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{{short description|Fractal named after mathematician Benoit Mandelbrot}}

{{Use dmy dates|date=February 2020}}

[[File:Mandel zoom 00 mandelbrot set.jpg|322px|thumb|The Mandelbrot set (black) within a continuously colored environment|alt=]]

-- Please don't remove it -->

-- 请不要把它拿走 --

[[File:Progressive infinite iterations of the 'Nautilus' section of the Mandelbrot Set.ogv|thumb|Progressive infinite iterations of the "Nautilus" section of the Mandelbrot Set rendered using webGL]]

Progressive infinite iterations of the "Nautilus" section of the Mandelbrot Set rendered using webGL

使用 webGL 呈现的 Mandelbrot 集的“ Nautilus”部分的渐进无限迭代

[[File:Animation of the growth of the Mandelbrot set as you iterate towards infinity.gif|thumb|Mandelbrot animation based on a static number of iterations per pixel]]

Mandelbrot animation based on a static number of iterations per pixel

基于每像素迭代次数的静态 Mandelbrot 动画

[[File:Mandelbrot set image.png|thumb|Mandelbrot set detail]]

Mandelbrot set detail

曼德布罗特集合细节

The '''Mandelbrot set''' is the [[set (mathematics)|set]] of [[complex number]]s <math>c</math> for which the function <math>f_c(z)=z^2+c</math> does not [[diverge (stability theory)|diverge]] when [[Iteration|iterated]] from <math>z=0</math>, i.e., for which the sequence <math>f_c(0)</math>, <math>f_c(f_c(0))</math>, etc., remains bounded in absolute value.

The Mandelbrot set is the set of complex numbers <math>c</math> for which the function <math>f_c(z)=z^2+c</math> does not diverge when iterated from <math>z=0</math>, i.e., for which the sequence <math>f_c(0)</math>, <math>f_c(f_c(0))</math>, etc., remains bounded in absolute value.

Mandelbrot 集是复数数学 c / math 的集合，其中函数数学 f c (z) z ^ 2 + c / math 在从数学 z0 / math 迭代时不会发散，也就是说，其数学顺序 f c (0) / math、 math f c (f c (0) / math 等在绝对值中保持有界。

[[File:Mandelbrot sequence new.gif|thumb|Zooming into the Mandelbrot set]]

Zooming into the Mandelbrot set

放大到曼德尔勃特集合

Its definition is credited to [[Adrien Douady]] who named it in tribute to the [[mathematician]] [[Benoit Mandelbrot]].<ref name="John H. Hubbard 1985">Adrien Douady and John H. Hubbard, ''Etude dynamique des polynômes complexes'', Prépublications mathémathiques d'Orsay 2/4 (1984 / 1985)</ref> The set is connected to a [[Julia set]], and related Julia sets produce similarly complex [[fractal]] shapes.

Its definition is credited to Adrien Douady who named it in tribute to the mathematician Benoit Mandelbrot. The set is connected to a Julia set, and related Julia sets produce similarly complex fractal shapes.

它的定义归功于数学家阿德里安 · 杜阿迪，杜阿迪将其命名为本华·曼德博。这个集合与一个 Julia 集合相连，相关的 Julia 集合产生类似的复杂的分形形状。

Mandelbrot set images may be created by sampling the complex numbers and testing, for each sample point <math>c</math>, whether the sequence <math>f_c(0), f_c(f_c(0)),\dotsc</math> [[Sequence#Bounded|goes to infinity]] (in practice – whether it leaves some predetermined bounded neighborhood of 0 after a predetermined number of iterations). Treating the [[Real numbers|real]] and [[Imaginary number|imaginary part]]s of <math>c</math> as [[image coordinate]]s on the [[complex plane]], pixels may then be coloured according to how soon the sequence <math>|f_c(0)|, |f_c(f_c(0))|,\dotsc</math> crosses an arbitrarily chosen threshold, with a special color (usually black) used for the values of <math>c</math> for which the sequence has not crossed the threshold after the predetermined number of iterations (this is necessary to clearly distinguish the Mandelbrot set image from the image of its complement). If <math>c</math> is held constant and the initial value of <math>z</math>—denoted by <math>z_0</math>—is variable instead, one obtains the [[Julia set#Quadratic polynomials|corresponding Julia set]] for each point <math>c</math> in the [[parameter space]] of the simple function.

Mandelbrot set images may be created by sampling the complex numbers and testing, for each sample point <math>c</math>, whether the sequence <math>f_c(0), f_c(f_c(0)),\dotsc</math> goes to infinity (in practice – whether it leaves some predetermined bounded neighborhood of 0 after a predetermined number of iterations). Treating the real and imaginary parts of <math>c</math> as image coordinates on the complex plane, pixels may then be coloured according to how soon the sequence <math>|f_c(0)|, |f_c(f_c(0))|,\dotsc</math> crosses an arbitrarily chosen threshold, with a special color (usually black) used for the values of <math>c</math> for which the sequence has not crossed the threshold after the predetermined number of iterations (this is necessary to clearly distinguish the Mandelbrot set image from the image of its complement). If <math>c</math> is held constant and the initial value of <math>z</math>—denoted by <math>z_0</math>—is variable instead, one obtains the corresponding Julia set for each point <math>c</math> in the parameter space of the simple function.

Mandelbrot 集合图像可以通过对复数进行采样和测试来创建，对于每个样本点的数学 c / math，是否序列数学 f c (0) ，f c (f c (0)) ， dotsc / math 到无穷大(实际上-是否在预定的迭代次数之后离开某个预定的有界邻域0)。将数学 c / math 的实部和虚部视为复平面上的图像坐标，然后可以根据序列 math | f c (0) | ，| f c (f c (0) | ， dotsc / math 跨越任意选择的阈值的速度对像素进行着色，并使用一种特殊颜色(通常为黑色)用于数学 c / math 的值，在预先确定的迭代次数之后，序列没有跨越阈值(这是清楚地区分 Mandelbrot 设置的图像和其补码的图像所必需的)。如果 math c / math 是常数，而 math z / math 的初始值是可变的，则在简单函数的参数空间中得到每个点的数学 c / math 的 Julia 集。

Images of the Mandelbrot set exhibit an elaborate and infinitely complicated [[Boundary (topology)|boundary]] that reveals progressively ever-finer [[Recursion|recursive]] detail at increasing magnifications. In other words, the boundary of the Mandelbrot set is a ''[[fractal|fractal curve]]''. The "style" of this repeating detail depends on the region of the set being examined. The set's boundary also incorporates smaller versions of the main shape, so the [[fractal]] property of [[self-similarity]] applies to the entire set, and not just to its parts.

Images of the Mandelbrot set exhibit an elaborate and infinitely complicated boundary that reveals progressively ever-finer recursive detail at increasing magnifications. In other words, the boundary of the Mandelbrot set is a fractal curve. The "style" of this repeating detail depends on the region of the set being examined. The set's boundary also incorporates smaller versions of the main shape, so the fractal property of self-similarity applies to the entire set, and not just to its parts.

曼德布洛特集合的图像展示了一个精致而无限复杂的边界，在不断增大的放大倍数下，逐渐展现出越来越精细的递归细节。换句话说，Mandelbrot 集合的边界是一条分形曲线。这个重复细节的“样式”取决于被检查的集合的区域。集合的边界也包含了主要形状的较小版本，因此自相似性的分形特性适用于整个集合，而不仅仅是它的部分。

The Mandelbrot set has become popular outside [[mathematics]] both for its aesthetic appeal and as an example of a complex structure arising from the application of simple rules. It is one of the best-known examples of [[mathematical visualization]] and [[mathematical beauty]].

The Mandelbrot set has become popular outside mathematics both for its aesthetic appeal and as an example of a complex structure arising from the application of simple rules. It is one of the best-known examples of mathematical visualization and mathematical beauty.

曼德布洛特集合因其美学吸引力和作为应用简单规则而产生的复杂结构的一个例子，已经成为数学界以外的流行集合。它是数学可视化和数学美的最著名的例子之一。

==History==

[[File:Mandel.png|322px|right|thumb|The first published picture of the Mandelbrot set, by [[Robert W. Brooks]] and Peter Matelski in 1978]]

The first published picture of the Mandelbrot set, by [[Robert W. Brooks and Peter Matelski in 1978]]

曼德尔布洛特集合的第一张出版图片，作者[[罗伯特 · w · 布鲁克斯和彼得 · 马特尔斯基，1978]]

The Mandelbrot set has its origin in [[complex dynamics]], a field first investigated by the [[French mathematicians]] [[Pierre Fatou]] and [[Gaston Julia]] at the beginning of the 20th century. This fractal was first defined and drawn in 1978 by [[Robert W. Brooks]] and Peter Matelski as part of a study of [[Kleinian group]]s.<ref>Robert Brooks and Peter Matelski, ''The dynamics of 2-generator subgroups of PSL(2,C)'', in {{cite book|url=http://www.math.harvard.edu/archive/118r_spring_05/docs/brooksmatelski.pdf|title=Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference|author=Irwin Kra|date=1 May 1981|publisher=Princeton University Press|others=[[Bernard Maskit]]|isbn=0-691-08267-7|editor=Irwin Kra|access-date=1 July 2019|archive-url=https://web.archive.org/web/20190728201429/http://www.math.harvard.edu/archive/118r_spring_05/docs/brooksmatelski.pdf|archive-date=28 July 2019|url-status=dead}}</ref> On 1 March 1980, at [[IBM]]'s [[Thomas J. Watson Research Center]] in [[Yorktown Heights, New York|Yorktown Heights]], [[New York (state)|New York]], [[Benoit Mandelbrot]] first saw a visualization of the set.<ref name="bf">{{cite web |url=http://sprott.physics.wisc.edu/pubs/paper311.pdf |title=Biophilic Fractals and the Visual Journey of Organic Screen-savers |author=R.P. Taylor & J.C. Sprott |accessdate=1 January 2009 |year=2008 |work=Nonlinear Dynamics, Psychology, and Life Sciences, Vol. 12, No. 1 |publisher=Society for Chaos Theory in Psychology & Life Sciences }}</ref>

The Mandelbrot set has its origin in complex dynamics, a field first investigated by the French mathematicians Pierre Fatou and Gaston Julia at the beginning of the 20th century. This fractal was first defined and drawn in 1978 by Robert W. Brooks and Peter Matelski as part of a study of Kleinian groups. On 1 March 1980, at IBM's Thomas J. Watson Research Center in Yorktown Heights, New York, Benoit Mandelbrot first saw a visualization of the set.

曼德尔布洛特集合起源于20世纪初由法国数学家 Pierre Fatou 和 Gaston Julia 首先研究的复动力学。1978年，罗伯特 · w · 布鲁克斯和彼得 · 马特尔斯基作为 Kleinian 群研究的一部分，首次定义并绘制了这种分形。1980年3月1日，在位于约克敦海茨的 IBM 汤玛士·Ｊ·华生研究中心，本华·曼德博第一次看到了这个场景的可视化。

Mandelbrot studied the [[parameter space]] of [[quadratic polynomial]]s in an article that appeared in 1980.<ref>Benoit Mandelbrot, ''Fractal aspects of the iteration of <math>z\mapsto\lambda z(1-z)</math> for complex <math>\lambda, z</math>'', ''Annals of the New York Academy of Sciences'' '''357''', 249/259</ref> The mathematical study of the Mandelbrot set really began with work by the mathematicians [[Adrien Douady]] and [[John H. Hubbard]] (1985),<ref name="John H. Hubbard 1985"/> who established many of its fundamental properties and named the set in honor of Mandelbrot for his influential work in [[fractal geometry]].

Mandelbrot studied the parameter space of quadratic polynomials in an article that appeared in 1980. The mathematical study of the Mandelbrot set really began with work by the mathematicians Adrien Douady and John H. Hubbard (1985), who established many of its fundamental properties and named the set in honor of Mandelbrot for his influential work in fractal geometry.

曼德布洛特在1980年的一篇文章中研究了二次多项式的参数空间。对曼德布洛特集合的数学研究真正开始于数学家阿德里安 · 杜阿迪和约翰 · h · 哈伯德(1985)的工作，他们确立了集合的许多基本性质，并以曼德布洛特在分形几何中的影响力为集合命名。

The mathematicians [[Heinz-Otto Peitgen]] and [[Peter Richter]] became well known for promoting the set with photographs, books (1986),<ref>{{cite book |title=The Beauty of Fractals |last=Peitgen |first=Heinz-Otto |author2=Richter Peter |year=1986 |publisher=Springer-Verlag |location=Heidelberg |isbn=0-387-15851-0 |title-link=The Beauty of Fractals }}</ref> and an internationally touring exhibit of the German [[Goethe-Institut]] (1985).<ref>[[Frontiers of Chaos]], Exhibition of the Goethe-Institut by H.O. Peitgen, P. Richter, H. Jürgens, M. Prüfer, D.Saupe. Since 1985 shown in over 40 countries.</ref><ref>{{cite book |title=Chaos: Making a New Science |last=Gleick |first=James |year=1987 |publisher=Cardinal |location=London |pages=229 |title-link=Chaos: Making a New Science }}</ref>

The mathematicians Heinz-Otto Peitgen and Peter Richter became well known for promoting the set with photographs, books (1986), and an internationally touring exhibit of the German Goethe-Institut (1985).

