信息生产:曼德布洛特集

信息内容

[图片1： The Mandelbrot set (black) within a continuously colored environment（图1）将不属于曼德布洛特集合内的点按照发散速度赋予不同的颜色所得到的图形/经过连续染色后曼德尔勃特集合(集合内染为黑色)]

[图片2：Progressive infinite iterations of the "Nautilus" section of the Mandelbrot Set rendered using webGL（图2）利用 " WebGL”（ 3D绘图协议）将曼德布洛特集的鹦鹉螺属部分进行渐进式无限迭代呈现出来的图形]

[图片3：Mandelbrot animation based on a static number of iterations per pixel（图3）基于不同像素静态迭代数的曼德布洛特集动画]

[图片4：Mandelbrot set detail （图4）曼德布洛特集的细节部分]

The Mandelbrot set is the set of complex numbers {\displaystyle c} for which the function {\displaystyle f_{c}(z)=z^{2}+c} does not diverge when iterated from {\displaystyle z=0}, i.e., for which the sequence {\displaystyle f_{c}(0)}, {\displaystyle f_{c}(f_{c}(0))}, etc., remains bounded in absolute value.

[图片5：Zooming into the Mandelbrot set （图5）放大后的曼德布洛特集]

Its definition is credited to Adrien Douady who named it in tribute to the mathematician Benoit Mandelbrot.[1] The set is connected to a Julia set, and related Julia sets produce similarly complex fractal shapes.

Mandelbrot set images may be created by sampling the complex numbers and testing, for each sample point {\displaystyle c}, whether the sequence {\displaystyle f_{c}(0),f_{c}(f_{c}(0)),\dotsc } 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 {\displaystyle c} as image coordinates on the complex plane, pixels may then be coloured according to how soon the sequence {\displaystyle |f_{c}(0)|,|f_{c}(f_{c}(0))|,\dotsc } crosses an arbitrarily chosen threshold, with a special color (usually black) used for the values of {\displaystyle c} 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 {\displaystyle c} is held constant and the initial value of {\displaystyle z}—denoted by {\displaystyle z_{0}}—is variable instead, one obtains the corresponding Julia set for each point {\displaystyle c} in the parameter space of the simple function. 通过选取不同的复数$\displaystyle{ C }$，观察其数列 $\displaystyle{ f_c(0), f_c(f_c(0)),\dotsc }$是否达到无穷大（即其是否在预设的迭代次数后离开之前预设好的某个含0在内的有界领域）。将$\displaystyle{ c }$的实部作为复平面图像 Complex Plane的横坐标，虚部作为复平面图像的纵坐标。 然后根据数列$\displaystyle{ |f_c(0)|,|f_c(f_c(0))|,\dotsc }$。 若数列发散，则在二维平面内将所有属于集合内的点标记为黑色，不属于集合内的点按照发散速度赋予不同的颜色，就可以得到经典的曼德布洛特集图像。 需要注意的是，判断在无限迭代后其数列是否保持有界，是区分曼德布洛特集与其补集图像的有效方法。令复数C保持不变，取某一z值（如$\displaystyle{ z=z_0 }$），可以得到数列$\displaystyle{ z_0,|f_c(z_0)|, |f_c(f_c(z_0))|,\dotsc }$。这一数列可能发散于无穷大或始终处于某一范围之内并收敛于某一值。我们将使得该数列收敛于某值的z值的集合称为朱利亚集。若令$\displaystyle{ z_0 }$取不同的值，则在简单函数的参数空间 Parameter Space中能够得到不同$\displaystyle{ z_0 }$值所对应的朱利亚集。

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. 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.

历史 History

[图片6：The first published picture of the Mandelbrot set, by Robert W. Brooks and Peter Matelski in 1978 （图6）罗伯特 · W· 布鲁克斯和彼得 · 马特尔斯基在1978年发表了第一张曼德尔布洛特集的图像]

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.[2] 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.[3]Mandelbrot studied the parameter space of quadratic polynomials in an article that appeared in 1980.[4]

The mathematical study of the Mandelbrot set really began with work by the mathematicians Adrien Douady and John H. Hubbard,[1] who established many of its fundamental properties and named the set in honor of Mandelbrot for his influential work in fractal geometry.

The mathematicians Heinz-Otto Peitgen and Peter Richter became well known for promoting the set with photographs, books,[5] and an internationally touring exhibit of the German Goethe-Institut.[6][7] 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 -0.909 + -0.275 and was created by Peitgen, et al.[8][9] 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.[10]

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,[11][12] Curt McMullen, John Milnor, Mitsuhiro Shishikura and Jean-Christophe Yoccoz.

Adrien Douady和 John H. Hubbard 的研究工作不断取得新的成果，感兴趣于复动力学和抽象数学 Abstract Mathematics领域的队伍快速壮大，自此，深入了解曼德布洛特集一直是这些领域的核心研究。包括米哈伊尔 · 柳比奇 Mikhail Lyubich,[11][12] 科特 · 麦克马伦 Curt McMullen, 约翰 · 米尔诺 John Milnor, 石仓光博 Mitsuhiro Shishikura and 让-克里斯托夫·约科兹Jean-Christophe Yoccoz在内的许多人在曼德布洛特集的研究工作中，都作出了大大小小的贡献。

--趣木木讨论） 2020年4月6日 (一) 04:12 (UTC)第一句中的coincided weith 相吻合 换了并列句进行叙述

正式定义 Formal definition

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{\displaystyle z_{n+1}=z_{n}^{2}+c}remains bounded.[13]

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. 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.

