查看“︁曼德尔球”︁的源代码

[[File:Power 8 mandelbulb fractal overview.jpg|thumb|一个[[光线跟踪]]的3维曼德尔球图像<br>
迭代于<math>z\mapsto z^8 + c</math>]]
'''曼德尔球'''（{{lang|en|Mandelbulb}}）是一个三维的[[曼德博集合]]的模拟，由丹尼尔·怀特和保罗·尼兰德采用[[球坐标系|球坐标]]构造。<ref>{{cite web
|url=http://www.bugman123.com/Hypercomplex/index.html
|title=Hypercomplex fractals
|accessdate=2013-03-05
|archive-date=2010-04-12
|archive-url=https://web.archive.org/web/20100412082051/http://bugman123.com/Hypercomplex/index.html
|dead-url=yes
}}</ref>

因为没有复数的二维空间的三维类似物，因此不存在规范的三维曼德博集合。可以使用[[四元数]]的4个维度构建曼德博集合。然而，这样构建的集合不能和二维的那样在所有尺度上表现细节。

怀特和尼兰德关于三维矢量<math>\langle x, y, z\rangle</math>的<math>n</math>次方公式为

<math>\langle x, y, z\rangle^n = r^n\langle\sin(n\theta)\cos(n\phi),\sin(n\theta)\sin(n\phi),\cos(n\theta)\rangle</math>

其中

<math>\begin{align}r&=\sqrt{x^2+y^2+z^2} \\
 \phi&=\arctan(y/x)=\arg (x+yi) \\
 {\rm and\ } \theta&=\arctan(\sqrt{x^2+y^2}/z)=\arccos(z/r).\end{align}</math>

他们使用迭代<math>z\mapsto z^n+c</math>，其中“<math> z^n </math>”定义如上，“<math> + </math>”是矢量相加。<ref>{{cite web
|url=http://www.skytopia.com/project/fractal/2mandelbulb.html
|title=Mandelbulb: The Unravelling of the Real 3D Mandelbrot Fractal
|accessdate=2013-03-05
|archive-date=2012-07-01
|archive-url=https://www.webcitation.org/68qK9biL5?url=http://www.skytopia.com/project/fractal/2mandelbulb.html
|dead-url=yes
}} see "formula" section</ref>当<math> n > 3 </math>，其结果是一个三维的球状物，有[[分形]]的表面和由参数<math>n</math>控制的叶子。他们的很多图形表现采用<math>n = 8</math>的设置。

== 参考资料==
{{Reflist}}

== 外部链接 ==
{{commons}}
* [http://www.skytopia.com/project/fractal/mandelbulb.html Mandelbulb: The Unravelling of the Real 3D Mandelbrot Fractal, on Daniel White's website] {{Wayback|url=http://www.skytopia.com/project/fractal/mandelbulb.html |date=20130329070945 }}
* [http://www.bugman123.com/Hypercomplex/index.html Several variants of the Mandelbulb, on Paul Nylander's website] {{Wayback|url=http://www.bugman123.com/Hypercomplex/index.html |date=20100412082051 }}
* [http://sourceforge.net/projects/mandelbulber/ An opensource fractal renderer that can be used to create images of the Mandelbulb] {{Wayback|url=http://sourceforge.net/projects/mandelbulber/ |date=20210218034327 }}
* [http://www.fractal.org/Formula-Mandelbulb.pdf Formula for Mandelbulb/Juliabulb/Juliusbulb by Jules Ruis] {{Wayback|url=http://www.fractal.org/Formula-Mandelbulb.pdf |date=20201127044926 }}
* [http://www.fractal.org/Mandelbulb.pdf Mandelbulb/Juliabulb/Juliusbulb with examples of real 3D objects] {{Wayback|url=http://www.fractal.org/Mandelbulb.pdf |date=20170822040632 }}
* [http://www.youtube.com/watch?v=82xDvdgrzx0 Video : View of the Mandelbulb ] {{Wayback|url=http://www.youtube.com/watch?v=82xDvdgrzx0 |date=20161210222851 }}
{{分形}}
[[Category:分形]]