Fractals from calculations of 2^I, 2^(2^I), 2^(2^(2^I).. a^(a^(...a^I)

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Fractals from calculations of 2^I, 2^(2^I), 2^(2^(2^I).. a^(a^(...a^I)

Gottfried Offline

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#17

05/25/2008, 10:48 AM (This post was last modified: 05/25/2008, 10:50 AM by Gottfried.)

Ivars Wrote:While studying this, I have become little puzzled about the way \( h(I) \) converges. When iterated from z_0=I with a any precision available in PARI, it shows a tri-cycle behaviour, as do many other imaginary numbers (e.g 2*I, 3*I etc. ), if I have made the calculations right.

Hi Ivars -
I've discussed this in some initial state earlier here, but I'm currently with my head elsewhere so too lazy to find the thread.
Here are some plots of that tri-furcation, see uncommented list below.
I find the bi-,tri- and multifurcation an interesting subject, as we ask: can we assign an individual value if the iteration oscillates/is furcated - since this is somehow related to the partial evaluation of non-convergent oscillating series, to which we assign a value anyway. But I don't have a special conclusion for this matter, yet, which were worth to write it down here.

Gottfried

http://go.helms-net.de/math/tetdocs/traj..._165_I.png
http://go.helms-net.de/math/tetdocs/traj..._175_I.png
http://go.helms-net.de/math/tetdocs/trajectories_I.png
http://go.helms-net.de/math/tetdocs/traj...rcated.png
http://go.helms-net.de/math/tetdocs/trif...on_1_2.png
http://go.helms-net.de/math/tetdocs/trif...on_1_8.png
http://go.helms-net.de/math/tetdocs/trif...quator.png
http://go.helms-net.de/math/tetdocs/trif..._bases.png
http://go.helms-net.de/math/tetdocs/trif...detail.png
http://go.helms-net.de/math/tetdocs/trif...bases1.png

Gottfried Helms, Kassel

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Messages In This Thread

Fractals from calculations of 2^I, 2^(2^I), 2^(2^(2^I).. a^(a^(...a^I) - by Ivars - 05/19/2008, 09:02 PM

RE: Calculations of 2^I, 2^(2^I), 2^(2^(2^I).. a^(a^(...a^I) - by bo198214 - 05/20/2008, 06:12 AM

RE: Calculations of 2^I, 2^(2^I), 2^(2^(2^I).. a^(a^(...a^I) - by Ivars - 05/20/2008, 06:43 AM

RE: Calculations of 2^I, 2^(2^I), 2^(2^(2^I).. a^(a^(...a^I) - by Ivars - 05/20/2008, 04:46 PM

RE: Calculations of 2^I, 2^(2^I), 2^(2^(2^I).. a^(a^(...a^I) - by Ivars - 05/20/2008, 04:59 PM

RE: Fractals from calculations of 2^I, 2^(2^I), 2^(2^(2^I).. a^(a^(...a^I) - by Ivars - 05/20/2008, 08:36 PM

RE: Fractals from calculations of 2^I, 2^(2^I), 2^(2^(2^I).. a^(a^(...a^I) - by Ivars - 05/20/2008, 09:08 PM

RE: Fractals from calculations of 2^I, 2^(2^I), 2^(2^(2^I).. a^(a^(...a^I) - by bo198214 - 05/21/2008, 06:33 AM

RE: Fractals from calculations of 2^I, 2^(2^I), 2^(2^(2^I).. a^(a^(...a^I) - by Ivars - 05/21/2008, 10:29 AM

RE: Fractals from calculations of 2^I, 2^(2^I), 2^(2^(2^I).. a^(a^(...a^I) - by Ivars - 05/22/2008, 08:13 PM

RE: Fractals from calculations of 2^I, 2^(2^I), 2^(2^(2^I).. a^(a^(...a^I) - by Ivars - 05/23/2008, 06:51 AM

RE: Fractals from calculations of 2^I, 2^(2^I), 2^(2^(2^I).. a^(a^(...a^I) - by bo198214 - 05/23/2008, 11:44 AM

RE: Fractals from calculations of 2^I, 2^(2^I), 2^(2^(2^I).. a^(a^(...a^I) - by Ivars - 05/24/2008, 08:13 AM

RE: Fractals from calculations of 2^I, 2^(2^I), 2^(2^(2^I).. a^(a^(...a^I) - by bo198214 - 05/24/2008, 08:55 AM

RE: Fractals from calculations of 2^I, 2^(2^I), 2^(2^(2^I).. a^(a^(...a^I) - by Ivars - 05/24/2008, 04:09 PM

RE: Fractals from calculations of 2^I, 2^(2^I), 2^(2^(2^I).. a^(a^(...a^I) - by Ivars - 05/25/2008, 08:28 AM

RE: Fractals from calculations of 2^I, 2^(2^I), 2^(2^(2^I).. a^(a^(...a^I) - by Gottfried - 05/25/2008, 10:48 AM

RE: Fractals from calculations of 2^I, 2^(2^I), 2^(2^(2^I).. a^(a^(...a^I) - by Ivars - 05/25/2008, 06:08 PM

RE: Fractals from calculations of 2^I, 2^(2^I), 2^(2^(2^I).. a^(a^(...a^I) - by Gottfried - 05/25/2008, 10:12 PM

RE: Fractals from calculations of 2^I, 2^(2^I), 2^(2^(2^I).. a^(a^(...a^I) - by Ivars - 05/26/2008, 07:43 AM

RE: Fractals from calculations of 2^I, 2^(2^I), 2^(2^(2^I).. a^(a^(...a^I) - by Gottfried - 05/26/2008, 09:01 AM

RE: Fractals from calculations of 2^I, 2^(2^I), 2^(2^(2^I).. a^(a^(...a^I) - by Ivars - 05/26/2008, 10:30 AM

RE: Fractals from calculations of 2^I, 2^(2^I), 2^(2^(2^I).. a^(a^(...a^I) - by Ivars - 05/26/2008, 01:11 PM

RE: Fractals from calculations of 2^I, 2^(2^I), 2^(2^(2^I).. a^(a^(...a^I) - by Gottfried - 05/25/2008, 10:51 PM

RE: Fractals from calculations of 2^I, 2^(2^I), 2^(2^(2^I).. a^(a^(...a^I) - by Ivars - 05/27/2008, 08:10 AM

RE: Fractals from calculations of 2^I, 2^(2^I), 2^(2^(2^I).. a^(a^(...a^I) - by Gottfried - 05/27/2008, 09:23 AM

RE: Fractals from calculations of 2^I, 2^(2^I), 2^(2^(2^I).. a^(a^(...a^I) - by Ivars - 05/27/2008, 11:45 AM

RE: Fractals from calculations of 2^I, 2^(2^I), 2^(2^(2^I).. a^(a^(...a^I) - by tommy1729 - 04/11/2014, 10:54 PM

RE: Fractals from calculations of 2^I, 2^(2^I), 2^(2^(2^I).. a^(a^(...a^I) - by Gottfried - 04/12/2014, 01:10 AM

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