[图片7：A mathematician's depiction of the Mandelbrot set .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. （图7）一位数学家对曼德布洛特集进行了以下描述: 曼德布洛特集仅代表图中的黑色区域，即属于曼德布洛特集的复数c在黑色区域中，不属于曼德布洛特集的复数c在周围的白色区域中。 Re [ c ]和 Im [ c ]分别表示复数 c 的实部和虚部。]

The Mandelbrot set can also be defined as the connectedness locus of a family of polynomials. 曼德布洛特集也可以定义为一族多项式 Polynomials连通轨迹 Connectedness Locus

基本性质 Basic properties

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 {\displaystyle c}{ displaystyle c } belongs to the Mandelbrot set if and only if$\displaystyle{ |P_c^n(0)|\leq 2 }$ for all {\displaystyle n\geq 0.}

In other words, if the absolute value of {\displaystyle P_{c}^{n}(0)} ever becomes larger than 2, the sequence will escape to infinity. 也就是说，若$\displaystyle{ |P_c^n(0)| }$大于2，则说明其函数值数列发散，所对应的复数$\displaystyle{ c }$不属于曼德布洛特集。

[图片8：Correspondence between the Mandelbrot set and the bifurcation diagram of the logistic map （图8）曼德布洛特集图像与逻辑斯蒂映射中的分叉图之间的关系。]

[图片9：With {\displaystyle z_{n}} iterates plotted on the vertical axis, the Mandelbrot set can be seen to bifurcate where the set is finite （图9）在y轴上对于Zn进行迭代，可以看到曼德布洛特集图像的收敛点处出现分叉]

The intersection of {\displaystyle M} 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。

--趣木木讨论） 2020年4月6日 (一) 04:12 (UTC)文中最后的logistic family链接与logistic map重定向，故统一译为Logistic Map。 其中，逻辑斯蒂映射 Logistic Map是一种二次的多项式映射（递归关系式），是一个由简单非线性方程序产生混沌现象的经典范例。 该句为补充

$\displaystyle{ x_{n+1} = r x_n(1-x_n),\quad r\in[1,4]. }$

The correspondence is given by{\displaystyle z=r\left({\frac {1}{2}}-x\right),\quad c={\frac {r}{2}}\left(1-{\frac {r}{2}}\right).} 其对应关系为 $\displaystyle{ z = r\left(\frac12 - x\right),\quad c = \frac{r}{2}\left(1-\frac{r}{2}\right). }$

In fact, this gives a correspondence between the entire parameter space of the logistic family and that of the Mandelbrot set. 事实上，这反映了逻辑斯蒂映射和曼德布洛特集的参数空间 Parameter Space之间的对应关系。

Douady and Hubbard have shown that the Mandelbrot set is connected. 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 {\displaystyle M} Upon further experiments, he revised his conjecture, deciding that {\displaystyle M} should be connected. There also exists a topological proof to the connectedness that was discovered in 2001 by Jeremy Kahn.[14]

Adrien Douady和 John H. Hubbard在曼德布洛特集的补集与闭合单位圆盘（以原点为中心，以1为半径做圆）的补集之间构造了一个显式的共形同构 Conformal Isomorphism ，由此证明了曼德布洛特集是连通的。 而由于当时计算机程序的局限性，导致程序无法检测到所生成的曼德布洛特集图形中连接不同细部的微小连线。Benoît B. Mandelbrot 最初猜测曼德布洛特集是不连通的。通过进一步的实验后，Benoît B. Mandelbrot 纠正了之前的看法，认为曼德布洛特集是连通的。 此外，杰里米 · 卡恩 Jeremy Kahn在2001年利用严格的拓扑证明，论证了曼德布洛特集的连通性。[14]

[图片10：External rays of wakes near the period 1 continent in the Mandelbrot set: （图10）在曼德布洛特集运行一周期的附近的外部尾迹射线]

The dynamical formula for the uniformisation of the complement of the Mandelbrot set, arising from Douady and Hubbard's proof of the connectedness of {\displaystyle M}，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.[15]

The boundary of the Mandelbrot set is exactly the bifurcation locus of the quadratic family; that is, the set of parameters {\displaystyle c} for which the dynamics changes abruptly under small changes of {\displaystyle c.} 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.

其他性质 Other properties

心脏形结构和圆盘形“芽苞”Main cardioid and period bulbs

[图片11：Periods of hyperbolic components（图11）双曲分量的周期]

Upon looking at a picture of the Mandelbrot set, one immediately notices the large cardioid-shaped region in the center. This main cardioid is the region of parameters {\displaystyle c} for which {\displaystyle P_{c}} has an attracting fixed point. It consists of all parameters of the form {\displaystyle c={\frac {\mu }{2}}\left(1-{\frac {\mu }{2}}\right)} for some {\displaystyle \mu } in the open unit disk. 观察曼德布洛特集的图像，会首先看到中间的心脏形结构。（即图11中的区域1）该心形曲线内部的参数$\displaystyle{ c }$若满足$\displaystyle{ c = \frac\mu2\left(1-\frac\mu2\right) }$， 其中 $\displaystyle{ \mu }$ 属于单位开圆盘； 则称这样的点$\displaystyle{ c }$，为复二次映射的周期点，记作$\displaystyle{ P_c }$

