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文件:Menger-Schwamm-farbig.png
An illustration of M4, the sponge after four iterations of the construction process

An illustration of M4, the sponge after four iterations of the construction process

图示 m < sub > 4 ,海绵经过四次迭代的施工过程

In mathematics, the Menger sponge (also known as the Menger cube, Menger universal curve, Sierpinski cube, or Sierpinski sponge)[1][2][3] is a fractal curve. It is a three-dimensional generalization of the one-dimensional Cantor set and two-dimensional Sierpinski carpet. It was first described by Karl Menger in 1926, in his studies of the concept of topological dimension.[4][5]

In mathematics, the Menger sponge (also known as the Menger cube, Menger universal curve, Sierpinski cube, or Sierpinski sponge) is a fractal curve. It is a three-dimensional generalization of the one-dimensional Cantor set and two-dimensional Sierpinski carpet. It was first described by Karl Menger in 1926, in his studies of the concept of topological dimension.

在数学上,门格尔海绵(也称门格尔立方体、门格尔万有曲线、赛尔宾斯基立方体或赛尔宾斯基海绵)是一条分形曲线。它是一维 Cantor 集和二维 Cantor 谢尔宾斯基地毯的三维推广。1926年,卡尔 · 门格尔在研究拓扑维度的概念时首次提出了这一概念。


Construction

文件:Mengerova houba.jpg
Image 3: A sculptural representation of iterations 0 (bottom) to 3 (top).

Image 3: A sculptural representation of iterations 0 (bottom) to 3 (top).

图3: 一个迭代0(底部)至3(顶部)的雕塑表现。


The construction of a Menger sponge can be described as follows:

The construction of a Menger sponge can be described as follows:

门格尔海绵的构造如下:

  1. Begin with a cube.
Begin with a cube.

从一个立方体开始。

  1. Divide every face of the cube into nine squares, like Rubik's Cube. This sub-divides the cube into 27 smaller cubes.
Divide every face of the cube into nine squares, like Rubik's Cube. This sub-divides the cube into 27 smaller cubes.

把立方体的每个面都分成九个正方形,就像魔方一样。这将立方体分成27个更小的立方体。

  1. Remove the smaller cube in the middle of each face, and remove the smaller cube in the very center of the larger cube, leaving 20 smaller cubes. This is a level-1 Menger sponge (resembling a void cube).
Remove the smaller cube in the middle of each face, and remove the smaller cube in the very center of the larger cube, leaving 20 smaller cubes. This is a level-1 Menger sponge (resembling a void cube).

移除每个正面中间的小立方体,移除大立方体正中间的小立方体,留下20个小立方体。这是一个1级门格尔海绵(类似于一个空白立方体)。

  1. Repeat steps two and three for each of the remaining smaller cubes, and continue to iterate ad infinitum.
Repeat steps two and three for each of the remaining smaller cubes, and continue to iterate ad infinitum.

对于剩下的每个较小的立方体,重复步骤2和3,并继续无限迭代。


The second iteration gives a level-2 sponge, the third iteration gives a level-3 sponge, and so on. The Menger sponge itself is the limit of this process after an infinite number of iterations.

The second iteration gives a level-2 sponge, the third iteration gives a level-3 sponge, and so on. The Menger sponge itself is the limit of this process after an infinite number of iterations.

第二次迭代提供了一个级别为2的海绵,第三次迭代提供了一个级别为3的海绵,以此类推。门格尔海绵本身就是这个过程经过无限次迭代后的极限。


文件:Menger sponge (Level 0-3).jpg
An illustration of the iterative construction of a Menger sponge up to M3, the third iteration

An illustration of the iterative construction of a Menger sponge up to M3, the third iteration

第三次迭代 m < sub > 3 的 Menger 海绵的迭代构造的例证

文件:Mengersponge.gif
Menger sponge animation through (4) recursion steps
Menger sponge animation through (4) recursion steps

通过(4)递归步骤实现门格尔海绵动画


Properties

文件:Menger sponge diagonal section 27.png
Hexagonal cross-section of a level-4 Menger sponge. See a series of cuts perpendicular to the space diagonal.

series of cuts perpendicular to the space diagonal.]]

