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Author Topic: Switch Mode  (Read 1082 times)
Description: Can the switch mode be used in deep zooms?
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tm2123
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« on: August 25, 2013, 07:42:19 PM »

Hello

Can the switch mode be used in deep zooms?  Are there any suggestions or tutorials about how to use the switch mode

tom
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Rychveldir
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« Reply #1 on: September 17, 2013, 04:06:24 PM »

Yes it can be used. But I can only explain how it works for Mandelbrot/Julia sets.

Mandelbrot Formula: z_n=z_{n-1}^2+z_0
Julia Formula: z_n=z_{n-1}^2+C

So while the Mandelbrot Formula will add the starting point for each iteration, the Julia Formula will add the same constant regardless where you started off. Switching mode switches from a Mandelbrot Set to a Julia Set using the complex value of the pixel you clicked as constant C for the new Julia Set. However, the switched fractal will never be zoomed. If you use switching during a deep zoom you will get a C with more significant digits (e.g. 0.48343407534095 instead of 0.483).

Since the formula for the switch function can be freely chosen when writing a formula, you can make it do almost anything which can be done by handing on parameters and/or calculating new ones from the existing ones. You can ofc write a formula where the switch function also hands the current location and zoom factor to the switched image, at least I think so. But since the new formula is very likely to produce a totally different shape you will end up in a space that is completely inside or outside of your fractal and does not contain any interesting pattern at all.
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tm2123
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« Reply #2 on: September 17, 2013, 05:41:52 PM »

Hi Rychveldir

Thank you for the reply, but do not understand your answer as I do art by site and feel.  Is there a very basic explanation or lesson(s) that can be follow to help me understand how to do a deep zoom switch mode.  Thanks again for the reply.

tom
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Alef
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« Reply #3 on: September 17, 2013, 05:44:35 PM »

Yes it work. But when you are in deep zoom you hardly can explore julia seed.
When you are in switch mode by moving cursor around image you change julia seed according to coordinates of cursor. Julia set will be like region of the seed (where cursour are). Say if it coloured green center of julia would be green. When in deep zoom every julia will be very simmilar.
« Last Edit: September 17, 2013, 05:50:59 PM by Alef » Logged

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aleph0
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« Reply #4 on: September 18, 2013, 11:30:14 PM »

So while the Mandelbrot Formula will add the starting point for each iteration, the Julia Formula will add the same constant regardless where you started off.

The fixed term z_0 in the Mandelbrot set formula isn't correct. The same iteration formula is used to determine membership of the Mandelbrot Set and all Julia sets and is usually written in this form (see for example p.161 of The Beauty of Fractals):

z_{n+1} = z_n^2 + c

That's equivalent to your Julia formula, but uses a different convention for the subscript numbering.

Each point in the complex plane is a 'candidate' for set membership and the iteration formula is applied repeatedly to determine whether it is or is not a set member. If the iterate remains bounded (modulus less than or equal to 2), the point is a member of the set. However for each set, different initialisation criteria are applied before starting the iteration sequence.

For the Mandelbrot Set:
- c is set to the coordinates of each candidate point.
- z_0 = 0 for all candidate points. Hence from the iteration formula z_1 = 0 + c = c, z_2 = c^2 + c, z_3 = (c^2 + c)^2 + c, and so on. So your comment that the starting point is added for each iteration is correct, but it's the result of application of the c term rather than the z_0 term.

For Julia sets:
- A fixed point is chosen from the complex plane. This choice establishes the 'identity' of the Julia set to be calculated.
- c is set to the coordinates of the chosen point and remains fixed for all candidate points (as you commented) and across all iterations for each candidate point.
- z_0 is set to the coordinates of each candidate point.

The consequences of these distinct initialisation criteria are that:
- There is only one Mandelbrot Set.
- There is an infinity of Julia sets, resulting from the free choice of fixed point. As Alef commented, when deeply zoomed in the Mandelbrot Set, small differences in choice of c are likely to result in Julia sets that are visually indistinguishable.

When drawing the sets using a pixel grid, the complex coordinate z_p of each pixel is used during initialisation:
- For the Mandelbrot Set, c = z_p.
- For Julia sets, z_0 = z_p.

Sometimes, descriptions of how the sets are generated fail to clearly address some aspect of the above, e.g. they might clearly state the iteration formula but fail to state the initialisation criteria for z_0.

John.
« Last Edit: September 19, 2013, 12:08:19 AM by aleph0 » Logged
aleph0
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« Reply #5 on: September 20, 2013, 07:21:02 PM »

As my post didn't specifically relate to Ultra Fractal, I've created a new thread here:
- http://www.fractalforums.com/general-discussion-b77/terms-z0-and-c-in-the-mandelbrotjulia-iteration-formula/

If there are any comments on what I wrote, please reply there.

Thanks,
John.
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David Makin
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« Reply #6 on: April 30, 2014, 12:58:14 AM »

Hi Rychveldir

Thank you for the reply, but do not understand your answer as I do art by site and feel.  Is there a very basic explanation or lesson(s) that can be follow to help me understand how to do a deep zoom switch mode.  Thanks again for the reply.

tom

Because switch mode relies on picking coordinates from the window that you're currently viewing the Mandelbrot in, when you are deep-zoomed it doesn't really work because the range of values available is restricted to the area of the complex plane displayed in the window.
Also in Ultra Fractal when you switch to a Julia the new Julia Set is always displayed at full scale i.e. not zoomed.
If you wish to do the equivalent of "switching" in Ultra Fractal then open a Julia Set and zoom-in, then right-click on the Julia Seed (or constant) parameter in the Julia formula and choose the "explore" tool - then use the explore window to "switch" - this will essentially do the same as switching but in the current Julia window *at the current zoom level*. The explore window has + and - gadgets to adjust the scale so "exploring" can work at any zoom depth. Unfortunately at the moment no-one has implemented a method that tries to automatically follow the interesting bits of the fractal when you do this and of course when really zoomed in parts of the fractal originally in view will morph out of the viewing window.
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fractalrebel
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« Reply #7 on: March 01, 2015, 09:44:15 PM »

This procedure works with my switch mode formula, and I presume also with Dave's. Zoom into the Mandelbrot wherever you want and copy the location. There is a button for doing that. Then switch and paste the location using the button on the location tab.
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xenodreambuie
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« Reply #8 on: March 01, 2015, 11:33:12 PM »

Switching between zoomed Mandelbrot and Julia, I find it worthwhile having the option of unzoomed or zoomed Julia, especially near minibrots. The appropriate zoom value for the Julia is not the same, but is the nth root of the Mandelbrot zoom where n is the degree of the formula (or just divide the power of 10 by the degree). I found this empirically; no idea about the theory.
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cKleinhuis
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« Reply #9 on: March 01, 2015, 11:35:03 PM »

this would be indeed a cool feature, i find the julias created from minibrots and zoomed julias quite interesting!!!
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