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===心脏形结构和圆盘形“芽苞”Main cardioid and period bulbs===
===心脏形结构和圆盘形“芽苞”Main cardioid and period bulbs===
[[File:P11_Mandelbrot_Set_–_Periodicities_coloured.png|300px|thumb|right|]]
[图片11：Periods of hyperbolic components（图11）双曲分量的周期]
[图片11：Periods of hyperbolic components（图11）双曲分量的周期]
则该点处存在一个与主心脏形结构曲线相切的"芽苞"，且记作“ <math>\tfrac{p}{q}</math>-芽苞”。
则该点处存在一个与主心脏形结构曲线相切的"芽苞"，且记作“ <math>\tfrac{p}{q}</math>-芽苞”。
[[File:P13_Animated_cycle.gif|300px|thumb|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)]
[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)]
<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>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 <math>\tfrac{p}{q}</math> 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<math>U_0,\dots,U_{q-1}</math>为逆时针方向，<math>P_c</math>将分量<math>U_j</math> 映射到分量<math>U_{j+p\,(\operatorname{mod} q)}</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>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 <math>\tfrac{p}{q}</math> 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<math>U_0,\dots,U_{q-1}</math>为逆时针方向，<math>P_c</math>将分量<math>U_j</math> 映射到分量<math>U_{j+p\,(\operatorname{mod} q)}</math>
[[File:P14_Juliacycles1.png||300px|thumb|right|]]
[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]
[[File:P15_Mandel_rays.jpg|300px|thumb|right|]]
[图片15：Cycle periods and antennae :（图15）周期和分枝]
[图片15：Cycle periods and antennae :（图15）周期和分枝]
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。
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。
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.
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.
===连通性 Local connectivity===
===连通性 Local connectivity===
[[File:P17_Cactus_model_of_Mandelbrot_set.svg.png|300px|thumb|right|]]
[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)））]
[[File:P18_Lavaurs-12.png|300px|thumb|right|]]
[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）]
[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）]
===自相似 Self-similarity===
===自相似 Self-similarity===
[[File:P19Mandelbrot_zoom.gif|300px|thumb|right|]]
[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 }.
[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) ，而视图从0.5x0.5放大到0.12x0.12，以接近 Feigenbaum 比率Ħẟ]
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) ，而视图从0.5x0.5放大到0.12x0.12，以接近 Feigenbaum 比率Ħẟ]
--[[用户:木子二月鸟|木子二月鸟]] the view magnifies from 0.5 × 0.5 to 0.12 × 0.12 调整为 “视图从0.5x0.5放大到0.12x0.12”
--[[用户:木子二月鸟|木子二月鸟]] the view magnifies from 0.5 × 0.5 to 0.12 × 0.12 调整为 “视图从0.5x0.5放大到0.12x0.12”
[[File:P20_Mandelzoom1.jpg|300px|thumb|right|]]
[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]
[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]
[[File:P21_Blue_Mandelbrot_Zoom.jpg|300px|thumb|right|]]
[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]
[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 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.
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[[File:P22_Relationship_between_Mandelbrot_sets_and_Julia_sets.png|300px|thumb|right|]]
[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]
[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===
===与朱利亚集之间的关系===
==几何结构Geometry==
==几何结构Geometry==
[[File:P23_Unrolled_main_cardioid_of_Mandelbrot_set_for_periods_8-14.png|300px|thumb|right|]]
[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]
[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]
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.
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<math>c</math>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。
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<math>c</math>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。
<center>
<gallery>
File:P24Mandel_zoom_00_mandelbrot_set.jpg|
[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]
File:P25Mandel_zoom_01_head_and_shoulder.jpg|
[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”]
File:P26Mandel_zoom_02_seehorse_valley.jpg|
[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26：Double-spirals on the left, "seahorses" on the right （图26）左边是双螺旋，右边是“海马”]
File:P27Mandel_zoom_03_seehorse.jpg|
[图片27："Seahorse" upside down（图27）颠倒过来的海马]
[图片27："Seahorse" upside down（图27）颠倒过来的海马]
</gallery>
</center>
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.