[一系列垂直于空间对角线的切割]

The nth stage of the Menger sponge, Mn, is made up of 20n smaller cubes, each with a side length of (1/3)n. The total volume of Mn is thus (20/27)n. The total surface area of Mn is given by the expression 2(20/9)n + 4(8/9)n.[6][7] Therefore the construction's volume approaches zero while its surface area increases without bound. Yet any chosen surface in the construction will be thoroughly punctured as the construction continues, so that the limit is neither a solid nor a surface; it has a topological dimension of 1 and is accordingly identified as a curve.

The nth stage of the Menger sponge, Mn, is made up of 20n smaller cubes, each with a side length of (1/3)n. The total volume of Mn is thus (20/27)n. The total surface area of Mn is given by the expression 2(20/9)n + 4(8/9)n. Therefore the construction's volume approaches zero while its surface area increases without bound. Yet any chosen surface in the construction will be thoroughly punctured as the construction continues, so that the limit is neither a solid nor a surface; it has a topological dimension of 1 and is accordingly identified as a curve.

门格尔海绵的第 n 级为 m < sub > n ,由20个较小的立方体组成,每个立方体的边长为(1/3) < sup > n 。因此,m < sub > n 的总体积为(20/27) < sup > n 。M < sub > n 的总表面积由表达式2(20/9) < sup > n + 4(8/9) < sup > n 给出。因此,建筑的体积接近零,而其表面积增加没有界限。然而,随着施工的进行,施工中任何选定的曲面都将被彻底戳穿,因此,这个极限既不是实体,也不是曲面; 它的拓扑维数为1,因此被确定为曲线。


Each face of the construction becomes a Sierpinski carpet, and the intersection of the sponge with any diagonal of the cube or any midline of the faces is a Cantor set. The cross section of the sponge through its centroid and perpendicular to a space diagonal is a regular hexagon punctured with hexagrams arranged in six-fold symmetry.[8] The number of these hexagrams, in descending size, is given by [math]\displaystyle{ a_n=9a_{n-1}-12a_{n-2} }[/math], with [math]\displaystyle{ a_0=1, \ a_1=6 }[/math][9].

Each face of the construction becomes a Sierpinski carpet, and the intersection of the sponge with any diagonal of the cube or any midline of the faces is a Cantor set. The cross section of the sponge through its centroid and perpendicular to a space diagonal is a regular hexagon punctured with hexagrams arranged in six-fold symmetry. The number of these hexagrams, in descending size, is given by [math]\displaystyle{ a_n=9a_{n-1}-12a_{n-2} }[/math], with [math]\displaystyle{ a_0=1, \ a_1=6 }[/math].

建筑的每个面都是一个谢尔宾斯基地毯,海绵与立方体的任何对角线或任何中线的交叉点是一个 Cantor 集合。海绵的横截面通过其质心和垂直于空间对角线是一个正六边形与六重对称排列。这些卦的数目,按下降的大小,是由 < math > a _ n = 9a _ { n-1}-12 a _ { n-2} </math > ,用 < math > a _ 0 = 1,a _ 1 = 6 </math > 。


The sponge's Hausdorff dimension is 模板:Sfrac ≅ 2.727. The Lebesgue covering dimension of the Menger sponge is one, the same as any curve. Menger showed, in the 1926 construction, that the sponge is a universal curve, in that every 模板:Interlanguage link multi is homeomorphic to a subset of the Menger sponge, where a curve means any compact metric space of Lebesgue covering dimension one; this includes trees and graphs with an arbitrary countable number of edges, vertices and closed loops, connected in arbitrary ways. In a similar way, the Sierpinski carpet is a universal curve for all curves that can be drawn on the two-dimensional plane. The Menger sponge constructed in three dimensions extends this idea to graphs that are not planar, and might be embedded in any number of dimensions.

The sponge's Hausdorff dimension is ≅ 2.727. The Lebesgue covering dimension of the Menger sponge is one, the same as any curve. Menger showed, in the 1926 construction, that the sponge is a universal curve, in that every is homeomorphic to a subset of the Menger sponge, where a curve means any compact metric space of Lebesgue covering dimension one; this includes trees and graphs with an arbitrary countable number of edges, vertices and closed loops, connected in arbitrary ways. In a similar way, the Sierpinski carpet is a universal curve for all curves that can be drawn on the two-dimensional plane. The Menger sponge constructed in three dimensions extends this idea to graphs that are not planar, and might be embedded in any number of dimensions.

