 it actually occured at e395.

nice

this makes things a little easier - I kept loosing track of when I really reached the doubling that I was aiming for.
and now the "find minibrot"-feature of kalles fraktaler realy becomes useful!

Thx again for sharing dinkyday!

[TheRedshiftRider]

Dinkydau generously shared this secret:

Now, I'm not sure, how do I calculate this?

Lets say, I leave the center at 3.84e258
When will the doubling occur? And how do I actually calculating it?
Just 258*1.5--> e387 for the first doubling? that easy?

I think it is like this:

If you first zoom to e100 for the base of your morph. And the start to go into the center, if you leave the center at e150 (or 100x1.5) the depth will be e150*1.5 . (4-fold)

At e175 the depth will be 175x1.75  (8-fold) and so on...

  too late.

[PieMan597]

Also, the depth of the final minibrot the depth of where you went off center last, doubled, approximately

[Dinkydau]

For a useful approximation, yes it's that easy! If you are at e100, and go offcenter, the symmetry increases will be:
2-fold at e150 = e(100*2 - 100/2)
4-fold at e175 = e(100*2 - 100/4)
8-fold at e187,5 = e(100*2 - 100/8)
16-fold at e193,75 = e(100*2 - 100/16)
...
infinity at e200 = e(100*2 - 100/infinity), which is a minibrot.

The numbers are always an upper bound. The numbers will be very close to reality if you go "totally" offcenter. If you leave the center and go into some structure that already has increased symmetry, you are basically skipping the part "already done" and the actual depth of the symmetry increases and the minibrot will be less. That's how I got my S containing trees and more S at not too much depth:
http://dinkydauset.deviantart.com/art/Mandelbrot-extremism-453112933

In the making of the main S, I made a few "circles" of high symmetry among the trees, which saved a lot of depth.