File:Golden Mean Quadratic Siegel Disc.png

DescriptionGolden Mean Quadratic Siegel Disc.png	Golden Mean Quadratic Siegel Disc
Source	Made with program Mandel by Wolf Jung
Author	--Adam majewski (talk) 16:26, 21 September 2011 (UTC)
Permission (Reusing this file)	This screenshot either does not contain copyright-eligible parts or visuals of copyrighted software, or the author has released it under a free license (which should be indicated beneath this notice), and as such follows the licensing guidelines for screenshots of Wikimedia Commons. You may use it freely according to its particular license. Free software license: This work is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or any later version. This work is distributed in the hope that it will be useful, but without any warranty; without even the implied warranty of merchantability or fitness for a particular purpose. See version 2 and version 3 of the GNU General Public License for more details.http://www.gnu.org/licenses/gpl.htmlGPLGNU General Public Licensetruetrue Note: if the screenshot shows any work that is not a direct result of the program code itself, such as a text or graphics that are not part of the program, the license for that work must be indicated separately. العربية ∙ български ∙ català ∙ čeština ∙ kaszëbsczi ∙ Deutsch ∙ Ελληνικά ∙ English ∙ British English ∙ Esperanto ∙ español ∙ فارسی ∙ suomi ∙ français ∙ galego ∙ עברית ∙ magyar ∙ Bahasa Indonesia ∙ italiano ∙ 日本語 ∙ 한국어 ∙ македонски ∙ മലയാളം ∙ Bahasa Melayu ∙ norsk bokmål ∙ Nederlands ∙ norsk ∙ polski ∙ português ∙ português do Brasil ∙ română ∙ русский ∙ sicilianu ∙ slovenčina ∙ slovenščina ∙ Simple English ∙ svenska ∙ தமிழ் ∙ ไทย ∙ Türkçe ∙ українська ∙ 简体中文‎ ∙ 繁體中文‎ ∙ +/−

• 
  Average velocity - color version
• 
  Average velocity - Gray and detailed version
• 
  Boundary made with MIIM
• 
  Boundary - Animated version

This image shows the filled Julia set of Kc for complex quadratic polynomials :^[1]

${\displaystyle f_{c}(z)=z^{2}+c\,}$

where the parameter c is located on the boundary of the main cardioid.

Here the rotation number ( or internal angle ) ${\displaystyle t\,}$

is irrational number, the Golden Mean :^[2]

${\displaystyle t={\frac {{\sqrt {5}}-1}{2}}\approx 0.618033988749895}$

The multiplier ${\displaystyle \lambda \,}$ of map ${\displaystyle f\,}$ at fixed point is :

${\displaystyle \lambda =\lambda (t)=e^{2*\Pi *t*i}\,}$

has magnitude=1 so point c is on the boundary of unit circle and can be mapped to the boundary of main cardioid of Mandelbrot set :

${\displaystyle c=c(\lambda )={\frac {\lambda }{2}}\left(1-{\frac {\lambda }{2}}\right)\approx -0.390540870218399-0.586787907346969*i}$

Inside filled Julia set Kc ( black color) there are some orbits ( white points ) craeting deformed disks.

Outermost disc is a critical orbit.

All discs are around the indifferent fixed point ${\displaystyle \alpha _{c}\,}$ ^[3]

${\displaystyle \alpha _{c}=\left\{z:\ f_{c}(z)=z\right\}\approx -0.3686844390391594-0.3377451471307621*i}$

See demo 2 page 7 for more description

• run program mandel^[4]
• go to a boundary point with the key b
• Draw the critical orbit with the key o which makes boundary of Siegel disc
• Draw some orbits inside disc made by critical orbit :
  • tell the program to draw all iterated points with key #
  • select a point inside Siegel disc with double click
  • make some forward iteration with key f
  • repeat it
• save image as png with key F7

One can find that fixed point by

• switching to dynamic plane ( key F2 )
• finding periodic point ( key x) and inputting period 1

1. ↑ complex quadratic polynomial at wikipedia
2. ↑ Golden_ratio#Golden_ratio_conjugate
3. ↑ Periodic points of complex quadratic mappings at wikipedia
4. ↑ Mandel: multiplatform and opensource software for real and complex dynamics by Wolf Jung

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