数学家海因茨-奥托 · 佩特根和彼得 · 里希特用照片、书籍(1986年)和德国歌德学院的国际巡回展览(1985年)来推广这套作品，从而闻名于世。

The cover article of the August 1985 ''[[Scientific American]]'' introduced a wide audience to the [[algorithm]] for computing the Mandelbrot set. The cover featured an image located at [https://mandelbrot-svelte.netlify.com/#{%22pos%22:{%22x%22:-0.909,%22y%22:-0.275},%22zoom%22:10000} -0.909 + -0.275] and was created by Peitgen et al.<ref>{{cite magazine |title= Computer Recreations, August 1985; A computer microscope zooms in for a look at the most complex object in mathematics |last=Dewdney |first=A. K. |year=1985 |magazine=Scientific American |url=https://www.scientificamerican.com/media/inline/blog/File/Dewdney_Mandelbrot.pdf}}</ref><ref>{{cite book |title=Fractals: The Patterns of Chaos |author=John Briggs |year=1992 |page=80}}</ref> The Mandelbrot set became prominent in the mid-1980s as a computer [[Demo (computer programming)|graphics demo]], when [[personal computer]]s became powerful enough to plot and display the set in high resolution.<ref>{{cite magazine |last=Pountain |first=Dick |date=September 1986 |title= Turbocharging Mandelbrot |url=https://archive.org/stream/byte-magazine-1986-09/1986_09_BYTE_11-09_The_68000_Family#page/n370/mode/1up |magazine= [[Byte (magazine) |Byte]] |access-date=11 November 2015 }}</ref>

The cover article of the August 1985 Scientific American introduced a wide audience to the algorithm for computing the Mandelbrot set. The cover featured an image located at [https://mandelbrot-svelte.netlify.com/#{%22pos%22:{%22x%22:-0.909,%22y%22:-0.275},%22zoom%22:10000} -0.909 + -0.275] and was created by Peitgen et al. The Mandelbrot set became prominent in the mid-1980s as a computer graphics demo, when personal computers became powerful enough to plot and display the set in high resolution.

1985年8月《科学美国人》的封面文章向广大读者介绍了计算 Mandelbrot 集的算法。封面上的图片位于[ https://mandelbrot-svelte.netlify.com/# {“ pos” : “ x” :-0.909，“ y” :-0.275} ，“ zoom” : 10000}-0.909 +-0.275] ，由 petgen 等人制作。在20世纪80年代中期，当个人电脑变得足够强大，可以以高分辨率绘制和显示这个集合时，曼德尔布洛特集作为计算机图形学的一个演示而变得突出。

The work of Douady and Hubbard coincided with a huge increase in interest in complex dynamics and [[abstract mathematics]], and the study of the Mandelbrot set has been a centerpiece of this field ever since. An exhaustive list of all who have contributed to the understanding of this set since then is long but would include [[Mikhail Lyubich]],<ref>{{cite journal

The work of Douady and Hubbard coincided with a huge increase in interest in complex dynamics and abstract mathematics, and the study of the Mandelbrot set has been a centerpiece of this field ever since. An exhaustive list of all who have contributed to the understanding of this set since then is long but would include Mikhail Lyubich,<ref>{{cite journal

Douady 和哈伯德的工作与人们对复动力学和抽象数学兴趣的巨大增长相吻合，从那时起，对曼德布洛特集合的研究一直是这个领域的核心。一个详尽的名单，所有的人都为理解这个集合做出了贡献，从那时起，这个名单是很长的，但是包括 Mikhail Lyubich，引用{ cite journal

| author = Lyubich, Mikhail

| author = Lyubich, Mikhail

作者 Lyubich，Mikhail

| title = Six Lectures on Real and Complex Dynamics

| title = Six Lectures on Real and Complex Dynamics

6个关于真实和复动力学的讲座

| version =

| version =

版本

| date = May–June 1999

| date = May–June 1999

| 日期1999年5月至6月

| url = http://citeseer.ist.psu.edu/cache/papers/cs/28564/http:zSzzSzwww.math.sunysb.eduzSz~mlyubichzSzlectures.pdf/

| url = http://citeseer.ist.psu.edu/cache/papers/cs/28564/http:zSzzSzwww.math.sunysb.eduzSz~mlyubichzSzlectures.pdf/

Http://citeseer.ist.psu.edu/cache/papers/cs/28564/http:zszzszwww.math.sunysb.eduzsz~mlyubichzszlectures.pdf/

| accessdate = 2007-04-04 }}</ref><ref>{{cite journal

| accessdate = 2007-04-04 }}</ref><ref>{{cite journal

| accessdate 2007-04-04} / ref { cite journal

| last = Lyubich

| last = Lyubich

最后一个 Lyubich

| first = Mikhail

| first = Mikhail

首先是米哈伊尔

| authorlink = Mikhail Lyubich

| authorlink = Mikhail Lyubich

作者: Mikhail Lyubich

| title = Regular and stochastic dynamics in the real quadratic family

| title = Regular and stochastic dynamics in the real quadratic family

| 标题实二次族中的正则和随机动力学

| journal = Proceedings of the National Academy of Sciences of the United States of America

| journal = Proceedings of the National Academy of Sciences of the United States of America

华尔街美国国家科学院院刊

| volume = 95

| volume = 95

第95卷

| issue =24

| issue =24

第24期

| pages = 14025–14027

| pages = 14025–14027

第14025-14027页

| date=November 1998

| date=November 1998

1998年11月

| url = http://www.pnas.org/cgi/reprint/95/24/14025.pdf

| url = http://www.pnas.org/cgi/reprint/95/24/14025.pdf

Http://www.pnas.org/cgi/reprint/95/24/14025.pdf

| doi = 10.1073/pnas.95.24.14025

| doi = 10.1073/pnas.95.24.14025

10.1073 / pnas. 95.24.14025

| accessdate = 2007-04-04

| accessdate = 2007-04-04

2007-04-04

| pmid = 9826646

| pmid = 9826646

9826646

| pmc = 24319 | bibcode =1998PNAS...9514025L

| pmc = 24319 | bibcode =1998PNAS...9514025L

24319 | bibcode 1998PNAS... 9514025L

}}</ref> [[Curtis T. McMullen|Curt McMullen]], [[John Milnor]], [[Mitsuhiro Shishikura]] and [[Jean-Christophe Yoccoz]].

}}</ref> Curt McMullen, John Milnor, Mitsuhiro Shishikura and Jean-Christophe Yoccoz.

}}</ref> Curt McMullen, John Milnor, Mitsuhiro Shishikura and Jean-Christophe Yoccoz.

==Formal definition==

The Mandelbrot set is the set of values of ''c'' in the [[complex plane]] for which the [[Orbit (dynamics)|orbit]] of 0 under [[Iterated function|iteration]] of the [[quadratic map]]

The Mandelbrot set is the set of values of c in the complex plane for which the orbit of 0 under iteration of the quadratic map

Mandelbrot 集是复平面上0的轨道在二次映射迭代下的 c 值集合

:<math>z_{n+1} = z_n^2 + c</math>

<math>z_{n+1} = z_n^2 + c</math>

数学 z { n + 1} n ^ 2 + c / math

remains [[Bounded sequence|bounded]].<ref>{{cite web|url=http://math.bu.edu/DYSYS/explorer/def.html|title=Mandelbrot Set Explorer: Mathematical Glossary|accessdate=2007-10-07}}</ref> Thus, a complex number ''c'' is a member of the Mandelbrot set if, when starting with ''z''0 = 0 and applying the iteration repeatedly, the [[absolute value]] of ''z''''n'' remains bounded for all ''n''>0.

remains bounded. Thus, a complex number c is a member of the Mandelbrot set if, when starting with z0 = 0 and applying the iteration repeatedly, the absolute value of zn remains bounded for all n>0.

保持有界。因此，复数 c 是 Mandelbrot 集合的一个成员，如果从 z sub 0 / sub 0开始并重复迭代，z sub n / sub 的绝对值对所有 n 0保持有界。

For example, for ''c''=1, the sequence is 0, 1, 2, 5, 26, ..., which tends to [[infinity]], so 1 is not an element of the Mandelbrot set. On the other hand, for ''c''=−1, the sequence is 0, −1, 0, −1, 0, ..., which is bounded, so −1 does belong to the set.

For example, for c=1, the sequence is 0, 1, 2, 5, 26, ..., which tends to infinity, so 1 is not an element of the Mandelbrot set. On the other hand, for c=−1, the sequence is 0, −1, 0, −1, 0, ..., which is bounded, so −1 does belong to the set.

例如，对于 c1，序列是0,1,2,5,26，... ，趋于无穷大，所以1不是 Mandelbrot 集合的元素。另一方面，对于 c-1，序列是0,-1,0,-1,0，... ，这是有界的，所以-1确实属于集合。

[[File:Mandelset hires.png|right|thumb|322px|A mathematician's depiction of the Mandelbrot set ''M''. A point ''c'' is colored black if it belongs to the set, and white if not. Re[c] and Im[c] denote the real and imaginary parts of ''c'', respectively.]]

A mathematician's depiction of the Mandelbrot set M. A point c is colored black if it belongs to the set, and white if not. Re[c] and Im[c] denote the real and imaginary parts of c, respectively.

一位数学家对曼德布罗特集 m 的描述。如果一个点 c 属于这个集合，它就是黑色的，如果不属于这个集合，它就是白色的。Re [ c ]和 Im [ c ]分别表示 c 的实部和虚部。

The Mandelbrot set can also be defined as the [[connectedness locus]] of a family of [[polynomial]]s.

The Mandelbrot set can also be defined as the connectedness locus of a family of polynomials.

Mandelbrot 集也可以定义为一族多项式的连通轨迹。

==Basic properties==

The Mandelbrot set is a [[compact set]], since it is [[closed set|closed]] and contained in the [[closed disk]] of radius 2 around the [[Origin (mathematics)|origin]]. More specifically, a point <math>c</math> belongs to the Mandelbrot set if and only if

The Mandelbrot set is a compact set, since it is closed and contained in the closed disk of radius 2 around the origin. More specifically, a point <math>c</math> belongs to the Mandelbrot set if and only if

Mandelbrot 集是一个紧集合，因为它是封闭的，并且包含在原点周围半径为2的封闭圆盘中。更具体地说，一个点数学 c / math 属于 Mandelbrot 集当且仅当

:<math>|P_c^n(0)|\leq 2</math> for all <math>n\geq 0.</math>

<math>|P_c^n(0)|\leq 2</math> for all <math>n\geq 0.</math>

数学 | p c ^ n (0) | leq 2 / math for all math n geq 0. / math

In other words, if the [[absolute value]] of <math>P_c^n(0)</math> ever becomes larger than 2, the sequence will escape to infinity.

In other words, if the absolute value of <math>P_c^n(0)</math> ever becomes larger than 2, the sequence will escape to infinity.

换句话说，如果 math 的 p c ^ n (0) / math 的绝对值大于2，这个序列将转义到无穷大。

[[File:Verhulst-Mandelbrot-Bifurcation.jpg|thumb|Correspondence between the Mandelbrot set and the [[bifurcation diagram]] of the [[logistic map]]]]

Correspondence between the Mandelbrot set and the [[bifurcation diagram of the logistic map]]

曼德布洛特集合与[[逻辑地图的分枝图]]之间的通信

[[File:Logistic Map Bifurcations Underneath Mandelbrot Set.gif|thumb|With <math>z_{n}</math> iterates plotted on the vertical axis, the Mandelbrot set can be seen to bifurcate where the set is finite]]

With <math>z_{n}</math> iterates plotted on the vertical axis, the Mandelbrot set can be seen to bifurcate where the set is finite

在垂直轴上绘制数学 z { n } / 数学迭代，可以看到 Mandelbrot 集在集合有限的地方分叉

The [[intersection (set theory)|intersection]] of <math>M</math> with the real axis is precisely the interval [−2, 1/4]. The parameters along this interval can be put in one-to-one correspondence with those of the real [[logistic map|logistic family]],

The intersection of <math>M</math> with the real axis is precisely the interval [−2, 1/4]. The parameters along this interval can be put in one-to-one correspondence with those of the real logistic family,

数学 m / math 与实轴的交点正是区间[-2,1 / 4]。这个区间上的参数可以和那些真正的逻辑斯蒂家族的参数放在一个双射里,

:<math>x_{n+1} = r x_n(1-x_n),\quad r\in[1,4].</math>

<math>x_{n+1} = r x_n(1-x_n),\quad r\in[1,4].</math>

数学 x { n + 1} r xn (1-xn) ，[1,4] . / math

The correspondence is given by

The correspondence is given by

这些信件是由

:<math>z = r\left(\frac12 - x\right),

<math>z = r\left(\frac12 - x\right),

左边(右边)

\quad

\quad

[咒语]

c = \frac{r}{2}\left(1-\frac{r}{2}\right).</math>

c = \frac{r}{2}\left(1-\frac{r}{2}\right).</math>

左(1- frac { r }{2}右) . / math

In fact, this gives a correspondence between the entire [[parameter space]] of the logistic family and that of the Mandelbrot set.

In fact, this gives a correspondence between the entire parameter space of the logistic family and that of the Mandelbrot set.

实际上，这给出了逻辑斯谛族的整个参数空间与 Mandelbrot 集的整个参数空间之间的对应关系。

Douady and Hubbard have shown that the Mandelbrot set is [[connected space|connected]]. In fact, they constructed an explicit [[holomorphic function|conformal isomorphism]] between the complement of the Mandelbrot set and the complement of the [[closed unit disk]]. Mandelbrot had originally conjectured that the Mandelbrot set is [[Disconnected (topology)|disconnected]]. This conjecture was based on computer pictures generated by programs that are unable to detect the thin filaments connecting different parts of <math>M</math>. Upon further experiments, he revised his conjecture, deciding that <math>M</math> should be connected. There also exists a [[Topology|topological]] proof to the connectedness that was discovered in 2001 by [[Jeremy Kahn]].<ref>{{Cite web|url=http://www.math.brown.edu/~kahn/mconn.pdf|title=The Mandelbrot Set is Connected: a Topological Proof|last=Kahn|first=Jeremy|date=8 August 2001}}</ref>

Douady and Hubbard have shown that the Mandelbrot set is connected. In fact, they constructed an explicit conformal isomorphism between the complement of the Mandelbrot set and the complement of the closed unit disk. Mandelbrot had originally conjectured that the Mandelbrot set is disconnected. This conjecture was based on computer pictures generated by programs that are unable to detect the thin filaments connecting different parts of <math>M</math>. Upon further experiments, he revised his conjecture, deciding that <math>M</math> should be connected. There also exists a topological proof to the connectedness that was discovered in 2001 by Jeremy Kahn.