[图片12：To the left of the main cardioid, attached to it at the point {\displaystyle c=-3/4}, a circular-shaped bulb is visible. This bulb consists of those parameters {\displaystyle c} for which {\displaystyle P_{c}} has an attracting cycle of period 2. This set of parameters is an actual circle, namely that of radius 1/4 around −1. （图12）在主心形线的左侧，点（-3/4，0）处，能够看到一个圆盘形的“芽苞”突起。该“芽苞”由一周期2的吸性周期循环的参数c的集合组成，其为曼德布洛特集的一个附属子集。该子集组成了以（-1，0）为中心，以1/4 为半径的圆形区域。]

There are infinitely many other bulbs tangent to the main cardioid: for every rational number {\displaystyle {\tfrac {p}{q}}}, with p and q coprime, there is such a bulb that is tangent at the parameter.This bulb is called the {\displaystyle {\tfrac {p}{q}}}-bulb of the Mandelbrot set. 类似这样的“芽苞”突起有无穷多个，它们都与主心脏形结构曲线相切。 且满足以下定义： 对于任意有理数 $\displaystyle{ \tfrac{p}{q} }$ ，p、q互素，若参数c满足：

$\displaystyle{ c_{\frac{p}{q}} = \frac{e^{2\pi i\frac pq}}2\left(1-\frac{e^{2\pi i\frac pq}}2\right). }$

[图片13：Attracting cycle in 2/5-bulb plotted over Julia set (animation):(图13）在 Julia 集上绘制一周期为2 / 5的吸性周期循环的“芽苞”(动画)]

It consists of parameters that have an attracting cycle of period {\displaystyle q} and combinatorial rotation number {\displaystyle {\tfrac {p}{q}}}. More precisely, the {\displaystyle q} periodic Fatou components containing the attracting cycle all touch at a common point (commonly called the {\displaystyle \alpha }-fixed point). If we label these components {\displaystyle U_{0},\dots ,U_{q-1}} in counterclockwise orientation, then {\displaystyle P_{c}} maps the component {\displaystyle U_{j}} to the component {\displaystyle U_{j+p\,(\operatorname {mod} q)}}.

$\displaystyle{ c_{\frac{p}{q}} = \frac{e^{2\pi i\frac pq}}2\left(1-\frac{e^{2\pi i\frac pq}}2\right) }$公式中的参数其由吸性循环周期的周期值q和组合旋转数 $\displaystyle{ \tfrac{p}{q} }$ 组成。包含周期为q的吸性周期循环的Factou 集 Fatou components 都在吸性不动点相交。若记分量$\displaystyle{ U_0,\dots,U_{q-1} }$为逆时针方向，$\displaystyle{ P_c }$将分量$\displaystyle{ U_j }$ 映射到分量$\displaystyle{ U_{j+p\,(\operatorname{mod} q)} }$

[图片14：（Attracting cycles and Julia sets for parameters in the 1/2, 3/7, 2/5, 1/3, 1/4, and 1/5 bulb （图14）吸性周期为 1/2, 3/7, 2/5, 1/3, 1/4, and 1/5 “芽苞”的朱利亚基]

[图片15：Cycle periods and antennae :（图15）周期和分枝]

The change of behavior occurring at {\displaystyle c_{\frac {p}{q}}} 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 {\displaystyle {\tfrac {p}{q}}}-bulb, the attracting fixed point turns into a repelling fixed point (the {\displaystyle \alpha }-fixed point), and the period q-cycle becomes attracting. $\displaystyle{ c_{\frac{p}{q}} }$分枝 Bifurcation点，在该点吸性的不动点与斥性的周期循环发生相反的变化。当经过分枝点进入$\displaystyle{ \tfrac{p}{q} }$-“芽苞”时，即$\displaystyle{ c_{\frac{p}{q}} }$ 从心脏形结构曲线内部变为开圆内部时，不动点$\displaystyle{ \alpha }$由吸性变为斥性，周期循环由斥性变为吸性。

双曲分量 Hyperbolic components

All the bulbs we encountered in the previous section were interior components of the Mandelbrot set in which the maps {\displaystyle P_{c}} have an attracting periodic cycle. Such components are called hyperbolic components. 我们上述所提及的圆盘形“芽苞”都是曼德布洛特集的内部分量，其中映射$\displaystyle{ P_{c} }$具有吸性周期循环，这样的分量称为双曲分量。

It is conjectured that these are the only interior regions of {\displaystyle M}. 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.[16] 推测这些双曲分量是否仅存在于曼德布洛特集内部区域，称为双曲密度问题。这或许也是复动力学领域中最值得关注的公开问题。曼德布洛特集中假设的非双曲分量通被称为“怪胎”或幽灵分量。

[17] 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，在1990年，双曲密度问题得到解决。柳比奇 Lyubich格拉奇克 Graczyk斯维亚特克 Świątek分别独立证明了双曲分量仅存在于曼德布洛特集的内部区域。

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). 并不是每个双曲分量都可由曼德布洛特集的主心脏形结构经过一系列的直接分叉即可得到。但像图（15）的双曲分量可由小的曼德布洛特集副本的主心脏结构曲线经过一系列的直接分叉得到。 [图片16：Cycle periods and antennae（图16）周期和天线]