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。
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。
<center>
<gallery>
File:P28Mandel_zoom_04_seehorse_tail.jpg|
[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。]
File:P29-Mandel_zoom_05_tail_part.jpg|
[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。]
File:P30-Mandel_zoom_06_double_hook.jpg|
[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。]
[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。]
File:P31Mandel_zoom_07_satellite.jpg|
[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]
File:P32Mandel_zoom_08_satellite_antenna.jpg|
[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。]
File:P33Mandel_zoom_09_satellite_head_and_shoulder.jpg|
[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]
File:P34Mandel_zoom_10_satellite_seehorse_valley.jpg|
[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）
[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。]
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。]
File:P35Mandel_zoom_11_satellite_double_spiral.jpg|
[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]
File:P36Mandel_zoom_12_satellite_spirally_wheel_with_julia_islands.jpg|
[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<math>J_c</math>.其中最大的岛结构可在右侧的“双钩”中心找到。
[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<math>J_c</math>.其中最大的岛结构可在右侧的“双钩”中心找到。
File:P37Mandel_zoom_13_satellite_seehorse_tail_with_julia_island.jpg|
[图片37：Part of the "double-hook"（图37）“双钩”的一部分]
[图片37：Part of the "double-hook"（图37）“双钩”的一部分]
File:P38Mandel_zoom_14_satellite_julia_island.jpg|
[图片38：Islands（图38）岛群]
[图片38：Islands（图38）岛群]
File:P39Mandel_zoom_15_one_island.jpg|
[图片39：Detail of one island （图39）一个岛屿的细节部分]
[图片39：Detail of one island （图39）一个岛屿的细节部分]
File:P40 Mandel_zoom_16_spiral_island.jpg|
[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]
[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]
</gallery>
</center>
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.
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.
[[File:P41Mandelbrot_set_3D_integer_iterations.jpg|300px|right|thumb|]]
[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]
[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]
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。
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。
[[File:P42Mandelbrot_set_3D_fractional_iterations.jpg|300px|thumb|right|]]
[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]
[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.
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。
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。
[[File:P43Mandelbrot_set_3D_Distance_Estimates.jpg|300px|thumb|right|]]
[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。]
[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。]
--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）下面的图形注释翻译与上文某处一致 只是对于不同的显示图形 渲染方式不同
--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）下面的图形注释翻译与上文某处一致 只是对于不同的显示图形 渲染方式不同
<center>
<gallery>
File:P44Mandel_zoom_00_mandelbrot_set_3D.jpg|
[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]
File:P45Mandel_zoom_01_head_and_shoulder_3D.jpg|
[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”]
File:P46Mandel_zoom_02_seehorse_valley_3D.jpg|
[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”]
File:P47Mandel_zoom_03_seehorse_3D.jpg|
[图片47：Zoom 03. "Seahorse" upside down （图47）颠倒后的“海马”]
[图片47：Zoom 03. "Seahorse" upside down （图47）颠倒后的“海马”]
File:P48Mandel_zoom_04_seehorse_tail_3D.jpg|