道迪和哈伯德已经证明了曼德尔布洛特集是连通的。实际上，他们在 Mandelbrot 集的补集和闭单位圆盘的补集之间构造了一个显式的共形同构。曼德布洛特最初猜测，曼德布洛特集合是不连贯的。这个猜想是基于程序生成的计算机图片，这些程序无法检测到连接数学 m / math 的不同部分的细丝。经过进一步的实验，他修改了他的猜想，决定数学 m / math 应该是连通的。2001年 Jeremy Kahn 发现的连通性也有一个拓扑证明。

[[File:Wakes near the period 1 continent in the Mandelbrot set.png|thumbnail|right|External rays of wakes near the period 1 continent in the Mandelbrot set]]

External rays of wakes near the period 1 continent in the Mandelbrot set

在曼德布洛特集合的1大陆周期附近的外部尾迹射线

The dynamical formula for the [[uniformization theorem|uniformisation]] of the complement of the Mandelbrot set, arising from Douady and Hubbard's proof of the connectedness of <math>M</math>, gives rise to [[external ray]]s of the Mandelbrot set. These rays can be used to study the Mandelbrot set in combinatorial terms and form the backbone of the [[Jean-Christophe Yoccoz#Mathematical work|Yoccoz parapuzzle]].<ref>''The Mandelbrot set, theme and variations''. Tan, Lei. Cambridge University Press, 2000. {{isbn|978-0-521-77476-5}}. Section 2.1, "Yoccoz para-puzzles", [https://books.google.com/books?id=-a_DsYXquVkC&pg=PA121 p. 121]</ref>

The dynamical formula for the uniformisation of the complement of the Mandelbrot set, arising from Douady and Hubbard's proof of the connectedness of <math>M</math>, gives rise to external rays of the Mandelbrot set. These rays can be used to study the Mandelbrot set in combinatorial terms and form the backbone of the Yoccoz parapuzzle.

曼德布洛特集合补集均匀化的动力学公式产生于 Douady 和哈伯德关于数学 m / math 连通性的证明，它产生了曼德布洛特集合的外部射线。这些射线可以用组合的方式来研究曼德布洛特集合，并形成 Yoccoz parapuzzle 的骨架。

The [[boundary (topology)|boundary]] of the Mandelbrot set is exactly the [[bifurcation locus]] of the quadratic family; that is, the set of parameters <math>c</math> for which the dynamics changes abruptly under small changes of <math>c.</math> It can be constructed as the limit set of a sequence of [[algebraic curves|plane algebraic curves]], the ''Mandelbrot curves'', of the general type known as [[polynomial lemniscate]]s. The Mandelbrot curves are defined by setting ''p''0 = ''z'', ''p''''n''+1 = ''p''''n''2 + ''z'', and then interpreting the set of points |''p''''n''(''z'')| = 2 in the complex plane as a curve in the real [[Cartesian coordinate system|Cartesian plane]] of degree 2''n''+1 in ''x'' and ''y''. These algebraic curves appear in images of the Mandelbrot set computed using the "escape time algorithm" mentioned below.

The boundary of the Mandelbrot set is exactly the bifurcation locus of the quadratic family; that is, the set of parameters <math>c</math> for which the dynamics changes abruptly under small changes of <math>c.</math> It can be constructed as the limit set of a sequence of plane algebraic curves, the Mandelbrot curves, of the general type known as polynomial lemniscates. The Mandelbrot curves are defined by setting p0 = z, pn+1 = pn2 + z, and then interpreting the set of points |pn(z)| = 2 in the complex plane as a curve in the real Cartesian plane of degree 2n+1 in x and y. These algebraic curves appear in images of the Mandelbrot set computed using the "escape time algorithm" mentioned below.

Mandelbrot 集的边界正是二次族的分岔轨迹，即参数数学 c / math 在数学 c / math 小的变化下动力学突变的集合，它可以构造成一系列平面代数曲线的极限集，即 Mandelbrot 曲线，一般称为多项式双曲线。通过设置 p 子0 / sub z，p 子 n + 1 / sub p 子 n / sub sup 2 / sup + z 来定义 Mandelbrot 曲线，然后将复平面上的点 | p 子 n / sub (z) | 2解释为 x 和 y 中实笛卡尔平面上的一条曲线。这些代数曲线出现在使用下面提到的“逃逸时间算法”计算出来的 Mandelbrot 集的图像中。

==Other properties==

===Main cardioid and period bulbs===





! -- douady rabbit 链接到这里 --

[[File:Mandelbrot Set – Periodicities coloured.png|right|thumb|Periods of hyperbolic components]]

Periods of hyperbolic components

双曲分量周期

Upon looking at a picture of the Mandelbrot set, one immediately notices the large [[cardioid]]-shaped region in the center. This ''main cardioid''

Upon looking at a picture of the Mandelbrot set, one immediately notices the large cardioid-shaped region in the center. This main cardioid

看到曼德布洛特集合的图片，人们会立即注意到中心的大心形区域。这是主心线

is the region of parameters <math>c</math> for which <math>P_c</math> has an [[Periodic points of complex quadratic mappings|attracting fixed point]]. It consists of all parameters of the form

is the region of parameters <math>c</math> for which <math>P_c</math> has an attracting fixed point. It consists of all parameters of the form

是参数数学 c / math 的区域，对于这个区域，数学 c / math 有一个吸引人的不动点。它由表单的所有参数组成

:<math> c = \frac\mu2\left(1-\frac\mu2\right)</math>

<math> c = \frac\mu2\left(1-\frac\mu2\right)</math>

数学 c frac mu2 left (1- frac mu2 right) / math

for some <math>\mu</math> in the [[open unit disk]].

for some <math>\mu</math> in the open unit disk.

在打开的单位磁盘上做数学题。

To the left of the main cardioid, attached to it at the point <math>c=-3/4</math>, a circular-shaped '''bulb''' is visible. This bulb consists of those parameters <math>c</math> for which <math>P_c</math> has an [[Periodic points of complex quadratic mappings|attracting cycle of period 2]]. This set of parameters is an actual circle, namely that of radius 1/4 around −1.

To the left of the main cardioid, attached to it at the point <math>c=-3/4</math>, a circular-shaped bulb is visible. This bulb consists of those parameters <math>c</math> for which <math>P_c</math> has an attracting cycle of period 2. This set of parameters is an actual circle, namely that of radius 1/4 around −1.

在主心线的左边，在数学 c-3 / 4 / math 的点上，可以看到一个圆形的灯泡。这个灯泡由那些数学 c / math 参数组成，数学 c / math 有一个周期2的吸引循环。这组参数是一个实际的圆，即半径1 / 4围绕-1的圆。

There are infinitely many other bulbs tangent to the main cardioid: for every rational number <math>\tfrac{p}{q}</math>, with ''p'' and ''q'' [[coprime]], there is such a bulb that is tangent at the parameter

There are infinitely many other bulbs tangent to the main cardioid: for every rational number <math>\tfrac{p}{q}</math>, with p and q coprime, there is such a bulb that is tangent at the parameter

有无穷多个与主心脏线相切的灯泡: 对于每个有理数 math tfrac { q } / math，有 p 和 q 的互素，有这样一个灯泡在参数处是相切的

:<math> c_{\frac{p}{q}} = \frac{e^{2\pi i\frac pq}}2\left(1-\frac{e^{2\pi i\frac pq}}2\right).</math>

<math> c_{\frac{p}{q}} = \frac{e^{2\pi i\frac pq}}2\left(1-\frac{e^{2\pi i\frac pq}}2\right).</math>

2(1- frac ^ 2 pi i frac pq }2 right) . / math

[[File:Animated cycle.gif|left|thumb|200px|Attracting cycle in 2/5-bulb plotted over [[Julia set]] (animation)]]This bulb is called the ''<math>\tfrac{p}{q}</math>-bulb'' of the Mandelbrot set. It consists of parameters that have an attracting cycle of period <math>q</math> and combinatorial rotation number <math>\tfrac{p}{q}</math>. More precisely, the <math>q</math> periodic [[Classification of Fatou components|Fatou components]] containing the attracting cycle all touch at a common point (commonly called the ''<math>\alpha</math>-fixed point''). If we label these components <math>U_0,\dots,U_{q-1}</math> in counterclockwise orientation, then <math>P_c</math> maps the component <math>U_j</math> to the component <math>U_{j+p\,(\operatorname{mod} q)}</math>.

Attracting cycle in 2/5-bulb plotted over [[Julia set (animation)]]This bulb is called the <math>\tfrac{p}{q}</math>-bulb of the Mandelbrot set. It consists of parameters that have an attracting cycle of period <math>q</math> and combinatorial rotation number <math>\tfrac{p}{q}</math>. More precisely, the <math>q</math> periodic Fatou components containing the attracting cycle all touch at a common point (commonly called the <math>\alpha</math>-fixed point). If we label these components <math>U_0,\dots,U_{q-1}</math> in counterclockwise orientation, then <math>P_c</math> maps the component <math>U_j</math> to the component <math>U_{j+p\,(\operatorname{mod} q)}</math>.

在2 / 5个灯泡中绘制出吸引周期[[[ Julia set (animation)]]]这个灯泡叫做 math tfrac { p }{ q } / math-bulb of the Mandelbrot set。它由具有周期数学 q / math 和组合旋转数学 tfrac { q }{ q } / math 的吸引循环的参数组成。更准确地说，包含吸引周期的 math q / math 周期 Fatou 分量在一个共同点(通常称为 math alpha / math-fixed point)全部接触。如果我们将这些组件标记为 math u 0， dots，u { q-1} / math，然后 math p / math 将组件 math u j / math 映射到组件 math u { j + p ，( operatorname { mod } q)} / math。

[[File:Juliacycles1.png|right|thumb|300px|Attracting cycles and [[Julia set]]s for parameters in the 1/2, 3/7, 2/5, 1/3, 1/4, and 1/5 bulbs]]

Attracting cycles and [[Julia sets for parameters in the 1/2, 3/7, 2/5, 1/3, 1/4, and 1/5 bulbs]]

吸引周期和[[ Julia 为1 / 2,3 / 7,2 / 5,1 / 3,1 / 4和1 / 5灯泡中的参数设置]]

[[File:Mandel rays.jpg|thumb|right|425px|Cycle periods and antennae]]

Cycle periods and antennae

周期和触角

The change of behavior occurring at <math>c_{\frac{p}{q}}</math> is known as a [[bifurcation theory|bifurcation]]: the attracting fixed point "collides" with a repelling period ''q''-cycle. As we pass through the bifurcation parameter into the <math>\tfrac{p}{q}</math>-bulb, the attracting fixed point turns into a repelling fixed point (the <math>\alpha</math>-fixed point), and the period ''q''-cycle becomes attracting.

The change of behavior occurring at <math>c_{\frac{p}{q}}</math> is known as a bifurcation: the attracting fixed point "collides" with a repelling period q-cycle. As we pass through the bifurcation parameter into the <math>\tfrac{p}{q}</math>-bulb, the attracting fixed point turns into a repelling fixed point (the <math>\alpha</math>-fixed point), and the period q-cycle becomes attracting.

在数学 c { frac { q }{ q } / math 中发生的行为变化被称为分支: 吸引不动点与排斥周期 q-cycle“碰撞”。当我们通过分岔参数进入 math { p }{ q } / math-bulb 时，吸引不动点变为排斥不动点(math alpha / math-fixed point) ，周期 q-cycle 变为吸引。

{{clear|left}}

===Hyperbolic components===

All the bulbs we encountered in the previous section were interior components of

All the bulbs we encountered in the previous section were interior components of

所有的灯泡，我们遇到在上一节的内部组成部分

the Mandelbrot set in which the maps <math>P_c</math> have an attracting periodic cycle. Such components are called ''hyperbolic components''.

the Mandelbrot set in which the maps <math>P_c</math> have an attracting periodic cycle. Such components are called hyperbolic components.

在曼德布洛特集合中，地图数学 p c / math 有一个吸引人的周期循环。这样的分量称为双曲分量。

It is conjectured that these are the ''only'' interior regions of <math>M</math>. This problem, known as ''density of hyperbolicity'', may be the most important open problem in the field of complex dynamics. Hypothetical non-hyperbolic components of the Mandelbrot set are often referred to as "queer" or ghost components.<ref>''Exploring the Mandelbrot set. The Orsay Notes'' by Adrien Douady and John H. Hubbard. page 12</ref><ref>Wolf Jung, March 2002, [http://www.mndynamics.com/papers/thesis.pdf Homeomorphisms on Edges of the Mandelbrot Set by Wolf Jung]</ref>

It is conjectured that these are the only interior regions of <math>M</math>. This problem, known as density of hyperbolicity, may be the most important open problem in the field of complex dynamics. Hypothetical non-hyperbolic components of the Mandelbrot set are often referred to as "queer" or ghost components.

推测这些是数学 m / math 唯一的内部区域。这个问题被称为双曲性密度，可能是复动力学领域中最重要的公开问题。曼德布洛特集合中假设的非双曲成分通常被称为“酷儿”或幽灵成分。

For ''real'' quadratic polynomials, this question was answered positively in the 1990s independently by Lyubich and by Graczyk and Świątek. (Note that hyperbolic components intersecting the real axis correspond exactly to periodic windows in the [[Bifurcation diagram|Feigenbaum diagram]]. So this result states that such windows exist near every parameter in the diagram.)

For real quadratic polynomials, this question was answered positively in the 1990s independently by Lyubich and by Graczyk and Świątek. (Note that hyperbolic components intersecting the real axis correspond exactly to periodic windows in the Feigenbaum diagram. So this result states that such windows exist near every parameter in the diagram.)

对于实二次多项式，这个问题在20世纪90年代由 Lyubich 和 Graczyk 以及 wi tek 独立给出了肯定的答案。(注意，与实轴相交的双曲分量精确对应于 Feigenbaum 图中的周期窗口。因此，这个结果表明，这样的窗口存在于图表中的每个参数附近

Not every hyperbolic component can be reached by a sequence of direct bifurcations from the main cardioid of the Mandelbrot set. However, such a component ''can'' be reached by a sequence of direct bifurcations from the main cardioid of a little Mandelbrot copy (see below).

Not every hyperbolic component can be reached by a sequence of direct bifurcations from the main cardioid of the Mandelbrot set. However, such a component can be reached by a sequence of direct bifurcations from the main cardioid of a little Mandelbrot copy (see below).