Each of the hyperbolic components has a center, which is a point c such that the inner Fatou domain for {\displaystyle P_{c}(z)} has a super-attracting cycle – that is, that the attraction is infinite. This means that the cycle contains the critical point 0, so that 0 is iterated back to itself after some iterations. 每个双曲分量都有一个中心，该中心记作点$\displaystyle{ c }$，使得$\displaystyle{ P_{c}（z） }$的内部Fatou区域具有一个超吸性周期循环，即其吸引力是无穷的。这意味着该循环包含临界点0，因此经过数次迭代后，0会回到本身。

We therefore have that {\displaystyle P_{c}}n{\displaystyle (0)=0} for some n. If we call this polynomial {\displaystyle Q^{n}(c)} (letting it depend on c instead of z), we have that {\displaystyle Q^{n+1}(c)=Q^{n}(c)^{2}+c} and that the degree of {\displaystyle Q^{n}(c)} is {\displaystyle 2^{n-1}}. We can therefore construct the centers of the hyperbolic components by successively solving the equations {\displaystyle Q^{n}(c)=0,n=1,2,3,...}. The number of new centers produced in each step is given by Sloane's OEIS: A00074 也就是说，将0带入$\displaystyle{ P_{c}（z） }$，存在n,使得$\displaystyle{ P_c }$n$\displaystyle{ (0) = 0 }$。 如果将公式的变量固定为$\displaystyle{ c }$而不是$\displaystyle{ z }$,得到 $\displaystyle{ Q^{n}(c) }$。 由$\displaystyle{ Q^{n+1}(c) = Q^{n}(c)^{2} + c }$，且令$\displaystyle{ Q^{n}(c)的纬度为2^{n-1} }$。 因此，我们可以通过依次求解方程$\displaystyle{ Q^{n}(c) = 0, n = 1, 2, 3, ... }$来构造双曲分量的中心。每一步产生的新中心的个数由斯隆的OEIS: A00074给出。OEIS的全称为：The On-Line Encyclopedia of Integer Sequences™ (OEIS™)，它是一个关于整数序列（数列）的专业型网站，是一个关于组合数学研究的重要的网站，里面包含了众多数列的研究成果。 趣木木讨论）解释一下OEIS的一些简单内容“ OEIS的全称为：The On-Line Encyclopedia of Integer Sequences™ (OEIS™)，它是一个关于整数序列（数列）的专业型网站，是一个关于组合数学研究的重要的网站，里面包含了众多数列的研究成果。”

连通性 Local connectivity

[图片17：Topological model of Mandelbrot set without mini Mandelbrot sets and Misiurewicz points (Cactus model)（图17）没有曼德布洛特集的微小副本和 Misiurewicz 点的曼德布洛特拓扑模型(Cactus 模型)））]

[图片18：Thurston model of Mandelbrot set (abstract Mandelbrot set) （图18）曼德布洛特集的Thurston模型(摘要曼德布洛特集）]

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. 在Adrien Douady 和 John H. Hubbard的研究工作中，提出了著名的MLC猜想：他们推测曼德布洛特集是局部连通的。这一猜想将曼德布洛特集简单抽象成一个“压缩圆盘”。 其中，它还体现了上述所提及的双曲分量的思想。

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.[18] Since then, local connectivity has been proved at many other points of {\displaystyle M}, but the full conjecture is still open. 让-克里斯托夫·约科兹 Jean-Christophe Yoccoz证明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，该种局部连通性只体现在有限多个小曼德布洛特集副本中。 [18]从那时起，在曼德布洛特集的其他参数点也体现了局部连通性。

自相似 Self-similarity

[图片19：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 {\displaystyle \delta }. 曼德布洛特集中的自相似性通过放大一个圆形“芽苞”，并将其中心往负x轴方向迁移来体现。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Ħẟ]

[图片20：Self-similarity around Misiurewicz point −0.1011 + 0.9563i. （图20）在Misiurewicz点−0.1011 + 0.9563i附近的自相似性]

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.[19][20] 在 Misiurewicz 点附近，对曼德布洛特集进行放大，能够观察到自相似性。将其收敛于一个极限集后，我们还推测在广义 Feigenbaum 点(例如-1.401155或-0.1528 + 1.0397 i)周围能够观察到自相似的特征。 [19][20]

[图片21：Quasi-self-similarity in the Mandelbrot set （图21）曼德布洛特集中的准自相似性] 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. 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. 总的来说，曼德布洛特集合并不是严格意义上的具有自相似特征的集合，但它具有准自相似性，因为在任意小的空间尺度上，都可以找到与自身略有不同的小副本。小副本之间的细微差别体现在它们与整体集合之间的连接细线上。

进一步的计算结果 Further results

The Hausdorff dimension of the boundary of the Mandelbrot set equals 2 as determined by a result of Mitsuhiro Shishikura.[21] It is not known whether the boundary of the Mandelbrot set has positive planar Lebesgue measure. Mitsuhiro Shishikura计算出曼德布洛特集分界线的豪斯多夫维数 Hausdorff dimension 为2。 现在还不知道曼德布洛特集分界线是否具有正平面上的勒贝格测度 Lebesgue Measure

