Fractals/Iterations in the complex plane/Mandelbrot set/mset distortion

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This page is about: warped midget, distortion ( parameter plane and Mandelbrot set ).


name[edit | edit source]

There are many names of mini copies of Mandelbrot set within itself:

  • Small Mandelbrot sets
  • mini Mandelbrot set
  • baby Mandelbrot set
  • baby-brot
  • Bug
  • Island : "word 'island' was introduced by Benoit Mandelbrot in his description of the Mandelbrot set in The Fractal Geometry of Nature" Robert Munafo
  • island mu-molecules
  • embedded copy of the Mandelbrot Set
  • Mandelbrotie
  • Midget
  • minibrot
  • primitive M-copies
  • slightly deformed copies of the original shape


Different names for distorion[1]

Names for distorted midgets:

  • warped midget

Examples[edit | edit source]

See also:


Sequence of locations from the elephant valley, with the period doubling ( by Claude Heiland-Allen )[4]

 # period center radius
     4 -1.565201668337550811e-01+ 1.032247108922831780e+00 i @ 1.697e-02
     8 4.048996651751222142e-01 + 1.458203637665893004e-01 i @ 2.743e-03
    16 2.925037532341934199e-01 + 1.492506899834379792e-02 i @ 3.484e-04
    32 2.602618199285006706e-01 + 1.667791320926506355e-03 i @ 4.113e-05
    64 2.524934589775105209e-01 + 1.971526796077277316e-04 i @ 4.920e-06
   128 2.506132008410751344e-01 + 2.396932642510365971e-05 i @ 5.997e-07
   256 2.501519680089798192e-01 + 2.954962325906881015e-06 i @ 7.398e-08
   512 2.500378219137852631e-01 + 3.668242052764790239e-07 i @ 9.185e-09
  1024 2.500094340031833728e-01 + 4.569478652064613658e-08 i @ 1.144e-09
  2048 2.500023558032561377e-01 + 5.701985912706845832e-09 i @ 1.428e-10
  4096 2.500005886128087162e-01 + 7.121326948562731412e-10 i @ 1.783e-11
  8192 2.500001471109009610e-01 + 8.897814201389539302e-11 i @ 2.228e-12


List by marcm200

cardioid       angle             
period p       p<-2p->4p         eccentricity   cardioid center
------------------------------------------------------------------------------------------------------
5              177.1             1.35           0.35925922475800742273-0.64251373713854231795*i
6              176.5             1.47           0.4433256333996235532-0.37296241666284651872*i
7              176.1             1.55           0.43237619264199450564-0.22675990443534863039*i
8              175.8             1.52           0.40489966517512215871-0.14582036376658927268*i
10             175.4             1.65           0.35681724849231194474-0.069452865466830299157*i
12             175.1             1.72           0.32558950955066034982-0.038047880934755723414*i
15             174.8             1.79           0.29844800890399547644-0.018383367322073254635*i
17             174.7             1.80           0.28756611704687790043-0.012281055409848253349*i
21             174.6             1.84           0.27436979919551240936-0.0062585874913430950342*i
25             174.5             1.87           0.2652783219046058732 -0.0037120599898783183101*i
31             174.4             1.90           0.26095224231422198269-0.0018410978175303128503*i
43             174.34            1.91           0.2556052938434967281 -0.00066797029763820438275*i                     


Locations:

  • 0,25000102515011806826817597033225524583655 + 0,0000000016387052819136931666219461i Zoom: 6,871947673*(10^10)
  • 0.2925294 + 0.0149698 i @ 0.0006675
  • c = 0.292503753234193 -0.014925068998344i period = 16, size of image +0.0005 [5]
  • by marcm200
    • period 7 c =0.43237619264199450564-0.22675990443534863039i
    • period 1 4 c=0.4325688150887696537-0.22873440581356344059i
    • period 28 c=0.43266973566541755414-0.22933968667591089763i

Self-similarity[edit | edit source]

Self-similarity in the Mandelbrot set shown by zooming in on a round feature while panning in the negative-x direction. The display center pans left from the fifth to the seventh round feature (-1.4002, 0) to (-1.4011, 0) while the view magnifies by a factor of 21.78 to approximate the square of the Feigenbaum ratio .

The Mandelbrot set is self-similar under magnification in the neighborhoods of the Misiurewicz points. It is also conjectured to be self-similar around generalized Feigenbaum points (e.g., −1.401155 or −0.1528 + 1.0397i), in the sense of converging to a limit set.[6][7] The Mandelbrot set in general is not strictly self-similar but it is quasi-self-similar, as small slightly different versions of itself can be found at arbitrarily small scales. These little copies of the Mandelbrot set are all slightly different, mostly because of the thin threads connecting them to the main body of the set.


"this combination of familiarity (small copies of the Mandelbrot set) and novelty (subtly distorted, differently decorated) that make the Mandelbrot set much more interesting than the fractals generated by IFS."[8]

Q&A[edit | edit source]

are they exact copies?[edit | edit source]

They are:

  • not exact copies
  • slightly deformed copies
  • approximately self-similar copies
  • quasiconformally equivalent to the whole Mandelbrot set
  • quasi-self-similar

Why does the Mandelbrot set contain copies of itself?[edit | edit source]

Why does the Mandelbrot set contain (slightly deformed) copies of itself?[9]

  • "renormalization explains why the baby Mandelbrot set appears: all of the quadratic polynomials in a baby Mandelbrot set renormalize and all renormalize in essentially the same way."
  • "the fractal is a reflection of reflections of reflections ..."
  • "iteration creates complexity. Mandelbrot set is based on checkin behaviour of infinite sequences. The behaviour of that sequences is very complex. In fact, it may be chaotic. We are in the presence of sensitive dependence on initial conditions (a.k.a. the butterfly effect.)" [10]
  • "While the process is very simple, the iterative nature of the process leads to very different results depending on the point on the complex plane at which you start."