并不是每一个双曲分量都可以由 Mandelbrot 集的主心脏线的一系列直接分岔得到。然而，这样的分量可以通过一系列的直接分叉从一个小的 Mandelbrot 副本(见下文)的主心脏线得到。

Each of the hyperbolic components has a ''center'', which is a point ''c'' such that the inner Fatou domain for <math>P_c(z)</math> has a super-attracting cycle – that is, that the attraction is infinite (see the image [[:commons:File:Centers8.png|here]]). This means that the cycle contains the critical point 0, so that 0 is iterated back to itself after some iterations. We therefore have that <math>P_c</math>n<math>(0) = 0</math> for some ''n''. If we call this polynomial <math>Q^{n}(c)</math> (letting it depend on ''c'' instead of ''z''), we have that <math>Q^{n+1}(c) = Q^{n}(c)^{2} + c</math> and that the degree of <math>Q^{n}(c)</math> is <math>2^{n-1}</math>. We can therefore construct the centers of the hyperbolic components by successively solving the equations <math>Q^{n}(c) = 0, n = 1, 2, 3, ...</math>. The number of new centers produced in each step is given by Sloane's {{oeis|A000740}}.

Each of the hyperbolic components has a center, which is a point c such that the inner Fatou domain for <math>P_c(z)</math> has a super-attracting cycle – that is, that the attraction is infinite (see the image here). This means that the cycle contains the critical point 0, so that 0 is iterated back to itself after some iterations. We therefore have that <math>P_c</math>n<math>(0) = 0</math> for some n. If we call this polynomial <math>Q^{n}(c)</math> (letting it depend on c instead of z), we have that <math>Q^{n+1}(c) = Q^{n}(c)^{2} + c</math> and that the degree of <math>Q^{n}(c)</math> is <math>2^{n-1}</math>. We can therefore construct the centers of the hyperbolic components by successively solving the equations <math>Q^{n}(c) = 0, n = 1, 2, 3, ...</math>. The number of new centers produced in each step is given by Sloane's .

每一个双曲分量都有一个中心，这个中心是一个点 c，使得用于数学计算的内部 Fatou 区域具有一个超级吸引周期——也就是说，吸引力是无限的(见这里的图像)。这意味着该循环包含临界点0，因此在一些迭代之后，0被迭代回到自身。因此，我们得到了某些 n 的数学公式。如果我们称这个多项式为数学 q ^ { n }(c) / math (让它依赖于 c 而不是 z) ，我们有数学 q ^ { n + 1}(c) q ^ { n }(c) ^ {2} + c / math，数学 q ^ { n }(c) / math 的次数是2 ^ { n-1} / math。因此，我们可以通过连续求解数学方程 q ^ { n }(c)0，n 1,2,3，... / 数学来构造双曲分量的中心。每一步产生的新中心的数量是由斯隆的。

===Local connectivity===

[[File:Cactus model of Mandelbrot set.svg|right|thumb|Topological model of Mandelbrot set without mini Mandelbrot sets and Misiurewicz points (Cactus model)]]

Topological model of Mandelbrot set without mini Mandelbrot sets and Misiurewicz points (Cactus model)

无 mini Mandelbrot 集和 Misiurewicz 点的 Mandelbrot 集拓扑模型(Cactus 模型)

[[File:Lavaurs-12.png|right|thumb|Thurston model of Mandelbrot set (abstract Mandelbrot set)]]

Thurston model of Mandelbrot set (abstract Mandelbrot set)

曼德布洛特集合的瑟斯顿模型(抽象曼德布洛特集合)

It is conjectured that the Mandelbrot set is [[locally connected]]. This famous conjecture is known as ''MLC'' (for ''Mandelbrot locally connected''). By the work of [[Adrien Douady]] and [[John H. Hubbard]], this conjecture would result in a simple abstract "pinched disk" model of the Mandelbrot set. In particular, it would imply the important ''hyperbolicity conjecture'' mentioned above.

It is conjectured that the Mandelbrot set is locally connected. This famous conjecture is known as MLC (for Mandelbrot locally connected). By the work of Adrien Douady and John H. Hubbard, this conjecture would result in a simple abstract "pinched disk" model of the Mandelbrot set. In particular, it would imply the important hyperbolicity conjecture mentioned above.

推测 Mandelbrot 集是局部连通的。这个著名的猜想被称为 MLC (对于 Mandelbrot 局部连通)。通过阿德里安 · 杜阿迪和约翰 · h · 哈伯德的工作，这一猜想将导致一个简单抽象的曼德尔布洛特集合的“压缩圆盘”模型。特别地，它意味着上面提到的重要的双曲性猜想。

The work of [[Jean-Christophe Yoccoz]] established local connectivity of the Mandelbrot set at all finitely [[Renormalization|renormalizable]] parameters; that is, roughly speaking those contained only in finitely many small Mandelbrot copies.<ref name="yoccoz">{{citation

The work of Jean-Christophe Yoccoz established local connectivity of the Mandelbrot set at all finitely renormalizable parameters; that is, roughly speaking those contained only in finitely many small Mandelbrot copies.<ref name="yoccoz">{{citation

让-克里斯托夫·约科兹的工作建立了在所有有限可重整化参数下的曼德布洛特集的局部连通性; 也就是说，粗略地说，这些连通性只包含在有限多个小的曼德布洛特副本中。文件名“ yoccos”{ citation

| last = Hubbard | first = J. H.

| last = Hubbard | first = J. H.

最后一个哈伯德 | 第一个 j. h。

| contribution = Local connectivity of Julia sets and bifurcation loci: three theorems of J.-C. Yoccoz

| contribution = Local connectivity of Julia sets and bifurcation loci: three theorems of J.-C. Yoccoz

| 贡献 Julia 集的局部连通性与分岔轨迹: J.-C. Yoccoz 的三个定理

| contribution-url = http://www.math.cornell.edu/~hubbard/hubbard.pdf

| contribution-url = http://www.math.cornell.edu/~hubbard/hubbard.pdf

| 贡献- http://www.math.cornell.edu/~hubbard/hubbard.pdf

| location = Houston, TX

| location = Houston, TX

地点: 休斯顿，TX

| mr = 1215974

| mr = 1215974

1215974先生

| pages = 467–511

| pages = 467–511

第467-511页

| publisher = Publish or Perish

| publisher = Publish or Perish

出版商出版或毁灭

| title = Topological methods in modern mathematics (Stony Brook, NY, 1991)

| title = Topological methods in modern mathematics (Stony Brook, NY, 1991)

| 题目现代数学中的拓扑方法(斯托尼布鲁克，纽约，1991)

| year = 1993}}. Hubbard cites as his source a 1989 unpublished manuscript of Yoccoz.</ref> Since then, local connectivity has been proved at many other points of <math>M</math>, but the full conjecture is still open.

| year = 1993}}. Hubbard cites as his source a 1989 unpublished manuscript of Yoccoz.</ref> Since then, local connectivity has been proved at many other points of <math>M</math>, but the full conjecture is still open.

1993年开始。哈伯德引用了1989年尚未出版的 Yoccoz 手稿作为他的资料来源。从那时起，局部连通性已经在数学 m / 数学的许多其他点上得到了证明，但是完整的猜想仍然是开放的。

===Self-similarity===

[[File:Mandelbrot zoom.gif|right|thumb|201px|[[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 from (−1, 0) to (−1.31, 0) while the view magnifies from 0.5 × 0.5 to 0.12 × 0.12 to approximate the [[Feigenbaum constants|Feigenbaum ratio]] <math>\delta</math>.]]

[[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 from (−1, 0) to (−1.31, 0) while the view magnifies from 0.5 × 0.5 to 0.12 × 0.12 to approximate the Feigenbaum ratio <math>\delta</math>.]]

[在 Mandelbrot 集合中的自相似性表现为放大一个圆形特征，同时平移负 x 方向。显示中心平移从(- 1,0)到(- 1.31,0) ，而视图放大从0.50.5到0.120.12，以接近 Feigenbaum 比率的 delta / math. ]

[[File:Mandelzoom1.jpg|left|thumb|280px|Self-similarity around Misiurewicz point −0.1011 + 0.9563i.]] The Mandelbrot set is [[self-similar]] under magnification in the neighborhoods of the [[Misiurewicz point]]s. It is also conjectured to be self-similar around generalized [[Feigenbaum point]]s (e.g., −1.401155 or −0.1528 + 1.0397''i''), in the sense of converging to a limit set.<ref>{{cite journal | last1 = Lei | year = 1990 | title = Similarity between the Mandelbrot set and Julia Sets | url = http://projecteuclid.org/euclid.cmp/1104201823| journal = Communications in Mathematical Physics | volume = 134 | issue = 3| pages = 587–617 | doi=10.1007/bf02098448| bibcode = 1990CMaPh.134..587L}}</ref><ref>{{cite book |author=J. Milnor |chapter=Self-Similarity and Hairiness in the Mandelbrot Set |editor=M. C. Tangora |location=New York |pages=211–257 |title=Computers in Geometry and Topology |url=https://books.google.com/books?id=wuVJAQAAIAAJ |year=1989|publisher=Taylor & Francis}})</ref>

Self-similarity around Misiurewicz point −0.1011 + 0.9563i. 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.

Misiurewicz 附近的自相似点 -0.1011 + 0.9563 i。在 Misiurewicz 点附近，曼德布洛特集合在放大倍率下是自相似的。在收敛到一个极限集的意义下，我们还推测在广义 Feigenbaum 点(例如-1.401155或-0.1528 + 1.0397 i)周围存在自相似性。

[[File:Blue Mandelbrot Zoom.jpg|left|thumb|340px|Quasi-self-similarity in the Mandelbrot set]]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.

Quasi-self-similarity in the Mandelbrot setThe 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.

Mandelbrot 集合中的准自相似性 Mandelbrot 集合通常不是严格的自相似集合，而是准自相似集合，因为在任意小的尺度上都可以找到自身略有不同的小版本。

The 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.

The 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.

曼德布洛特集合的小副本都略有不同，主要是因为它们与集合主体之间的细线连接。

===Further results===

The [[Hausdorff dimension]] of the [[boundary (topology)|boundary]] of the Mandelbrot set equals 2 as determined by a result of [[Mitsuhiro Shishikura]].<ref name="shishikura"/> It is not known whether the boundary of the Mandelbrot set has positive planar [[Lebesgue measure]].

The Hausdorff dimension of the boundary of the Mandelbrot set equals 2 as determined by a result of Mitsuhiro Shishikura. It is not known whether the boundary of the Mandelbrot set has positive planar Lebesgue measure.

曼德布洛集合边界的豪斯多夫维数等于2，这是由 Mitsuhiro Shishikura 的结果决定的。现在还不知道曼德布洛特集的边界是否具有正的平面勒贝格测度。

In the [[Blum-Shub-Smale]] model of [[real computation]], the Mandelbrot set is not computable, but its complement is [[Recursively enumerable set|computably enumerable]]. However, many simple objects (''e.g.'', the graph of exponentiation) are also not computable in the BSS model. At present, it is unknown whether the Mandelbrot set is computable in models of real computation based on [[computable analysis]], which correspond more closely to the intuitive notion of "plotting the set by a computer". Hertling has shown that the Mandelbrot set is computable in this model if the hyperbolicity conjecture is true.[[File:Relationship between Mandelbrot sets and Julia sets.PNG|3220x|left|thumb|A zoom into the Mandelbrot set illustrating a Julia "island" and a similar Julia set.]]

In the Blum-Shub-Smale model of real computation, the Mandelbrot set is not computable, but its complement is computably enumerable. However, many simple objects (e.g., the graph of exponentiation) are also not computable in the BSS model. At present, it is unknown whether the Mandelbrot set is computable in models of real computation based on computable analysis, which correspond more closely to the intuitive notion of "plotting the set by a computer". Hertling has shown that the Mandelbrot set is computable in this model if the hyperbolicity conjecture is true.A zoom into the Mandelbrot set illustrating a Julia "island" and a similar Julia set.

在实数计算的 Blum-Shub-Smale 模型中，Mandelbrot 集是不可计算的，但其补集是可计算可枚举的。然而，许多简单的对象(例如，求幂图)在 BSS 模型中也是不可计算的。目前，还不清楚在基于可计算性分析的实际计算模型中，Mandelbrot 集是否是可计算的，这更接近于直观的概念“由计算机绘制集合”。赫特林证明了如果双曲性猜想是正确的，那么在这个模型中 Mandelbrot 集是可计算的。

===Relationship with Julia sets===

As a consequence of the definition of the Mandelbrot set, there is a close correspondence between the geometry of the Mandelbrot set at a given point and the structure of the corresponding [[Julia set]]. For instance, a point is in the Mandelbrot set exactly when the corresponding Julia set is connected.

As a consequence of the definition of the Mandelbrot set, there is a close correspondence between the geometry of the Mandelbrot set at a given point and the structure of the corresponding Julia set. For instance, a point is in the Mandelbrot set exactly when the corresponding Julia set is connected.