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. 利用可在实数域内计算的 Blum-Shub-Smale 模型 Blum-Shub-Smale Model （简称BBS模型），可得出以下结论：虽然无法对于曼德布洛特集进行计算，但可计算其补集。BBS模型也不可计算包括求幂图在内的许多比较简单的图像。目前，在基于可计算性分析 Computable Analysis的实际计算模型中，曼德布洛特集是否可算尚未有定论。由该种模型计算集合的方式更接近于直观概念上的“由计算机绘制集合”。 赫特林 Hertling 证明了若双曲性猜想是正确的，则该模型中的曼德布洛特也是可计算的。

[图片22：A zoom into the Mandelbrot set illustrating a Julia "island" and a similar Julia set.将曼德布洛特集进行放大，可观察到朱利亚岛和一个与朱利亚集很相似的结构]

与朱利亚集之间的关系

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. 由曼德布洛特集的定义，在给定点处曼德布洛特集的几何结构与相应的朱利亚集的几何结构有着密切的对应关系。 例如，当一个点在曼德布洛特集聚集时，相应的朱利亚集是连通的。

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.[21] Similarly, Yoccoz first proved the local connectivity of Julia sets, before establishing it for the Mandelbrot set at the corresponding parameters.[18] Adrien Douady phrases this principle as: Plough in the dynamical plane, and harvest in parameter space. 事实上，该原理被运用到曼德布洛特集的所有深层结果中。 例如，Shishikura 证明了对于在曼德布洛特集分界线上的一组稠密的参数，相对应的朱利亚集 Julia set的豪斯多夫维数为2，然后将这些信息传递到参数平面上。 [21]同样，Yoccoz 首先证明了朱利亚集的局部连通性，然后在相应的参数点处进一步证明曼德布洛特集的局部连通性。 [18]Adrien Douady将这一原则表述为:在动力平面上耕耘，在参数空间中收获。

几何结构Geometry

[图片23：Components on main cardioid for periods 8–14 with antennae 7–13(图23)主心脏形结构上带有7-13个天线的8-14个周期图案]

For every rational number {\displaystyle {\tfrac {p}{q}}}, 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 {\displaystyle {\tfrac {1}{q^{2}}}}. The best current estimate known is the Yoccoz-inequality, which states that the size tends to zero like {\displaystyle {\tfrac {1}{q}}}. 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. 对于任意有理数$\displaystyle{ \tfrac{p}{q} }$，其中pq互素，周期 q 的一个双曲分量从主心脏形结构分支出来。 在该分枝点上与主心脏形结构相连的曼德布洛特集部分称为 p / q-limb。 计算机实验表明，分枝体半径像$\displaystyle{ \tfrac{1}{q^2} }$一样趋于0，目前最佳的拟合形式是约科兹不等式 Yoccoz-inequality，其尺寸像$\displaystyle{ \tfrac{1}{q} }$一样趋于0。 周期为$\displaystyle{ q }$的分枝体顶端有$\displaystyle{ q-1 }$个天线。我们可以通过计数这些触角反过来推算该特定分枝体的周期为多少。

曼德布洛特集中的πPi 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 = {\displaystyle -{\tfrac {3}{4}}+i\epsilon } ({\displaystyle -{\tfrac {3}{4}}} being the location thereof). As the series doesn't diverge for the exact value of z = {\displaystyle -{\tfrac {3}{4}}}, 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 $\displaystyle{ -\tfrac{3}{4} + i\epsilon }$ ($\displaystyle{ -\tfrac{3}{4} }$ iterations is 31415928 and the product is 3.1415928.[22]

曼德布洛特集中的斐波那契数列Fibonacci sequence in 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. 可以看出，斐波那契数列 Fibonacci Sequence位于曼德布洛特集内，且与主心脏形结构曲线和法雷数列图存在关联。

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. 将主心脏结构曲线映射到一个圆盘上时，会观察到由下一个最大的双曲分量延伸出来的天线，其位于之前选择的两个分量之间。天线的数目变化是一个斐波那契数列。

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 {\displaystyle {\frac {1}{3}}}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. 天线数目也与法雷数列图有关，由于n级法雷数列是指对任意给定的一个自然数n，将分母小于等于n的不可约的真分数按升序排列，并且在第一个分数之前加上0/1，在最后一个分数之后加上1/1所得到的数列，以$\displaystyle{ F_n }$表示。如$\displaystyle{ F_5 }$为：0/1, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1/1。

不同缩放比率下的图像库Image gallery of a zoom sequence

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. 当一个人看得越近或放大图像时，达到的放大效果可以让曼德布洛特集显示出更复杂的细节。当将下图不断的放大，缩放到选定的$\displaystyle{ c }$值形成一个反映其变化的图集时，会给人一种不同几何结构中蕴藏着无限想象力的感觉。下也对于它们的一些典型规则进行说明。 [图片24：Start. Mandelbrot set with continuously colored environment.（图24）开始：将曼德布洛特集进行着色，以便于观察]

[图片25：Gap between the "head" and the "body", also called the "seahorse valley"(图25）主心脏形结构和第二大圆盘之间的空隙称为“海马谷”]

[图片26：Double-spirals on the left, "seahorses" on the right （图26）左边是双螺旋，右边是“海马”]

[图片27："Seahorse" upside down（图27）颠倒过来的海马]

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. 图27相对于图24的放大倍率之比约为1010：1。以普通的显示器显示出该图像的完整图形，其表示直径为400万公里的的曼德布洛特集合的一部分，其分界线将显示出数量庞大的不同分形结构。 趣木木讨论）以显示器有关 觉得应该是指 按照图27的缩放倍率将整个图像进行填充 讨论最后成型的图形