海绵的豪斯多夫维数是2.727。门格尔海绵的拓朴维数是一个,和任何曲线都是一样的。在1926年的构造中 Menger 证明了海绵是一条普遍的曲线,因为每一个都同胚于 Menger 海绵的一个子集,其中一条曲线意味着拓朴维数的任意紧致度量空间; 这包括树和图,它们的边、顶点和闭环的任意可数个数目,以任意方式连接。类似地,谢尔宾斯基地毯是一条适用于所有可以在二维平面上绘制的曲线的通用曲线。构建在三维空间中的门格尔海绵将这一思想扩展到了非平面的图形,并且可以嵌入任意数量的维度。


The Menger sponge is a closed set; since it is also bounded, the Heine–Borel theorem implies that it is compact. It has Lebesgue measure 0. Because it contains continuous paths, it is an uncountable set.

The Menger sponge is a closed set; since it is also bounded, the Heine–Borel theorem implies that it is compact. It has Lebesgue measure 0. Because it contains continuous paths, it is an uncountable set.

门格尔海绵是一个闭集合,因为它也是有界的,海涅-博雷尔定理暗示它是紧的。它的勒贝格测度是0。因为它包含连续的路径,所以是一个不可数的集合。


Experiments also showed that for the same material, cubes with a Menger sponge structure could dissipate shocks five times better than a cubes without any pores.[10]

Experiments also showed that for the same material, cubes with a Menger sponge structure could dissipate shocks five times better than a cubes without any pores.

实验还表明,对于同样的材料,具有门格尔海绵结构的立方体消除冲击的效果比没有任何毛孔的立方体好五倍。

文件:Menger fractal structures after shockwave loading.jpg
Cubes with Menger fractal structures after shockwave loading. The color indicates the temperature rise associated with plastic deformation.[10]

Cubes with Menger fractal structures after shockwave loading. The color indicates the temperature rise associated with plastic deformation. In 2014, twenty level-three Menger sponges were constructed, which combined would form a distributed level-four Menger sponge. The name comes from a face of the cube resembling a [[Jerusalem cross pattern.

冲击波加载后具有门格尔分形结构的立方体。这个颜色表示与塑性变形有关的温度上升。2014年,二十层门格尔海绵被制造出来,这些海绵组合成一个分布式的4层门格尔海绵。这个名字来源于一个类似于[耶路撒冷十字图案]的立方体的正面。


Formal definition

The construction of the Jerusalem cube can be described as follows:

耶路撒冷立方体的建造可以说明如下:

Formally, a Menger sponge can be defined as follows:

Start with a cube.

从一个立方体开始。


Cut a cross through each side of the cube, leaving eight cubes (of rank +1) at the corners of the original cube, as well as twelve smaller cubes (of rank +2) centered on the edges of the original cube between cubes of rank +1.

在立方体的每一边切一个十字,在原立方体的角落留下8个立方体(秩 + 1) ,还有十二个较小的立方体(秩 + 2)集中在原立方体的边缘之间的秩 + 1立方体。

[math]\displaystyle{ M := \bigcap_{n\in\mathbb{N}} M_n }[/math]
Repeat the process on the cubes of rank 1 and 2.

在秩1和2的立方体上重复这个过程。


where M0 is the unit cube and

3D-printed model Jerusalem cube

3d 打印模型耶路撒冷立方体


[math]\displaystyle{ M_{n+1} := \left\{\begin{matrix} Each iteration adds eight cubes of rank one and twelve cubes of rank two, a twenty-fold increase. (Similar to the Menger sponge but with two different-sized cubes.) Iterating an infinite number of times results in the Jerusalem cube. 每次迭代增加8个秩为1的立方体和十二个秩为2的立方体,增加了20倍。(类似门格尔海绵,但有两个不同大小的立方体。)迭代无限次将导致耶路撒冷立方体。 (x,y,z)\in\mathbb{R}^3: & \begin{matrix}\exists i,j,k\in\{0,1,2\}: (3x-i,3y-j,3z-k)\in M_n \\ \mbox{and at most one of }i,j,k\mbox{ is equal to 1}\end{matrix} [[File:Sierpinskisnowflake.gif|thumb|right|326x326px| Sierpinski-Menger snowflake. Eight corner cubes and the one central cube are kept [文件: Sierpinskisnowflake.gif | thumb | right | 326x326px | Sierpinski-Menger 雪花。保留了八个角立方体和一个中心立方体 \end{matrix}\right\}. }[/math]