由于 Mandelbrot 集的定义，在给定点上 Mandelbrot 集的几何与相应的 Julia 集的结构有着密切的对应关系。例如，当一个点在 Mandelbrot 集中时，相应的 Julia 集是连通的。

This principle is exploited in virtually all deep results on the Mandelbrot set. For example, Shishikura proved that, for a dense set of parameters in the boundary of the Mandelbrot set, the Julia set has [[Hausdorff dimension]] two, and then transfers this information to the parameter plane.<ref name="shishikura">{{citation

This principle is exploited in virtually all deep results on the Mandelbrot set. For example, Shishikura proved that, for a dense set of parameters in the boundary of the Mandelbrot set, the Julia set has Hausdorff dimension two, and then transfers this information to the parameter plane.<ref name="shishikura">{{citation

事实上，这一原理在曼德布洛特集合的所有深层结果中都得到了利用。例如，Shishikura 证明了，对于在 Mandelbrot 集边界上的一组稠密的参数，Julia 集有2个豪斯多夫维数，然后将这些信息传递到参数平面上。文献名称“ shishikura”{ citation

| last = Shishikura | first = Mitsuhiro

| last = Shishikura | first = Mitsuhiro

| last = Shishikura | first = Mitsuhiro

| arxiv = math.DS/9201282

| arxiv = math.DS/9201282

| arxiv math.DS / 9201282

| doi = 10.2307/121009

| doi = 10.2307/121009

| doi 10.2307 / 121009

| issue = 2

| issue = 2

第二期

| journal = Annals of Mathematics

| journal = Annals of Mathematics

华尔街数学纪事

| mr = 1626737

| mr = 1626737

1626737先生

| pages = 225–267

| pages = 225–267

第225-267页

| series = Second Series

| series = Second Series

系列第二季

| title = The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets

| title = The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets

曼德布洛特集合与茱莉亚集合边界的豪斯多夫维数

| volume = 147

| volume = 147

第147卷

| year = 1998| jstor = 121009

| year = 1998| jstor = 121009

1998年 jstor 121009

}}.</ref> Similarly, Yoccoz first proved the local connectivity of Julia sets, before establishing it for the Mandelbrot set at the corresponding parameters.<ref name="yoccoz"/> [[Adrien Douady]] phrases this principle as:

}}.</ref> Similarly, Yoccoz first proved the local connectivity of Julia sets, before establishing it for the Mandelbrot set at the corresponding parameters. Adrien Douady phrases this principle as:

}}. 同样，Yoccoz 首先证明了 Julia 集的局部连通性，然后在相应的参数处建立了 Mandelbrot 集的局部连通性。阿德里安 · 杜阿迪将这一原则表述为:

:{{quote|Plough in the dynamical plane, and harvest in parameter space.}}

==Geometry==





! -- douady rabbit 链接到这里 --

[[File:Unrolled main cardioid of Mandelbrot set for periods 8-14.png|thumbnail|right|Components on main cardioid for periods 8–14 with antennae 7–13]]

Components on main cardioid for periods 8–14 with antennae 7–13

主心脏线上的成分，周期8-14，触角7-13

For every rational number <math>\tfrac{p}{q}</math>, where ''p'' and ''q'' are [[relatively prime]], a hyperbolic component of period ''q'' bifurcates from the main cardioid. The part of the Mandelbrot set connected to the main cardioid at this bifurcation point is called the '''''p''/''q''-limb'''. Computer experiments suggest that the [[diameter]] of the limb tends to zero like <math>\tfrac{1}{q^2}</math>. The best current estimate known is the ''[[Jean-Christophe Yoccoz|Yoccoz-inequality]]'', which states that the size tends to zero like <math>\tfrac{1}{q}</math>.

For every rational number <math>\tfrac{p}{q}</math>, where p and q are relatively prime, a hyperbolic component of period q bifurcates from the main cardioid. The part of the Mandelbrot set connected to the main cardioid at this bifurcation point is called the p/q-limb. Computer experiments suggest that the diameter of the limb tends to zero like <math>\tfrac{1}{q^2}</math>. The best current estimate known is the Yoccoz-inequality, which states that the size tends to zero like <math>\tfrac{1}{q}</math>.

对于每个有理数 math tfrac { q } / math，其中 p 和 q 是相对素数，周期 q 的一个双曲分支从主心形分出。在这个分叉点上与主心脏相连的 Mandelbrot 集部分称为 p / q-limb。计算机实验表明，肢体的直径趋于零，就像 math [1][ q ^ 2][ math ]一样。目前已知的最佳估计是 Yoccoz-inequality，它表示大小趋于零，就像 math tfrac {1}{ q } / math。

A period-''q'' limb will have ''q'' − 1 "antennae" at the top of its limb. We can thus determine the period of a given bulb by counting these antennas.

A period-q limb will have q − 1 "antennae" at the top of its limb. We can thus determine the period of a given bulb by counting these antennas.

一个周期-q 的肢体将有 q-1“触角”在它的肢体顶部。因此，我们可以通过计算这些天线来确定给定灯泡的周期。

=== Pi in the Mandelbrot set ===

In an attempt to demonstrate that the thickness of the ''p''/''q''-limb is zero, David Boll carried out a computer experiment in 1991, where he computed the number of iterations required for the series to diverge for z = <math>-\tfrac{3}{4} + i\epsilon</math> (<math>-\tfrac{3}{4}</math> being the location thereof). As the series doesn't diverge for the exact value of z = <math>-\tfrac{3}{4}</math>, the number of iterations required increases with a small ε. It turns out that multiplying the value of ε with the number of iterations required yields an approximation of π that becomes better for smaller ε. For example, for ε = 0.0000001 the number of iterations is 31415928 and the product is 3.1415928.<ref>Gary William Flake, ''The Computational Beauty of Nature'', 1998. p. 125. {{isbn|978-0-262-56127-3}}.</ref>

In an attempt to demonstrate that the thickness of the p/q-limb is zero, David Boll carried out a computer experiment in 1991, where he computed the number of iterations required for the series to diverge for z = <math>-\tfrac{3}{4} + i\epsilon</math> (<math>-\tfrac{3}{4}</math> being the location thereof). As the series doesn't diverge for the exact value of z = <math>-\tfrac{3}{4}</math>, the number of iterations required increases with a small ε. It turns out that multiplying the value of ε with the number of iterations required yields an approximation of π that becomes better for smaller ε. For example, for ε = 0.0000001 the number of iterations is 31415928 and the product is 3.1415928.

为了证明 p / q-limb 的厚度为零，David Boll 在1991年进行了一个计算机实验，他计算了 z math-tfrac {4} + i-epsilon / math (math-tfrac {4} / math 是其位置)的数列发散所需的迭代次数。由于序列没有偏离 z math- tfrac {3}{4} / math 的精确值，所需的迭代次数随着。结果表明，将值乘以所需的迭代次数，得到的近似值越小越好。例如，对于0.0000001，迭代次数是31415928，产品是3.1415928。

=== Fibonacci sequence in the Mandelbrot set ===

It can be shown that the [[Fibonacci Sequence|Fibonacci sequence]] is located within the Mandelbrot Set and that a relation exists between the main cardioid and the [[Farey sequence|Farey Diagram]]. Upon mapping the main cardioid to a disk, one can notice that the amount of antennae that extends from the next largest Hyperbolic component, and that is located between the two previously selected components, follows suit with the Fibonacci sequence. The amount of antennae also correlates with the Farey Diagram and the denominator amounts within the corresponding fractional values, of which relate to the distance around the disk. Both portions of these fractional values themselves can be summed together after <math>\frac{1}{3}</math>to produce the location of the next Hyperbolic component within the sequence. Thus, the Fibonacci sequence of 1, 2, 3, 5, 8, 13, and 21 can be found within the Mandelbrot set.

It can be shown that the Fibonacci sequence is located within the Mandelbrot Set and that a relation exists between the main cardioid and the Farey Diagram. Upon mapping the main cardioid to a disk, one can notice that the amount of antennae that extends from the next largest Hyperbolic component, and that is located between the two previously selected components, follows suit with the Fibonacci sequence. The amount of antennae also correlates with the Farey Diagram and the denominator amounts within the corresponding fractional values, of which relate to the distance around the disk. Both portions of these fractional values themselves can be summed together after <math>\frac{1}{3}</math>to produce the location of the next Hyperbolic component within the sequence. Thus, the Fibonacci sequence of 1, 2, 3, 5, 8, 13, and 21 can be found within the Mandelbrot set.

结果表明，Fibonacci 数列位于 Mandelbrot 集内，主心线与 Farey 图之间存在一定的关系。将主心线映射到一个圆盘时，人们可以注意到从下一个最大的双曲线分量延伸出来的天线的数量，这个天线位于之前选定的两个分量之间，这与斐波纳契数列相同。天线的数量也与 Farey 图和分母数量在相应的小数值内相关，这些小数值与圆盘周围的距离相关。在 math frac {1}{3} / math 之后，这些小数值的两个部分可以相加，以产生序列中下一个双曲分量的位置。因此，可以在 Mandelbrot 集合中找到1,2,3,5,8,13和21的斐波那契序列。

===Image gallery of a zoom sequence===

The Mandelbrot set shows more intricate detail the closer one looks or [[magnification|magnifies]] the image, usually called "zooming in". The following example of an image sequence zooming to a selected ''c'' value gives an impression of the infinite richness of different geometrical structures and explains some of their typical rules.

The Mandelbrot set shows more intricate detail the closer one looks or magnifies the image, usually called "zooming in". The following example of an image sequence zooming to a selected c value gives an impression of the infinite richness of different geometrical structures and explains some of their typical rules.

曼德布洛特集合显示了更复杂的细节，一个人看起来越近或放大图像，通常称为“放大”。下面的图像序列放大到一个选定的 c 值的例子给人一种印象，不同的几何结构的无限丰富，并解释了它们的一些典型规则。

The magnification of the last image relative to the first one is about 1010 to 1. Relating to an ordinary monitor, it represents a section of a Mandelbrot set with a diameter of 4 million kilometers. Its border would show an astronomical number of different fractal structures.

The magnification of the last image relative to the first one is about 1010 to 1. Relating to an ordinary monitor, it represents a section of a Mandelbrot set with a diameter of 4 million kilometers. Its border would show an astronomical number of different fractal structures.

最后一幅图像相对于第一幅图像的放大率约为10 / 10 / 1。与一个普通的显示器相关，它代表了一个直径为400万公里的曼德布洛特集合的一部分。它的边界将显示天文数字的不同分形结构。

{{Clear}}

<gallery mode="packed">

<gallery mode="packed">

美术馆模式”包装”

File:Mandel zoom 00 mandelbrot set.jpg|Start. Mandelbrot set with continuously colored environment.

File:Mandel zoom 00 mandelbrot set.jpg|Start. Mandelbrot set with continuously colored environment.

文件: Mandel zoom 00 mandelbrot set.jpg | Start。具有连续彩色环境的曼德勃特集合。

File:Mandel zoom 01 head and shoulder.jpg|Gap between the "head" and the "body", also called the "seahorse valley"

File:Mandel zoom 01 head and shoulder.jpg|Gap between the "head" and the "body", also called the "seahorse valley"

文件: Mandel zoom 01 head and shoulder.jpg | “ head”和“ body”之间的间隙，也叫“ seahhorse valley”

File:Mandel zoom 02 seehorse valley.jpg|Double-spirals on the left, "seahorses" on the right

File:Mandel zoom 02 seehorse valley.jpg|Double-spirals on the left, "seahorses" on the right

文件: Mandel zoom 02 seethhorse valley.jpg | Double-spirals on the left，“ sea horses” on the right

File:Mandel zoom 03 seehorse.jpg|"Seahorse" upside down

File:Mandel zoom 03 seehorse.jpg|"Seahorse" upside down

File:Mandel zoom 03 seehorse.jpg|"Seahorse" upside down

</gallery>

</gallery>

/ 画廊

The seahorse "body" is composed by 25 "spokes" consisting of two groups of 12 "spokes" each and one "spoke" connecting to the main cardioid. These two groups can be attributed by some kind of metamorphosis to the two "fingers" of the "upper hand" of the Mandelbrot set; therefore, the number of "spokes" increases from one "seahorse" to the next by 2; the "hub" is a so-called [[Misiurewicz point]]. Between the "upper part of the body" and the "tail" a distorted small copy of the Mandelbrot set called satellite may be recognized.

The seahorse "body" is composed by 25 "spokes" consisting of two groups of 12 "spokes" each and one "spoke" connecting to the main cardioid. These two groups can be attributed by some kind of metamorphosis to the two "fingers" of the "upper hand" of the Mandelbrot set; therefore, the number of "spokes" increases from one "seahorse" to the next by 2; the "hub" is a so-called Misiurewicz point. Between the "upper part of the body" and the "tail" a distorted small copy of the Mandelbrot set called satellite may be recognized.

海马“身体”是由25个“辐条”组成的两组12个“辐条”每个和一个“辐条”连接到主心脏线。这两个群体可以通过某种形式的蜕变归因于 Mandelbrot 集合的”上手”的两个”手指” ; 因此，”辐条”的数量从一个”海马”增加到下一个”海马” ，增加了2个; ”中心”是所谓的 Misiurewicz 点。在“身体的上半部分”和“尾巴”之间，可以辨认出一个扭曲的曼德布罗特集合的小复制品，这个集合叫做“卫星”。

<gallery mode="packed">

<gallery mode="packed">

美术馆模式”包装”

File:Mandel zoom 04 seehorse tail.jpg|The central endpoint of the "seahorse tail" is also a [[Misiurewicz point]].

File:Mandel zoom 04 seehorse tail.jpg|The central endpoint of the "seahorse tail" is also a Misiurewicz point.

文件: Mandel zoom 04 seethhorse tail.jpg | “海马尾巴”的中心端点也是 Misiurewicz 点。

File:Mandel zoom 05 tail part.jpg|Part of the "tail" — there is only one path consisting of the thin structures that lead through the whole "tail". This zigzag path passes the "hubs" of the large objects with 25 "spokes" at the inner and outer border of the "tail"; thus the Mandelbrot set is a [[Simply connected space|simply connected]] set, which means there are no islands and no loop roads around a hole.

File:Mandel zoom 05 tail part.jpg|Part of the "tail" — there is only one path consisting of the thin structures that lead through the whole "tail". This zigzag path passes the "hubs" of the large objects with 25 "spokes" at the inner and outer border of the "tail"; thus the Mandelbrot set is a simply connected set, which means there are no islands and no loop roads around a hole.

文件: Mandel zoom 05 tail Part.jpg | Part of the“ tail”ーー只有一条由薄结构组成的路径通过整个“ tail”。这条曲折的路径穿过大型物体的”中心” ，在”尾部”的内外边界上有25个”辐条” ; 因此 Mandelbrot 集是一个单连通集，这意味着在一个洞周围没有岛屿和环路。

File:Mandel zoom 06 double hook.jpg|Satellite. The two "seahorse tails" are the beginning of a series of concentric crowns with the satellite in the center. [https://mandelbrot-svelte.netlify.com/#{%22pos%22:{%22x%22:-0.743904874255535,%22y%22:-0.1317119067802009},%22zoom%22:7502494.442311305} Open this location in an interactive viewer.]

File:Mandel zoom 06 double hook.jpg|Satellite. The two "seahorse tails" are the beginning of a series of concentric crowns with the satellite in the center. [https://mandelbrot-svelte.netlify.com/#{%22pos%22:{%22x%22:-0.743904874255535,%22y%22:-0.1317119067802009},%22zoom%22:7502494.442311305} Open this location in an interactive viewer.]