Why there are copies of Mandelbrot set in other fractals ?[edit | edit source]

The Mandelbrot set is universal[11] so it can be find in:

  • an iterated complex mapping[12]
  • the bifurcation locus for a parametrized family of holomorphic functions: rational families of functions, transcendental families [13]
  • the bifurcation locus of any holomorphic family of rational maps[14]
  • quadratic-like maps of Douady and Hubbard [15]

How to describe minibrot ?[edit | edit source]

Measures

How to compute size and orientation ?[edit | edit source]

Size and orientation[edit | edit source]

Scaling for period-n windows of chaotic dynamical systems


The formula for the size and orientation:

 

Input and related values:

  • nucleus of a minibrot's main pseudocardioid
  • period of a minibrot's main pseudocardioid

Output:

  • complex valued size and orientation estimation r ( complex number). To compute it just use arg and abs on the result


where







How to use it ?

"if the nucleus of the baby is c and the complex size is r , there is another miniature copy near the baby around  with size approximately " 	Claude Heiland-Allen

Example minibrots:

  • period 3 near −2
  • period 4 near i
  • period 5 near −1.5+0.5i


See also c function m_size from mandelbrot-numerics library by Claude Heiland-Allen

// mandelbrot-numerics -- numerical algorithms related to the Mandelbrot set
// Copyright (C) 2015-2018 Claude Heiland-Allen
// License GPL3+ http://www.gnu.org/licenses/gpl.html

#include <mandelbrot-numerics.h>

extern double _Complex m_d_size(double _Complex nucleus, int period) {
  double _Complex l = 1;
  double _Complex b = 1;
  double _Complex z = 0;
  for (int i = 1; i < period; ++i) {
    z = z * z + nucleus;
    l = 2 * z * l;
    b = b + 1 / l;
  }
  return 1 / (b * l * l);
}

area[edit | edit source]

The area of the island of period p+1 is approximated by the formula by Robert Munafo:[16]

How to measure the distorsion ?[edit | edit source]

Measure

  • express these in terms of the derivatives of the iteration at the preimage of 0[17]
  • the ratios of radii of the satellites to the main cardioid
  • angular positions of secondary ( child) components
  • in degrees(or radians) because the parts of the minibrot are simply rotated ( conformal transformations )
  • the ratio of two distances to the cusp[18]


How distorted can a minibrot be?[edit | edit source]

What is maxima distortion of minibrot ?[19]


How to find minibrot ?[edit | edit source]


How to find distorted minibrot ?[edit | edit source]

See also[edit | edit source]

References[edit | edit source]

  1. Distortion (optics) in wikipedia
  2. Image_warping in wikipedia
  3. 2D Mesh Warping by Yurong Sun
  4. fractalforums.com  : how-distorted-can-a-minibrot-be ?
  5. Distortion by Robert P. Munafo, 2010 Oct 20. From the Mandelbrot Set Glossary and Encyclopedia
  6. Lei (1990). "Similarity between the Mandelbrot set and Julia Sets". Communications in Mathematical Physics 134 (3): 587–617. doi:10.1007/bf02098448. Bibcode1990CMaPh.134..587L. http://projecteuclid.org/euclid.cmp/1104201823. 
  7. J. Milnor (1989). "Self-Similarity and Hairiness in the Mandelbrot Set". in M. C. Tangora. Computers in Geometry and Topology. New York: Taylor & Francis. pp. 211–257. ISBN 9780824780319. https://books.google.com/books?id=wuVJAQAAIAAJ. )
  8. The Mandelbrot Set and Julia Sets .The Mandelbrot Set - Small Copies from Fractal Geometry Yale University Michael Frame, Benoit Mandelbrot (1924-2010), and Nial Neger November 24, 2022
  9. math.stackexchange question: why-does-the-mandelbrot-set-contain-slightly-deformed-copies-of-itself ?
  10. math.stackexchange question: mandelbrot-fractal-how-is-it-possible ?
  11. The Mandelbrot set is universal by Curtis T. McMullen
  12. quora : Why-is-the-Mandelbrot-Set-important?
  13. math.stackexchange" question: why-does-the-mandelbrot-set-appear-when-i-use-newtons-method-to-find-the-invers
  14. math.stackexchange question: are-mini-mandelbrots-known-to-be-found-in-any-fractals-other-than-the-mandelbrot ?
  15. Universality of the Mandelbrot set from Fractal Geometry Yale University Michael Frame, Benoit Mandelbrot (1924-2010), and Nial Neger
  16. Main Sequence by Robert P. Munafo, 2009 Nov 5. from the Mandelbrot Set Glossary and Encyclopedia
  17. math.stackexchange question: mini-mandelbrots-are-they-exact-copies ?
  18. fractalforums.org : period-doubling-in-minibrots
  19. fractalforums.com : how-distorted-can-a-minibrot-be ?
  20. math.stackexchange question " help-locating-mini-mandelbrots?