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. 海马的“身体”由25个“辐条”组成。25个“辐条”被分为三组，其中两组中各含有12个“辐条”，另一组单独由一个连接到主心脏形结构的“辐条”组成。这各含有12个“辐条”的两组可以通过某种变形变集合的“上臂”为曼德布洛特集的两根“手指”；中心是通常说的 Misiurewicz 点。在海马的“身体的上半部分”和“尾巴”之间，可发现一个扭曲的曼德布洛特集的小副本。该集合称为“卫星集”，也就是附属集。 [图片28：The central endpoint of the "seahorse tail" is also a Misiurewicz point.(图28)“海马尾巴”的中心端点也是 Misiurewicz 点。]

[图片29：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. （图29）海马“尾巴”的一部分ーー只有一条路径是由贯穿整个“尾巴”的细小结构组成的。 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” ; 因此曼德布洛特集是一个单连通集，这意味着在洞周围没有岛屿和环路。]

[图片30：Satellite. The two "seahorse tails" are the beginning of a series of concentric crowns with the satellite in the center. Open this location in an interactive viewer.（图30）卫星集。这两个“海马尾”是一系列同心冠的源头，冠的中心是卫星集。 在交互式查看器中打开此位置。] 趣木木讨论）考虑将最后一句删去

[图片31：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". （图31）每个冠都由类似的“海马尾”组成; 它们的数量随着2的幂数增加而增加，这是卫星环境中的典型现象。通向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]

[图片32："Antenna" of the satellite. Several satellites of second order may be recognized.（图32）卫星的“天线”。可以辨认出几颗二级卫星。]

[图片33：The "seahorse valley" of the satellite. All the structures from the start of the zoom reappear.（图33）卫星的“海马谷”。所有的结构以一开始的图形样式再次出现]

[图片34：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.（图34） 双螺旋和“海马”——与第二张图片开始时不同，它们有附属结构，如“海马尾” ; 这展示了 n + 1个不同结构在 n 阶卫星环境中的典型连接，最简单的情况是 n = 1。]

[图片35：Double-spirals with satellites of second order – analogously to the "seahorses", the double-spirals may be interpreted as a metamorphosis of the "antenna"（图35）具有二级卫星的双螺旋——类似于“海马” ，双螺旋由“天线”演化而来]

[图片36：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（图36）能够辨认出附属集的外部岛结构，它们的形状类似于朱利亚集$\displaystyle{ J_c }$.其中最大的岛结构可在右侧的“双钩”中心找到。

[图片37：Part of the "double-hook"（图37）“双钩”的一部分]

[图片38：Islands（图38）岛群]

[图片39：Detail of one island （图39）一个岛屿的细节部分]

[图片40：Detail of the spiral. Open this location in an interactive viewer.（图40）其中一个螺旋的细节部分，利用交互式查看器中打开]

The islands above seem to consist of infinitely many parts like Cantor sets, as is[clarification needed] 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. 上面的岛屿似乎是由无限多的部分组成，就像康托集 Cantor Sets一样。但实际上这是对应的朱利亚集$\displaystyle{ J_c }$的情况。然而，它们通过微小的结构进行连接，其整体代表了一个单连通集。

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. 这些微小的结构在中间的卫星集处相遇，在这样的放大倍率下，该卫星集太小以至于无法被识别。相应的$\displaystyle{ J_c }$$\displaystyle{ c }$值不是该图像中心所对应的$\displaystyle{ c }$值，而是第六个缩放步骤中，相对于曼德布洛特集的主体，其中心所示的相同位置上的卫星集所对应的$\displaystyle{ c }$值。

曼德布洛特集和朱利亚集的3D图像 3D images of Mandelbrot and Julia sets

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. 除了能绘制出曼德布洛特集的二维图像外，还可以利用各种技术绘制出曼德布洛特集和朱利亚集的三维图像。具体做法为：将其二维图像中的每个像素赋予一个高度值，即可生成三维图像。

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. 最简单的三维图像渲染方法是将每个像素的迭代值作为高度值，从而生成具有不同高度值的“台阶”的图像。

[图片41：Mandelbrot set rendered in 3D using integer iterations （图41）利用整数迭代渲染曼德布洛特集的三维图像]

If instead you use the fractional iteration value (also known as the potential function[23] 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[23])来计算每个点的高度值，则可以避免在生成的图像中执行步骤。 然而，使用分数迭代数据渲染的三维图像看起来比较粗糙且不太美观。

[图片42：Mandelbrot set rendered in 3D using fractional iteration values （图41）利用分数迭代渲染曼德布洛特集的三维图像]

An alternative approach is to use Distance Estimate[24] (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[24](DE)数据来计算高度值。利用指数函数对距离估计值进行非线性映射，可以得到感观较好的图像。使用 DE 数据绘制的图像往往在视觉上更引人关注。更重要的是，三维图像使得连接图像中各点的细“卷须”更易于观察。 在“分形图像的科学”第121页的图29、30显示了使用外部距离估计绘制的二维和三维图像。

[图片43：Mandelbrot set rendered in 3D using Distance Estimates（图43）利用距离估计渲染曼德布洛特集的三维图像。]