文件: Mandel zoom 06 double hook.jpg | Satellite。这两个“海马尾巴”是一系列同心王冠的开始，王冠的中心是卫星。[ https://mandelbrot-svelte.netlify.com/# {“ pos” : “ x” :-0.743904874255535，“ y” :-0.1317119067802009} ，“ zoom” : 7502494.442311305}在一个交互式查看器中打开这个位置]

File:Mandel zoom 07 satellite.jpg|Each of these crowns consists of similar "seahorse tails"; their number increases with powers of 2, a typical phenomenon in the environment of satellites. The unique path to the spiral center passes the satellite from the groove of the cardioid to the top of the "antenna" on the "head".

File:Mandel zoom 07 satellite.jpg|Each of these crowns consists of similar "seahorse tails"; their number increases with powers of 2, a typical phenomenon in the environment of satellites. The unique path to the spiral center passes the satellite from the groove of the cardioid to the top of the "antenna" on the "head".

每个皇冠都由类似的“海马尾巴”组成; 它们的数量随着功率的增加而增加，这是卫星环境中的典型现象。通往螺旋中心的独特路径将卫星从心形的凹槽传递到“头”上的“天线”顶部。

File:Mandel zoom 08 satellite antenna.jpg|"Antenna" of the satellite. Several satellites of second order may be recognized.

File:Mandel zoom 08 satellite antenna.jpg|"Antenna" of the satellite. Several satellites of second order may be recognized.

文件: Mandel zoom 08 satellite Antenna.jpg | “ Antenna” of the satellite。可以辨认出几颗二阶卫星。

File:Mandel zoom 09 satellite head and shoulder.jpg|The "seahorse valley" of the satellite. All the structures from the start of the zoom reappear.

File:Mandel zoom 09 satellite head and shoulder.jpg|The "seahorse valley" of the satellite. All the structures from the start of the zoom reappear.

文件: Mandel zoom 09 satellite head and shoulder.jpg | The“ seahhorse valley” of The satellite。从缩放开始的所有结构都会重新出现。

File:Mandel zoom 10 satellite seehorse valley.jpg|Double-spirals and "seahorses" – unlike the 2nd image from the start, they have appendices consisting of structures like "seahorse tails"; this demonstrates the typical linking of ''n'' + 1 different structures in the environment of satellites of the order ''n'', here for the simplest case ''n'' = 1.

File:Mandel zoom 10 satellite seehorse valley.jpg|Double-spirals and "seahorses" – unlike the 2nd image from the start, they have appendices consisting of structures like "seahorse tails"; this demonstrates the typical linking of n + 1 different structures in the environment of satellites of the order n, here for the simplest case n = 1.

文件: Mandel zoom 10 satellite seethhorse valley.jpg | Double-spirals and“ sea horses”——与第二张图片不同，它们有附属结构，如“海马尾巴” ; 这表明了 n + 1不同结构在 n 阶卫星环境中的典型连接，这里是 n 1最简单的情况。

File:Mandel zoom 11 satellite double spiral.jpg|Double-spirals with satellites of second order – analogously to the "seahorses", the double-spirals may be interpreted as a metamorphosis of the "antenna"

File:Mandel zoom 11 satellite double spiral.jpg|Double-spirals with satellites of second order – analogously to the "seahorses", the double-spirals may be interpreted as a metamorphosis of the "antenna"

文件: Mandel zoom 11 satellite double spiral.jpg | Double-spirals with satellite of second order-alias to the“ seashorses” ，the Double-spirals may be interpreted of the“ antenna”

File:Mandel zoom 12 satellite spirally wheel with julia islands.jpg|In the outer part of the appendices, islands of structures may be recognized; they have a shape like [[Julia set]]s ''Jc''; the largest of them may be found in the center of the "double-hook" on the right side

File:Mandel zoom 12 satellite spirally wheel with julia islands.jpg|In the outer part of the appendices, islands of structures may be recognized; they have a shape like Julia sets Jc; the largest of them may be found in the center of the "double-hook" on the right side

文件: Mandel zoom 12 satellite spirally wheel with Julia islands.jpg | 在附录的外部，结构的岛屿可能被识别，它们的形状像 Julia set j sub c / sub，最大的可能在右侧的“双钩”中心

File:Mandel zoom 13 satellite seehorse tail with julia island.jpg|Part of the "double-hook"

File:Mandel zoom 13 satellite seehorse tail with julia island.jpg|Part of the "double-hook"

文件: Mandel zoom 13 satellite seehorse tail with julia island.jpg | Part of the“ double-hook”

File:Mandel zoom 14 satellite julia island.jpg|Islands

File:Mandel zoom 14 satellite julia island.jpg|Islands

文件: Mandel zoom 14 satellite julia island.jpg | Islands

File:Mandel zoom 15 one island.jpg|Detail of one island

File:Mandel zoom 15 one island.jpg|Detail of one island

文件: Mandel zoom 15 one island.jpg | Detail of one island

File:Mandel zoom 16 spiral island.jpg|Detail of the spiral. [https://guciek.github.io/web_mandelbrot.html#-0.7436439049875745;0.13182591455433018;2.2351741790771485e-13;7000 Open this location in an interactive viewer.]

File:Mandel zoom 16 spiral island.jpg|Detail of the spiral. [https://guciek.github.io/web_mandelbrot.html#-0.7436439049875745;0.13182591455433018;2.2351741790771485e-13;7000 Open this location in an interactive viewer.]

16 spiral island.jpg | Detail of the spiral.[在一个交互式查看器中打开这个位置 https://guciek.github.io/web_mandelbrot.html#-0.7436439049875745;0.13182591455433018;2.2351741790771485e-13;7000]

</gallery>

</gallery>

/ 画廊

The islands above seem to consist of infinitely many parts like [[Cantor set]]s, as is{{Clarify|date=May 2010}} actually the case for the corresponding Julia set ''Jc''. However, they are connected by tiny structures, so that the whole represents a simply connected set. The tiny structures meet each other at a satellite in the center that is too small to be recognized at this magnification. The value of ''c'' for the corresponding ''Jc'' is not that of the image center but, relative to the main body of the Mandelbrot set, has the same position as the center of this image relative to the satellite shown in the 6th zoom step.

The islands above seem to consist of infinitely many parts like Cantor sets, as is actually the case for the corresponding Julia set Jc. However, they are connected by tiny structures, so that the whole represents a simply connected set. The tiny structures meet each other at a satellite in the center that is too small to be recognized at this magnification. The value of c for the corresponding Jc is not that of the image center but, relative to the main body of the Mandelbrot set, has the same position as the center of this image relative to the satellite shown in the 6th zoom step.

上面的岛屿看起来是由无限多个部分组成的，就像 Cantor 集一样，实际上就是相应的 Julia 集 j sub c / sub。然而，它们是通过微小的结构连接起来的，因此整体代表了一个单连通的集合。这些微小的结构在中心的卫星上相遇，在这样的放大倍率下，它们太小以至于无法被识别。相应的 j 子 c / sub 的 c 值不是图像中心的 c 值，而是相对于 Mandelbrot 集合的主体，与第6个缩放步骤中显示的卫星图像的中心位置相同。

===3D images of Mandelbrot and Julia sets===

{{More citations needed section|talk=3D images|date=January 2020}} 



! -- 一旦有足够的引用被添加到这个区域 b，那么前面的模板就可以被删除 --

In addition to creating two dimensional images of the Mandelbrot set, various techniques can be used to render Mandelbrot and Julia sets as 3D [[Heightmap]] images, where each pixel in a 2D image is given a height value, and the resulting image is rendered as a 3D graphic.

In addition to creating two dimensional images of the Mandelbrot set, various techniques can be used to render Mandelbrot and Julia sets as 3D Heightmap images, where each pixel in a 2D image is given a height value, and the resulting image is rendered as a 3D graphic.

除了创建 Mandelbrot 集合的二维图像，还可以使用各种技术将 Mandelbrot 和 Julia 集合渲染成3 d Heightmap 图像，其中2 d 图像中的每个像素都被赋予一个高度值，并将生成的图像渲染成3 d 图形。

The simplest approach to 3D rendering uses the iteration value for each pixel as a height value. This produces images with distinct "steps" in the height value.

The simplest approach to 3D rendering uses the iteration value for each pixel as a height value. This produces images with distinct "steps" in the height value.

最简单的3D 渲染方法是使用每个像素的迭代值作为高度值。这将产生高度值中具有明显“台阶”的图像。

[[File: Mandelbrot set 3D integer iterations.jpg|thumb|none|Mandelbrot set rendered in 3D using integer iterations]]

Mandelbrot set rendered in 3D using integer iterations

使用整数迭代在3 d 中渲染 Mandelbrot 集

If instead you use the fractional iteration value (also known as the potential function<ref>[[Julia set#The potential function and the real iteration number|potential function]])</ref> to calculate the height value for each point, you avoid steps in the resulting image. However, images rendered in 3D using fractional iteration data still look rather bumpy and visually noisy.

If instead you use the fractional iteration value (also known as the potential function to calculate the height value for each point, you avoid steps in the resulting image. However, images rendered in 3D using fractional iteration data still look rather bumpy and visually noisy.

如果使用分数迭代值(也称为势函数)来计算每个点的高度值，则可以避免最终图像中的步骤。然而，使用分数迭代数据渲染的3 d 图像看起来仍然相当颠簸和视觉噪音。

[[File: Mandelbrot set 3D fractional iterations.jpg|thumb|none| Mandelbrot set rendered in 3D using fractional iteration values]]

Mandelbrot set rendered in 3D using fractional iteration values

使用分数迭代值在3D 中渲染 Mandelbrot 集

An alternative approach is to use Distance Estimate<ref>{{cite book |title=The Science of Fractal Images |last=Peitgen |first=Heinz-Otto |author2=Saupe Dietmar |year=1988 |publisher=Springer-Verlag |location=New York |isbn=0-387-96608-0 |pages=121, 196–197 |title-link=The Beauty of Fractals }}</ref> (DE) data for each point to calculate a height value. Non-linear mapping of distance estimate value using an exponential function can produce visually pleasing images. Images plotted using DE data are often visually striking, and more importantly, the 3D shape makes it easy to visualize the thin "tendrils" that connect points of the set. Color plates 29 and 30 on page 121 of "The Science of Fractal Images" show a 2D and 3D image plotted using External Distance Estimates.

An alternative approach is to use Distance Estimate (DE) data for each point to calculate a height value. Non-linear mapping of distance estimate value using an exponential function can produce visually pleasing images. Images plotted using DE data are often visually striking, and more importantly, the 3D shape makes it easy to visualize the thin "tendrils" that connect points of the set. Color plates 29 and 30 on page 121 of "The Science of Fractal Images" show a 2D and 3D image plotted using External Distance Estimates.

另一种方法是使用每个点的距离估计(DE)数据来计算高度值。非线性映射的距离估计值使用指数函数可以产生视觉上令人满意的图像。使用 DE 数据绘制的图像往往在视觉上引人注目，而且更重要的是，3D 形状使得连接图像中各点的细“卷须”可视化变得更加容易。在“分形图像的科学”第121页的29和30色板显示了使用外部距离估计绘制的2D 和3D 图像。

[[File: Mandelbrot set 3D Distance Estimates.jpg |thumb|none| Mandelbrot set rendered in 3D using Distance Estimates]]

Mandelbrot set rendered in 3D using Distance Estimates

基于距离估计的 Mandelbrot 集三维渲染

Below is a 3D version of the "Image gallery of a zoom sequence" gallery above, rendered as height maps using Distance Estimate data, and using similar cropping and coloring.

Below is a 3D version of the "Image gallery of a zoom sequence" gallery above, rendered as height maps using Distance Estimate data, and using similar cropping and coloring.

下面是一个3D 版本的“图像画廊的缩放序列”画廊上面，渲染为高度地图使用距离估计数据，并使用类似的裁剪和着色。

<gallery mode="packed">

<gallery mode="packed">

美术馆模式”包装”

File:Mandel zoom 00 mandelbrot set 3D.jpg|[[:File: Mandel zoom 00 mandelbrot set.jpg|Zoom 00]]. Start. Mandelbrot set with continuously colored environment.

File:Mandel zoom 00 mandelbrot set 3D.jpg|Zoom 00. Start. Mandelbrot set with continuously colored environment.

文件: Mandel Zoom 00 mandelbrot set 3 d. jpg | Zoom 00。开始。具有连续彩色环境的曼德勃特集合。

File:Mandel zoom 01 head and shoulder 3D.jpg|[[:File: Mandel zoom 01 head and shoulder.jpg|Zoom 01]]. Gap between the "head" and the "body", also called the "seahorse valley"

File:Mandel zoom 01 head and shoulder 3D.jpg|Zoom 01. Gap between the "head" and the "body", also called the "seahorse valley"

文件: Mandel Zoom 01 head and shoulder 3D. jpg | Zoom 01。“头”和“身体”之间的空隙，也称为“海马谷”

File:Mandel zoom 02 seehorse valley 3D.jpg|[[:File: Mandel zoom 02 seehorse valley.jpg|Zoom 02]]. Double-spirals on the left, "seahorses" on the right

File:Mandel zoom 02 seehorse valley 3D.jpg|Zoom 02. Double-spirals on the left, "seahorses" on the right

文件: Mandel Zoom 02 seehorse valley 3 d. jpg | Zoom 02。左边是双螺旋，右边是“海马”

File:Mandel zoom 03 seehorse 3D.jpg|[[:File: Mandel zoom 03 seehorse.jpg|Zoom 03]]. "Seahorse" upside down

File:Mandel zoom 03 seehorse 3D.jpg|Zoom 03. "Seahorse" upside down

文件: Mandel Zoom 03 seehorse 3D. jpg | Zoom 03。"Seahorse" upside down

</gallery>

</gallery>

/ 画廊

<gallery mode="packed">

<gallery mode="packed">

美术馆模式”包装”

File:Mandel zoom 04 seehorse tail 3D.jpg|[[:File: Mandel zoom 04 seehorse tail.jpg|Zoom 04]]. A "seahorse tail".

File:Mandel zoom 04 seehorse tail 3D.jpg|Zoom 04. A "seahorse tail".

文件: Mandel Zoom 04 seethhorse tail 3 d. jpg | Zoom 04。A "seahorse tail".

File:Mandel zoom 05 tail part 3D.jpg|[[:File: Mandel zoom 05 tail part.jpg|Zoom 05]]. Part of the "tail".

File:Mandel zoom 05 tail part 3D.jpg|Zoom 05. Part of the "tail".

