# name

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

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

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 ${\displaystyle \delta }$.

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

## are they exact copies?

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?

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 ?

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]

Measures

## How to compute size and orientation ?

### Size and orientation

Scaling for period-n windows of chaotic dynamical systems

The formula for the size and orientation:

${\displaystyle r={\frac {1}{\beta \Lambda _{p}^{2}}}}$



Input and related values:

• nucleus ${\displaystyle c}$ of a minibrot's main pseudocardioid
• period ${\displaystyle p}$ of a minibrot's main pseudocardioid
• ${\displaystyle \beta }$
• ${\displaystyle \Lambda _{n}}$
• ${\displaystyle \lambda _{n}}$

Output:

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

where

${\displaystyle z_{1}=0;z_{n+1}=z_{n}^{2}+c}$

${\displaystyle \lambda _{n}=2z_{n}}$

${\displaystyle \Lambda _{n}=\lambda _{2}\lambda _{3}\cdots \lambda _{n}}$

${\displaystyle \beta =1+\Lambda _{2}^{-1}+\Lambda _{3}^{-1}+\dots +\Lambda _{p}^{-1}}$



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 ${\displaystyle c+rc}$ with size approximately ${\displaystyle r^{2}}$" 	Claude Heiland-Allen



Example minibrots:

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

// mandelbrot-numerics -- numerical algorithms related to the Mandelbrot set
// Copyright (C) 2015-2018 Claude Heiland-Allen

#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

The area ${\displaystyle A}$ of the island of period p+1 is approximated by the formula by Robert Munafo:[16]

${\displaystyle A_{p+1}\approx 0.05{\frac {sin^{2}({\frac {\pi }{p}})}{p^{4}}}}$

## How to measure the distorsion ?

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]

${\displaystyle ecc={\frac {cusp_{B}}{cusp_{A}}}}$

## How distorted can a minibrot be?

What is maxima distortion of minibrot ?[19]