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“一直缩放的三维曼德布洛特集图集”，在高度值图中利用距离估计将其进行渲染，并使用类似的裁剪和着色。 趣木木讨论）height maps可以理解为高度值组成的图吗？

[图片44：Zoom 00. Start. Mandelbrot set with continuously colored environment.（图44）开始，曼德布洛特集进行一系列的着色]

[图片45：Zoom 01. Gap between the "head" and the "body", also called the "seahorse valley"（图45）“头部”和“身体”之间的空隙，也称为“海马谷”]

[图片46：Zoom 02. Double-spirals on the left, "seahorses" on the right (图46)左边是双螺旋，右边是“海马”]

[图片47：Zoom 03. "Seahorse" upside down （图47）颠倒后的“海马”]

[图片48：Zoom 04. A "seahorse tail".（图48）海马尾]

[图片49：Zoom 05. Part of the "tail".（图49）海马尾的一部分]

[图片50：Zoom 06. Satellite with twin "Seahorse tails."（图50）拥有一对“海马尾”的卫星集]

[图片51：Zoom 07. Satellite closeup. （图51）卫星集的特写]

[图片52：Zoom 08. "Antenna" of the satellite. Several satellites of second order may be recognized.（图52）卫星的“天线”，可以识别出几颗二级卫星。]

[图片53：Zoom 09. The "seahorse valley" of the satellite. All the structures from the start of the zoom reappear.（图53）卫星的“海马谷”。所有的结构以一开始的图形样式再次出现]

[图片54：Zoom 10. Double-spirals and "seahorses" （图54）双螺旋和“海马”]

[图片55：Zoom 11. Double-spirals with satellites of second order. (图55)二级卫星的双螺旋]

[图片56：Zoom 12.（图56）]

[图片57：Zoom 13. Part of the "double-hook"（图57）“双钩”的一部分]

[图片58：Zoom 14. Islands （图58）岛群]

[图片59：Zoom 15. Detail of one island（图59）一个岛的细节部分]

[图片60：Zoom 16. Detail of the spiral.（图60）螺旋的细节部分]

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"[25] using a visually similar color scheme that shows the details of the plot in 3D. 下图类似于上图中的“图49”，它来自于《分形之美》这本书的第85页图44的3D图像版本。

[图片61：A 3D version of the Mandelbrot set plot "Map 44" from the book "The Beauty of Fractals 《分形之美》这本书的第85页图44的曼德布洛特集的3D图像版本]

推广 Generalizations

[图片62：Animations of the Multibrot set for d from 0 to 5 (left) and from 0.05 to 2 (right).（图62）左从0到5，右从0.05到2的多重曼德布洛特集的动画。

[图片63：A 4D Julia set may be projected or cross-sectioned into 3D, and because of this a 4D Mandelbrot is also possible.（图63）由于四维曼德布洛特集可以通过投影或横切成三维，故四维朱利亚集也可以进行该种变换]

多重曼德布洛特集 Multibrot sets

Multibrot sets are bounded sets found in the complex plane for members of the general monic univariate polynomial family of recursions {\displaystyle z\mapsto z^{d}+c.\ } 在数学中，Multibrot集是复平面上的有限集，集中的元素取自一般一元多项式递归族中。如： $\displaystyle{ z \mapsto z^d + c.\ }$ 这个名字是多重和曼德布洛特集的混合词。同样的也适用于朱利亚集合，称为多重朱利亚集 Multi-Julia set

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 {\displaystyle z\mapsto z^{3}+3kz+c}, 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.[26] 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. 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.

Click to play video of multibrot set with d changing from 0 to 8 点击观看d从0到8的多重曼德布洛特集视频

更高维下的曼德布洛特集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 quaternions, that creates a perfect extension of the Mandelbrot set and the Julia sets into 4 dimensions.[27] These can then be either cross-sectioned or projected into a 3D structure.

其他非解析性质的映射Other, non-analytic, mappings

[图片64：Image of the Tricorn / Mandelbar fractal （图64）三角分形图像] 趣木木讨论）由于查询后 tricorn fractal是 John Milnor 米诺尔发现的 在网上 尝试 三角分形 、独角兽分形、三角骨分形都没有查到结果 暂定直译为三角分形再添加英文

[图片65：Image of the burning ship fractal （图65）燃烧船分形图像]

Of particular interest is the tricorn fractal, the connectedness locus of the anti-holomorphic family $\displaystyle{ z \mapsto \bar{z}^2 + c. }$ 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. Another non-analytic generalization is the Burning Ship fractal, which is obtained by iterating the following : $\displaystyle{ z \mapsto (|\Re \left(z\right)|+i|\Im \left(z\right)|)^2 + c. }$

$\displaystyle{ z \mapsto \bar{z}^2 + c. }$

$\displaystyle{ z \mapsto (|\Re \left(z\right)|+i|\Im \left(z\right)|)^2 + c. }$

利用电脑绘制曼德布洛特集Computer drawings

Main article: Plotting algorithms for the Mandelbrot set 主要文章:绘制曼德尔勃特集合的算法

[图片66：Still image of a movie of increasing magnification on 0.001643721971153 − 0.822467633298876i（图66）放大率在0.001643721971153 − 0.822467633298876i处动态视频截图]

[图片67：Still image of an animation of increasing magnification （图67）放大倍数增大后的动态视频截图]