文件: Mandel Zoom 05 tail part 3 d. jpg | Zoom 05。“尾巴”的一部分。

File:Mandel zoom 06 double hook 3D.jpg|[[:File: Mandel zoom 06 double hook.jpg|Zoom 06]]. Satellite with twin "Seahorse tails."

File:Mandel zoom 06 double hook 3D.jpg|Zoom 06. Satellite with twin "Seahorse tails."

文件: Mandel Zoom 06 double hook 3D. jpg | Zoom 06。带有双“海马尾巴”的卫星

File:Mandel zoom 07 satellite 3D.jpg|[[:File: Mandel zoom 07 satellite.jpg|Zoom 07]]. Satellite closeup.

File:Mandel zoom 07 satellite 3D.jpg|Zoom 07. Satellite closeup.

文件: Mandel Zoom 07 satellite 3 d. jpg | Zoom 07。卫星特写镜头。

File:Mandel zoom 08 satellite antenna 3D.jpg|[[:File: Mandel zoom 08 satellite antenna.jpg|Zoom 08]]. "Antenna" of the satellite. Several satellites of second order may be recognized.

File:Mandel zoom 08 satellite antenna 3D.jpg|Zoom 08. "Antenna" of the satellite. Several satellites of second order may be recognized.

文件: Mandel Zoom 08 satellite antenna 3D. jpg | Zoom 08。卫星的“天线”。可以辨认出几颗二阶卫星。

File:Mandel zoom 09 satellite head and shoulder 3D.jpg|[[:File: Mandel zoom 09 satellite head and shoulder.jpg|Zoom 09]]. The "seahorse valley" of the satellite. All the structures from the start of the zoom reappear.

File:Mandel zoom 09 satellite head and shoulder 3D.jpg|Zoom 09. The "seahorse valley" of the satellite. All the structures from the start of the zoom reappear.

文件: Mandel Zoom 09 satellite head and shoulder 3D. jpg | Zoom 09。卫星的“海马谷”。从缩放开始的所有结构都会重新出现。

File:Mandel zoom 10 satellite seehorse valley 3D.jpg|[[:File: Mandel zoom 10 satellite seehorse valley.jpg|Zoom 10]]. Double-spirals and "seahorses"

File:Mandel zoom 10 satellite seehorse valley 3D.jpg|Zoom 10. Double-spirals and "seahorses"

文件: Mandel Zoom 10 satellite seethhorse valley 3 d. jpg | Zoom 10。双螺旋和“海马”

File:Mandel zoom 11 satellite double spiral 3D.jpg|[[:File: Mandel zoom 11 satellite double spiral.jpg|Zoom 11]]. Double-spirals with satellites of second order.

File:Mandel zoom 11 satellite double spiral 3D.jpg|Zoom 11. Double-spirals with satellites of second order.

11 satellite double spiral 3D. jpg | Zoom 11.具有二阶卫星的双螺旋。

File:Mandel zoom 12 satellite spirally wheel with julia islands 3D.jpg|[[:File: Mandel zoom 12 satellite spirally wheel with julia islands.jpg|Zoom 12]].

File:Mandel zoom 12 satellite spirally wheel with julia islands 3D.jpg|Zoom 12.

文件: Mandel Zoom 12 satellite spirally wheel with julia islands 3 d. jpg | Zoom 12。

File:Mandel zoom 13 satellite seehorse tail with julia island 3D.jpg|[[:File: Mandel zoom 13 satellite seehorse tail with julia island.jpg|Zoom 13]]. Part of the "double-hook"

File:Mandel zoom 13 satellite seehorse tail with julia island 3D.jpg|Zoom 13. Part of the "double-hook"

13 satellite seehorse tail with julia island 3 d. jpg | Zoom 13.”双钩”的一部分

File:Mandel zoom 14 satellite julia island 3D.jpg|[[:File: Mandel zoom 14 satellite julia island.jpg|Zoom 14]]. Islands

File:Mandel zoom 14 satellite julia island 3D.jpg|Zoom 14. Islands

14 satellite julia island 3 d. jpg | Zoom 14.离岛

File:Mandel zoom 15 one island 3D.jpg|[[:File: Mandel zoom 15 one island.jpg|Zoom 15]]. Detail of one island

File:Mandel zoom 15 one island 3D.jpg|Zoom 15. Detail of one island

文件: Mandel Zoom 15 one island 3 d. jpg | Zoom 15。一个岛的细节

File:Mandel zoom 16 spiral island 3D.jpg|[[:File: Mandel zoom 16 spiral island.jpg|Zoom 16]]. Detail of the spiral.

File:Mandel zoom 16 spiral island 3D.jpg|Zoom 16. Detail of the spiral.

文件: Mandel Zoom 16 spiral island 3D. jpg | Zoom 16。螺旋的细节。

</gallery>

</gallery>

/ 画廊

The image below is similar to "zoom 5", above, but is an attempt to create a 3D version of the image "Map 44" from page 85 of the book "The Beauty of Fractals"<ref>{{cite book |title=The Beauty of Fractals |last=Peitgen |first=Heinz-Otto |author2=Richter Peter |year=1986 |publisher=Springer-Verlag |location=Heidelberg |isbn=0-387-15851-0 |pages=[https://archive.org/details/beautyoffractals0000peit/page/85 85] |title-link=The Beauty of Fractals }}</ref> using a visually similar color scheme that shows the details of the plot in 3D.

The image below is similar to "zoom 5", above, but is an attempt to create a 3D version of the image "Map 44" from page 85 of the book "The Beauty of Fractals" using a visually similar color scheme that shows the details of the plot in 3D.

下面的图像类似于上面的“放大5” ，但是试图创建一个3D 版本的图像“地图44”从书“分形的美丽”第85页使用一个视觉上相似的配色方案，显示了3D 的情节细节。

[[File:A 3D version of the Mandelbrot set plot "Map 44" from the book "The Beauty of Fractals".jpg|thumb|none|A 3D version of the Mandelbrot set plot "Map 44" from the book "The Beauty of Fractals]]

A 3D version of the Mandelbrot set plot "Map 44" from the book "The Beauty of Fractals

曼德尔布洛特集合的3D 版本绘制了《分形之美》一书中的“地图44”

==Generalizations==

{{multiple image

{{multiple image

{多重图像

| image1 = Mandelbrot Set Animation 1280x720.gif

| image1 = Mandelbrot Set Animation 1280x720.gif

1 Mandelbrot Set Animation 1280x720. gif

| image2 = Mandelbrot set from powers 0.05 to 2.webm

| image2 = Mandelbrot set from powers 0.05 to 2.webm

2 Mandelbrot set from powers 0.05 to 2. webm

| width2 = 150

| width2 = 150

2150

| footer = Animations of the Multibrot set for ''d'' from 0 to 5 (left) and from 0.05 to 2 (right).

| footer = Animations of the Multibrot set for d from 0 to 5 (left) and from 0.05 to 2 (right).

从0到5(左)和从0.05到2(右)的 d 的 Multibrot 集的页脚动画。

}}

}}

}}

[[File:Quaternion Julia x=-0,75 y=-0,14.jpg|thumb|A 4D Julia set may be projected or cross-sectioned into 3D, and because of this a 4D Mandelbrot is also possible.]]

A 4D Julia set may be projected or cross-sectioned into 3D, and because of this a 4D Mandelbrot is also possible.

一个4D 的 Julia 集合可以被投影或者交叉切割成3D，正因为如此，一个4D 的 Mandelbrot 也是可能的。

===Multibrot sets===

[[Multibrot set]]s are bounded sets found in the complex plane for members of the general monic univariate [[polynomial]] family of recursions

Multibrot sets are bounded sets found in the complex plane for members of the general monic univariate polynomial family of recursions

多布罗特集是一般一元多项式递归族的成员在复平面上找到的有界集

:<math> z \mapsto z^d + c.\ </math>

<math> z \mapsto z^d + c.\ </math>

数学，地图，地图，地图，数学

For an integer d, these sets are connectedness loci for the Julia sets built from the same formula. The full cubic connectedness locus has also been studied; here one considers the two-parameter recursion <math> z \mapsto z^3 + 3kz + c </math>, whose two [[critical point (mathematics)|critical points]] are the [[complex square root]]s of the parameter ''k''. A parameter is in the cubic connectedness locus if both critical points are stable.<ref>[[Rudy Rucker]]'s discussion of the CCM: [http://www.cs.sjsu.edu/faculty/rucker/cubic_mandel.htm CS.sjsu.edu]</ref> For general families of [[holomorphic function]]s, the ''boundary'' of the Mandelbrot set generalizes to the [[bifurcation locus]], which is a natural object to study even when the connectedness locus is not useful.

For an integer d, these sets are connectedness loci for the Julia sets built from the same formula. The full cubic connectedness locus has also been studied; here one considers the two-parameter recursion <math> z \mapsto z^3 + 3kz + c </math>, whose two critical points are the complex square roots of the parameter k. A parameter is in the cubic connectedness locus if both critical points are stable. For general families of holomorphic functions, the boundary of the Mandelbrot set generalizes to the bifurcation locus, which is a natural object to study even when the connectedness locus is not useful.

对于整数 d，这些集合是由同一公式构造的 Julia 集合的连通轨迹。研究了全三次连通轨迹，这里考虑了双参数递归数学 z mapsto ^ 3 + 3kz + c / math，其中两个临界点是参数 k 的复平方根。如果两个临界点都是稳定的，则参数在三次连通轨迹中。对于一般的全纯函数族，Mandelbrot 集的边界推广到分支轨迹，即使连通轨迹不存在，也是一个自然的研究对象。

The [[Multibrot set]] is obtained by varying the value of the exponent ''d''. The [[Multibrot set|article]] has a video that shows the development from ''d'' = 0 to 7, at which point there are 6 i.e. (''d'' − 1) lobes around the perimeter. A similar development with negative exponents results in (1 − ''d'') clefts on the inside of a ring.

The Multibrot set is obtained by varying the value of the exponent d. The article has a video that shows the development from d = 0 to 7, at which point there are 6 i.e. (d − 1) lobes around the perimeter. A similar development with negative exponents results in (1 − d) clefts on the inside of a ring.

通过改变指数 d 的值，得到了多布罗特集。这篇文章有一个视频，展示了从 d 0到7的发展，在这一点上有6个例子。(d-1)周围的叶片。类似的负指数发展也会导致(1-d)裂缝出现在环的内侧。

===Higher dimensions===

There is no perfect extension of the Mandelbrot set into 3D. This is because there is no 3D analogue of the complex numbers for it to iterate on. However, there is an extension of the complex numbers into 4 dimensions, called the [[quaternion]]s, that creates a perfect extension of the Mandelbrot set and the Julia sets into 4 dimensions.<ref>http://archive.bridgesmathart.org/2010/bridges2010-247.pdf retrieved 19 August 2018</ref> These can then be either [[cross section (geometry)|cross-sectioned]] or [[Projection mapping|projected]] into a 3D structure.

There is no perfect extension of the Mandelbrot set into 3D. This is because there is no 3D analogue of the complex numbers for it to iterate on. However, there is an extension of the complex numbers into 4 dimensions, called the quaternions, that creates a perfect extension of the Mandelbrot set and the Julia sets into 4 dimensions. These can then be either cross-sectioned or projected into a 3D structure.

曼德布洛特集没有完美的3 d 扩展。这是因为它没有复数的3 d 模拟来进行迭代。然而，有一种将复数扩展到4维的方法，称为四元数，它创建了一个完美的 Mandelbrot 集和 Julia 集的扩展到4维。然后，这些可以被截断或投影到一个3D 结构。

===Other, non-analytic, mappings===

[[File:Mandelbar fractal from XaoS.PNG|left|thumb|Image of the Tricorn / Mandelbar fractal]]

Image of the Tricorn / Mandelbar fractal

三角 / 扁桃体分形图像

[[File:BurningShip01.png|thumb|Image of the burning ship fractal]]

Image of the burning ship fractal

燃烧船舶的分形图像

Of particular interest is the [[tricorn (mathematics)|tricorn]] fractal, the connectedness locus of the anti-holomorphic family

Of particular interest is the tricorn fractal, the connectedness locus of the anti-holomorphic family

特别有趣的是三角形分形，反全纯族的连通轨迹

:<math> z \mapsto \bar{z}^2 + c.</math>

<math> z \mapsto \bar{z}^2 + c.</math>

2 + c / math

The tricorn (also sometimes called the ''Mandelbar'') was encountered by [[John Milnor|Milnor]] in his study of parameter slices of real [[Cubic function|cubic polynomials]]. It is ''not'' locally connected. This property is inherited by the connectedness locus of real cubic polynomials.

The tricorn (also sometimes called the Mandelbar) was encountered by Milnor in his study of parameter slices of real cubic polynomials. It is not locally connected. This property is inherited by the connectedness locus of real cubic polynomials.

米尔诺在研究实三次多项式的参数切片时遇到了三角形(有时也称为曼德尔巴)。它不是局部连接的。这一性质是由实三次多项式的连通轨迹继承的。

Another non-analytic generalization is the [[Burning Ship fractal]], which is obtained by iterating the following :

Another non-analytic generalization is the Burning Ship fractal, which is obtained by iterating the following :

另一个非分析性的概括是燃烧的船只分形，它是通过迭代以下得到的:

:<math> z \mapsto (|\Re \left(z\right)|+i|\Im \left(z\right)|)^2 + c.</math>

<math> z \mapsto (|\Re \left(z\right)|+i|\Im \left(z\right)|)^2 + c.</math>

Math z mapsto (| Re left (z right) | + i | Im left (z right) |) ^ 2 + c / math

==Computer drawings==

{{Main|Plotting algorithms for the Mandelbrot set}}

[[File:Fractal-zoom-1-03-Mandelbrot Buzzsaw.png|right|thumbnail|Still image of [//upload.wikimedia.org/wikipedia/commons/0/07/Fractal-zoom-1-03-Mandelbrot_Buzzsaw.ogv a movie of increasing magnification] on 0.001643721971153 − 0.822467633298876''i'']]

Still image of [//upload.wikimedia.org/wikipedia/commons/0/07/Fractal-zoom-1-03-Mandelbrot_Buzzsaw.ogv a movie of increasing magnification] on 0.001643721971153 − 0.822467633298876i

0.001643721971153-0.822467633298876 i 的静止图像[ / / upload.wikimedia.org/wikipedia/commons/0/07/fractal-zoom-1-03-mandelbrot_buzzsaw.ogv 放大倍数增加的电影]

[[File:Mandelbrot sequence new still.png|right|thumbnail|Still image of [//upload.wikimedia.org/wikipedia/commons/5/51/Mandelbrot_sequence_new.webm an animation of increasing magnification]]]

Still image of [//upload.wikimedia.org/wikipedia/commons/5/51/Mandelbrot_sequence_new.webm an animation of increasing magnification]

静止图像[ / / upload.wikimedia.org/wikipedia/commons/5/51/mandelbrot_sequence_new.webm / 增大放大的动画]





<！ ——有许多程序和算法用于生成 Mandelbrot 集和其他分形，其中一些用分形生成软件进行了描述。这些程序使用多种算法来确定单个像素的颜色，从而实现高效的计算。-->

There exist a multitude of various algorithms for plotting the Mandelbrot set via a computing device. Here, the most widely used and simplest algorithm will be demonstrated, namely, the naïve "escape time algorithm".