[图片48：Zoom 04. A "seahorse tail".（图48）海马尾]
[图片48：Zoom 04. A "seahorse tail".（图48）海马尾]
File:P49Mandel_zoom_05_tail_part_3D.jpg|
[图片49：Zoom 05. Part of the "tail".（图49）海马尾的一部分]
[图片49：Zoom 05. Part of the "tail".（图49）海马尾的一部分]
File:P50Mandel_zoom_06_double_hook_3D.jpg|
[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50：Zoom 06. Satellite with twin "Seahorse tails."（图50）拥有一对“海马尾”的卫星集]
File:P51Mandel_zoom_07_satellite_3D.jpg|
[图片51：Zoom 07. Satellite closeup. （图51）卫星集的特写]
[图片51：Zoom 07. Satellite closeup. （图51）卫星集的特写]
File:P52Mandel_zoom_08_satellite_antenna_3D.jpg|
[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。]
File:P53Mandel_zoom_09_satellite_head_and_shoulder_3D.jpg|
[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]
File:P54Mandel_zoom_10_satellite_seehorse_valley_3D.jpg|
[图片54：Zoom 10. Double-spirals and "seahorses" （图54）双螺旋和“海马”]
[图片54：Zoom 10. Double-spirals and "seahorses" （图54）双螺旋和“海马”]
File:P55Mandel_zoom_11_satellite_double_spiral_3D.jpg|
[图片55：Zoom 11. Double-spirals with satellites of second order. (图55)二级卫星的双螺旋]
[图片55：Zoom 11. Double-spirals with satellites of second order. (图55)二级卫星的双螺旋]
File:P56Mandel_zoom_12_satellite_spirally_wheel_with_julia_islands_3D.jpg|
[图片56：Zoom 12.（图56）]
[图片56：Zoom 12.（图56）]
File:P57Mandel_zoom_13_satellite_seehorse_tail_with_julia_island_3D.jpg|
[图片57：Zoom 13. Part of the "double-hook"（图57）“双钩”的一部分]
[图片57：Zoom 13. Part of the "double-hook"（图57）“双钩”的一部分]
File:P58Mandel_zoom_14_satellite_julia_island_3D.jpg|
[图片58：Zoom 14. Islands （图58）岛群]
[图片58：Zoom 14. Islands （图58）岛群]
File:P59Mandel_zoom_15_one_island_3D.jpg|
[图片59：Zoom 15. Detail of one island（图59）一个岛的细节部分]
[图片59：Zoom 15. Detail of one island（图59）一个岛的细节部分]
File:P60Mandel_zoom_16_spiral_island_3D.jpg|
[图片60：Zoom 16. Detail of the spiral.（图60）螺旋的细节部分]
[图片60：Zoom 16. Detail of the spiral.（图60）螺旋的细节部分]
</gallery>
</center>
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。
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。
[[File:P61A_3D_version_of_the_Mandelbrot_set_plot__Map_44__from_the_book__The_Beauty_of_Fractals_.jpg|300px|thumb|right|]]
[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]
[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 ==
==推广 Generalizations ==
[[File:P62Mandelbrot_Set_Animation_1280x720.gif|300px|thumb|right|]]
[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。
[[File:P63-Quaternion_Julia_x=-0,75_y=-0,14.jpg|300px|thumb|right|]]
[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]
[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]
===其他非解析性质的映射Other, non-analytic, mappings===
===其他非解析性质的映射Other, non-analytic, mappings===
[[File:P64Mandelbar_fractal_from_XaoS.png|300px|thumb|right|]]
[图片64：Image of the Tricorn / Mandelbar fractal （图64）三角分形图像]
[图片64：Image of the Tricorn / Mandelbar fractal （图64）三角分形图像]
--[[用户:趣木木|趣木木]]（[[用户讨论: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
--[[用户:趣木木|趣木木]]（[[用户讨论: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
--[[用户:木子二月鸟|木子二月鸟]]此处需要专家确定。
--[[用户:木子二月鸟|木子二月鸟]]此处需要专家确定。
[[File:P65BurningShip01.png|300px|thumb|right|]]
[图片65：Image of the burning ship fractal （图65）燃烧船分形图像]
[图片65：Image of the burning ship fractal （图65）燃烧船分形图像]
主要文章:绘制曼德尔勃特集合的算法
主要文章:绘制曼德尔勃特集合的算法
[[File:P66Fractal-zoom-1-03-Mandelbrot_Buzzsaw.png|300px|thumb|right|]]
[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]
[[File:P67Mandelbrot_sequence_new_still.png|300px|thumb|right|]]
[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]
[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]