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". 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. 可以利用多种算法通过计算设备来绘制曼德布洛特集。在这里，展示应用最广泛且最简单的算法：即朴素的“逃逸时间算法”。 在逃逸算法中，对于绘图区域中每一个（“x”，“y”)点重复计算，根据计算后的不同表现为某一像素区域选择不同的颜色。

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. 每个点（“x”,“y”）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测这些值，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，选择像素的颜色并绘制，并检测下一个（“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. 每个点的颜色值反应了逃逸点处的逃逸速度。通常，黑色用于显示在迭代限制之前未能逃逸的值，而逐渐逃逸的点使用较亮的颜色表示。这很直观的表示了在达到逃逸条件之前需要多少个循环。

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 {\displaystyle c} be the midpoint of that pixel. We now iterate the critical point 0 under {\displaystyle P_{c}}, checking at each step whether the orbit point has modulus larger than 2. When this is the case, we know that {\displaystyle c} 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，考虑将复平面区域分割成一定数量的点区域。要为任意点区域着色，设置$\displaystyle{ c }$作为其点区域的中点。在$\displaystyle{ P_{c} }$的前提下迭代临界点0，在每一步检查轨道点的模量是否大于2。（即$\displaystyle{ x^2+y^2 }$$\displaystyle{ 2^2 }$的关系）在这种情况下，我们知道$\displaystyle{ c }$不属于曼德布洛特集。根据判断模量是否大于2所用的迭代次数为点区域着色。否则，我们将继续进行迭代直到遇到逃逸点，然后确定我们的参数“可能”在曼德布洛特集中，或者至少非常接近它，并将点区域涂成黑色。 趣木木讨论）将像素译为点区域（可再讨论）

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. 在虚拟程序代码中，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，使用两个实数手动对复数进行模拟。如果编程语言包含复杂数据类型的操作，则程序可以简化。

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

x0 = scaled x coordinate of pixel (scaled to lie in the Mandelbrot X scale (-2.5, 1))
y0 = scaled y coordinate of pixel (scaled to lie in the Mandelbrot Y scale (-1, 1))
x := 0.0
y := 0.0
iteration := 0
max_iteration := 1000
while (x×x + y×y ≤ 2×2 AND iteration < max_iteration) do
xtemp := x×x - y×y + x0
y := 2×x×y + y0
x := xtemp
iteration := iteration + 1

color := palette[iteration]
plot(Px, Py, color)

• $\displaystyle{ z = x + iy\ }$
• $\displaystyle{ z^2 = x^2 +i2xy - y^2\ }$
• $\displaystyle{ c = x_0 + i y_0\ }$

and so, as can be seen in the pseudocode in the computation of x and y: 因此，在计算“x”和“y”的虚拟程序代码中可以看出:

• $\displaystyle{ x = \mathop{\mathrm{Re}}(z^2+c) = x^2-y^2 + x_0 }$ and $\displaystyle{ y = \mathop{\mathrm{Im}}(z^2+c) = 2xy + y_0.\ }$

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.). 要获得曼德布洛特集的彩色图像，可以使用各种函数(线性、指数等)中的一个来为执行程序所需不同的迭代次数所对应的点区域分配颜色。

大众文化中的引用 References in popular culture

The Mandelbrot set is considered by many the most popular fractal,[28][29] 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[28][29] ，并在流行文化中被多次提及。

The Jonathan Coulton song "Mandelbrot Set" is a tribute to both the fractal itself and to its discoverer Benoit Mandelbrot.[30] The second book of the Mode series by Piers Anthony, Fractal Mode, describes a world that is a perfect 3D model of the set.[31] The Piers Anthony novel The Ghost from the Grand Banks features an artificial lake made to replicate the shape of the Mandelbrot set.[32] 乔纳森·库尔顿 Jonathan Coulton的歌曲《曼德尔布洛特集》既是对分形本身的赞颂，也是对它的发现者 Benoît B. Mandelbrot的赞颂 皮尔斯·安东尼 Piers Anthony的《时尚》系列的第二本书《分形模式》描述了一个完美的三维模型世界。 [31] 亚瑟·查理斯·克拉克 Piers Anthony的小说《来自大浅滩的幽灵》描绘了一个曼德布洛特集图形复刻版的人工湖。 [32]

参考文献

• [[1]John W. Milnor, Dynamics in One Complex Variable (Third Edition), Annals of Mathematics Studies 160, (Princeton University Press, 2006), ISBN 0-691-12488-4 约翰米尔诺，《一个复杂变量中的动力学》(第三版) ，数学纪事研究160，(普林斯顿大学出版社，2006) ，ISBN 0-691-12488-4]

(First appeared in 1990 as a Stony Brook IMS Preprint, available as arXiV:math.DS/9201272 ) (首次出现于1990年，作为斯托尼布鲁克 IMS 预印本，可作为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 奈杰尔 · 莱斯莫尔-戈登，《无限的颜色: 分形的美、力量和感觉》 ，ISBN 1-904555-05-5

(includes a DVD featuring Arthur C. Clarke and David Gilmour) (包括一张 DVD，主角是亚瑟·查理斯·克拉克 Arthur C. Clarke大卫·吉尔摩 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 海因茨-奥托·佩特根 Heinz-Otto Peitgen ，哈特穆特·尤尔根斯 Hartmut Jürgens，迪特马尔·索普 Dietmar Saupe，混沌与分形:科学的新前沿(施普林格，纽约，1992,2004)，ISBN 0-387-20229-3