There exist a multitude of various algorithms for plotting the Mandelbrot set via a computing device. Here, the most widely used and simplest algorithm will be demonstrated, namely, the naïve "escape time algorithm".

通过计算设备绘制 Mandelbrot 集合有多种算法。在这里，将演示最广泛使用和最简单的算法，即天真的“逃逸时间算法”。 <！ ——用于在计算机上绘制曼德布洛特集的最简单算法之一被称为“逃逸时间算法”。-- ! -- 空间反走样链接直接在这里 --

In the escape time algorithm, a repeating calculation is performed for each ''x'', ''y'' point in the plot area and based on the behavior of that calculation, a color is chosen for that pixel.

In the escape time algorithm, a repeating calculation is performed for each x, y point in the plot area and based on the behavior of that calculation, a color is chosen for that pixel.

在逃逸时间算法中，对绘图区域中的每个 x，y 点执行重复计算，并根据该计算的行为为该像素选择颜色。

The ''x'' and ''y'' locations of each point are used as starting values in a repeating, or iterating calculation (described in detail below). The result of each iteration is used as the starting values for the next. The values are checked during each iteration to see whether they have reached a critical "escape" condition, or "bailout". If that condition is reached, the calculation is stopped, the pixel is drawn, and the next ''x'', ''y'' point is examined.

The x and y locations of each point are used as starting values in a repeating, or iterating calculation (described in detail below). The result of each iteration is used as the starting values for the next. The values are checked during each iteration to see whether they have reached a critical "escape" condition, or "bailout". If that condition is reached, the calculation is stopped, the pixel is drawn, and the next x, y point is examined.

每个点的 x 和 y 位置用作重复计算或迭代计算中的起始值(下面将详细描述)。每次迭代的结果用作下一次迭代的起始值。在每次迭代期间都会检查这些值，以查看它们是否已经达到了关键的“逃逸”条件或“跳出”条件。如果达到该条件，则停止计算，绘制像素，并检查下一个 x，y 点。

The color of each point represents how quickly the values reached the escape point. Often black is used to show values that fail to escape before the iteration limit, and gradually brighter colors are used for points that escape. This gives a visual representation of how many cycles were required before reaching the escape condition.

The color of each point represents how quickly the values reached the escape point. Often black is used to show values that fail to escape before the iteration limit, and gradually brighter colors are used for points that escape. This gives a visual representation of how many cycles were required before reaching the escape condition.

每个点的颜色表示值到达转义点的速度。通常，黑色用于显示在迭代限制之前未能转义的值，并逐渐为转义的点使用较亮的颜色。这给出了在达到逃逸条件之前需要多少个循环的直观表示。

To render such an image, the region of the complex plane we are considering is subdivided into a certain number of [[pixel]]s. To color any such pixel, let <math>c</math> be the midpoint of that pixel. We now iterate the critical point 0 under <math>P_c</math>, checking at each step whether the orbit point has modulus larger than 2. When this is the case, we know that <math>c</math> does not belong to the Mandelbrot set, and we color our pixel according to the number of iterations used to find out. Otherwise, we keep iterating up to a fixed number of steps, after which we decide that our parameter is "probably" in the Mandelbrot set, or at least very close to it, and color the pixel black.

To render such an image, the region of the complex plane we are considering is subdivided into a certain number of pixels. To color any such pixel, let <math>c</math> be the midpoint of that pixel. We now iterate the critical point 0 under <math>P_c</math>, checking at each step whether the orbit point has modulus larger than 2. When this is the case, we know that <math>c</math> does not belong to the Mandelbrot set, and we color our pixel according to the number of iterations used to find out. Otherwise, we keep iterating up to a fixed number of steps, after which we decide that our parameter is "probably" in the Mandelbrot set, or at least very close to it, and color the pixel black.

为了渲染这样的图像，我们正在考虑的复平面的区域被再分割成一定数量的像素。要给任何这样的像素着色，让 math c / math 作为该像素的中点。我们现在用数学公式迭代临界点0，在每一步检查轨道点的模是否大于2。在这种情况下，我们知道 math c / math 不属于 Mandelbrot 集合，我们根据迭代次数给像素涂色。否则，我们继续迭代固定数量的步骤，然后我们决定我们的参数“可能”在 Mandelbrot 集中，或者至少非常接近它，并将像素涂成黑色。

In [[pseudocode]], this algorithm would look as follows. The algorithm does not use complex numbers and manually simulates complex-number operations using two real numbers, for those who do not have a [[complex data type]]. The program may be simplified if the programming language includes complex-data-type operations.

In pseudocode, this algorithm would look as follows. The algorithm does not use complex numbers and manually simulates complex-number operations using two real numbers, for those who do not have a complex data type. The program may be simplified if the programming language includes complex-data-type operations.

在伪代码中，该算法如下所示。该算法不使用复数，对于那些没有复数类型的数据，使用两个实数手动模拟复数操作。如果编程语言包含复杂数据类型的操作，则程序可以简化。





! -- 注意 xtemp 是必需的，否则 y 将使用新的 x 计算，这将是错误的。还要注意，必须绘制(x 小于0 / 小于，y 小于0 / 小于) ，而不是(x，y)。-->

'''for each''' pixel (Px, Py) on the screen '''do'''

for each pixel (Px, Py) on the screen do

对于屏幕上的每个像素(Px，Py)做

x0 = scaled x coordinate of pixel (scaled to lie in the Mandelbrot X scale (-2.5, 1))

x0 = scaled x coordinate of pixel (scaled to lie in the Mandelbrot X scale (-2.5, 1))

X0缩放 x 像素坐标(缩放为曼德布洛特 x 标度(- 2.5,1))

y0 = scaled y coordinate of pixel (scaled to lie in the Mandelbrot Y scale (-1, 1))

y0 = scaled y coordinate of pixel (scaled to lie in the Mandelbrot Y scale (-1, 1))

Y0像素的缩放 y 坐标(缩放到 Mandelbrot y 标度中(- 1,1))

x := 0.0

x := 0.0

0.0

y := 0.0

y := 0.0

0.0

iteration := 0

iteration := 0

迭代: 0

max_iteration := 1000

max_iteration := 1000

Max 迭代: 1000

'''while''' (x×x + y×y ≤ 2×2 AND iteration < max_iteration) '''do'''

while (x×x + y×y ≤ 2×2 AND iteration < max_iteration) do

而(x + y ≤22和迭代最大迭代)则可以得到

xtemp := x×x - y×y + x0

xtemp := x×x - y×y + x0

xtemp := x×x - y×y + x0

y := 2×x×y + y0

y := 2×x×y + y0

y := 2×x×y + y0

x := xtemp

x := xtemp

Xtemp

iteration := iteration + 1

iteration := iteration + 1

1. 迭代

color := palette[iteration]

color := palette[iteration]

调色板[迭代]

plot(Px, Py, color)

plot(Px, Py, color)

绘图(Px，Py，color)

Here, relating the pseudocode to <math>c</math>, <math>z</math> and <math>P_c</math>:

Here, relating the pseudocode to <math>c</math>, <math>z</math> and <math>P_c</math>:

在这里，将伪代码与 math c / math、 math z / math 和 math p c / math 联系起来:

* <math>z = x + iy\ </math>

* <math>z^2 = x^2 +i2xy - y^2\ </math>

* <math>c = x_0 + i y_0\ </math>

and so, as can be seen in the pseudocode in the computation of ''x'' and ''y'':

and so, as can be seen in the pseudocode in the computation of x and y:

所以，就像在 x 和 y 的计算中看到的伪代码:

* <math>x = \mathop{\mathrm{Re}}(z^2+c) = x^2-y^2 + x_0</math> and <math>y = \mathop{\mathrm{Im}}(z^2+c) = 2xy + y_0.\ </math>

To get colorful images of the set, the assignment of a color to each value of the number of executed iterations can be made using one of a variety of functions (linear, exponential, etc.).

To get colorful images of the set, the assignment of a color to each value of the number of executed iterations can be made using one of a variety of functions (linear, exponential, etc.).

为了得到集合的彩色图像，可以使用各种函数(线性、指数等)为每个已执行迭代次数的值分配一种颜色。).

==References in popular culture==

The Mandelbrot set is considered by many the most popular fractal,<ref>Mandelbaum, Ryan F. (2018). [https://gizmodo.com/this-trippy-music-video-is-made-of-3d-fractals-1822168809 "This Trippy Music Video Is Made of 3D Fractals."] Retrieved 17 January 2019</ref><ref>Moeller, Olga de. (2018).[https://thewest.com.au/lifestyle/kids/what-are-fractals-ng-b88838072z "what are Fratals?"] Retrieved 17 January 2019.</ref> and has been referenced several times in popular culture.

The Mandelbrot set is considered by many the most popular fractal, and has been referenced several times in popular culture.

曼德布洛特集合被许多人认为是最流行的分形，并在流行文化中多次被提及。

* The [[Jonathan Coulton]] song "Mandelbrot Set" is a tribute to both the fractal itself and to its discoverer Benoit Mandelbrot.<ref name="JoCopedia">{{cite web|title=Mandelbrot Set|url=http://www.jonathancoulton.com/wiki/Mandelbrot_Set|website=JoCopeda|accessdate=15 January 2015}}</ref>

* The second book of the ''[[Mode series]]'' by [[Piers Anthony]], ''Fractal Mode'', describes a world that is a perfect 3D model of the set.<ref name="Anthony1992">{{cite book|author=Piers Anthony|title=Fractal Mode|url=https://books.google.com/books?id=XdUyAAAACAAJ|year=1992|publisher=HarperCollins|isbn=978-0-246-13902-3}}</ref>

* The [[Arthur C. Clarke]] novel ''[[The Ghost from the Grand Banks]]'' features an artificial lake made to replicate the shape of the Mandelbrot set.<ref name="Clarke2011">{{cite book|author=Arthur C. Clarke|title=The Ghost From The Grand Banks|url=https://books.google.com/books?id=6ELsYigmXNoC|date=29 September 2011|publisher=Orion|isbn=978-0-575-12179-9}}</ref>

==See also==

* [[Buddhabrot]]

* [[Collatz fractal]]

* [[Fractint]]

* [[Gilbreath permutation]]

* [[Mandelbox]]

* [[Mandelbulb]]

* [[Menger Sponge]]

* [[Newton fractal]]

* [[Orbit portrait]]

* [[Orbit trap]]

* [[Pickover stalk]]

==References==

{{Reflist|30em}}

==Further reading==

* [[John W. Milnor]], ''Dynamics in One Complex Variable'' (Third Edition), Annals of Mathematics Studies 160, (Princeton University Press, 2006), {{isbn|0-691-12488-4}} (First appeared in 1990 as a [https://web.archive.org/web/20060424085751/http://www.math.sunysb.edu/preprints.html Stony Brook IMS Preprint], available as [http://www.arxiv.org/abs/math.DS/9201272 arXiV:math.DS/9201272] )

* Nigel Lesmoir-Gordon, ''The Colours of Infinity: The Beauty, The Power and the Sense of Fractals'', {{isbn|1-904555-05-5}} (includes a DVD featuring [[Arthur C. Clarke]] and [[David Gilmour]])

* [[Heinz-Otto Peitgen]], [[Hartmut Jürgens]], [[Dietmar Saupe]], ''Chaos and Fractals: New Frontiers of Science'' (Springer, New York, 1992, 2004), {{isbn|0-387-20229-3}}

==External links==

{{Wikibooks|Fractals }}

{{commons}}











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* {{Curlie|Science/Math/Chaos_and_Fractals|Chaos and Fractals}}

* [http://classes.yale.edu/Fractals/MandelSet/welcome.html The Mandelbrot Set and Julia Sets by Michael Frame, Benoit Mandelbrot, and Nial Neger]

* [http://vimeo.com/12185093 Video: Mandelbrot fractal zoom to 6.066 e228]

* [https://www.youtube.com/watch?v=NGMRB4O922I Relatively simple explanation of the mathematical process, by [[Holly Krieger|Dr Holly Krieger]], MIT]

* [https://mandelbrot-svelte.netlify.com Mandelbrot set images online rendering]

* [https://www.rosettacode.org/wiki/Mandelbrot_set Various algorithms for calculating the Mandelbrot set] (on [[Rosetta Code]])

* [https://github.com/pkulchenko/ZeroBraneEduPack/blob/master/fractal-samples/zplane.lua Fractal calculator written in Lua by Deyan Dobromiroiv, Sofia, Bulgaria]

{{Fractal software}}

{{Fractals}}

{{DEFAULTSORT:Mandelbrot Set}}

[[Category:Fractals]]

Category:Fractals

分类: 分形

[[Category:Articles containing video clips]]

Category:Articles containing video clips

类别: 包含视频剪辑的文章

[[Category:Articles with example pseudocode]]

Category:Articles with example pseudocode

类别: 带有伪代码示例的文章

[[Category:Complex dynamics]]

Category:Complex dynamics

类别: 复动力学

<noinclude>

This page was moved from [[wikipedia:en:Mandelbrot set]]. Its edit history can be viewed at [[曼德布罗特集/edithistory]]</noinclude>

[[Category:待整理页面]]

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