each time at the lower and lower recursion steps. This peculiar three dimensional fractal has the Hausdorff dimension of the natively two dimensional object like the plane i.e. =2]]

每次在较低的递归步骤。这种特殊的三维分形具有像平面一样的原生二维物体的豪斯多夫维数。=2]]


MegaMenger

文件:Cmglee Cambridge Science Festival 2015 Menger sponge.jpg
A model of a tetrix viewed through the centre of the Cambridge Level-3 MegaMenger at the 2015 Cambridge Science Festival
文件:Megamenger Bath.jpg
One of the MegaMengers, at the University of Bath

MegaMenger was a project aiming to build the largest fractal model, pioneered by Matt Parker of Queen Mary University of London and Laura Taalman of James Madison University. Each small cube is made from six interlocking folded business cards, giving a total of 960 000 for a level-four sponge. The outer surfaces are then covered with paper or cardboard panels printed with a Sierpinski carpet design to be more aesthetically pleasing.[11] In 2014, twenty level-three Menger sponges were constructed, which combined would form a distributed level-four Menger sponge.[12]


Similar fractals

Jerusalem cube

文件:Cube de Jérusalem, itération 3.png
Third iteration Jerusalem cube

模板:Anchor


A Jerusalem cube is a fractal object described by Eric Baird in 2011. It is created by recursively drilling Greek cross-shaped holes into a cube.[13][14] The name comes from a face of the cube resembling a Jerusalem cross pattern.


The construction of the Jerusalem cube can be described as follows:

  1. Start with a cube.
  1. Cut a cross through each side of the cube, leaving eight cubes (of rank +1) at the corners of the original cube, as well as twelve smaller cubes (of rank +2) centered on the edges of the original cube between cubes of rank +1.
  1. Repeat the process on the cubes of rank 1 and 2.


文件:Jerusalem Cube.jpg
3D-printed model Jerusalem cube


Each iteration adds eight cubes of rank one and twelve cubes of rank two, a twenty-fold increase. (Similar to the Menger sponge but with two different-sized cubes.) Iterating an infinite number of times results in the Jerusalem cube.


Others

文件:Sierpinskisnowflake.gif
Sierpinski-Menger snowflake. Eight corner cubes and the one central cube are kept each time at the lower and lower recursion steps. This peculiar three dimensional fractal has the Hausdorff dimension of the natively two dimensional object like the plane i.e. 模板:Sfrac=2


  • A tetrix is a tetrahedron-based fractal made from four smaller copies, arranged in a tetrahedron.[16]


See also

Category:Iterated function system fractals

分类: 迭代函数系统

Category:Curves

类别: 曲线


Category:Topological spaces

范畴: 拓扑空间

References

Category:Cubes

类别: 立方体

  1. Beck, Christian; Schögl, Friedrich (1995) (in en). Thermodynamics of Chaotic Systems: An Introduction. Cambridge University Press. pp. 97. ISBN 9780521484510. https://books.google.com/books?id=GyPpZ-Lg6KAC&pg=PA97. 
  2. Bunde, Armin; Havlin, Shlomo (2013) (in en). Fractals in Science. Springer. p. 7. ISBN 9783642779534. https://books.google.com/books?id=dh7rCAAAQBAJ&pg=PA7. 
  3. Menger, Karl (2013) (in en). Reminiscences of the Vienna Circle and the Mathematical Colloquium. Springer Science & Business Media. pp. 11. ISBN 9789401111027. https://books.google.com/books?id=BKIyBwAAQBAJ&pg=PR11. 
  4. Menger, Karl (1928), Dimensionstheorie, B.G Teubner Publishers
  5. Menger, Karl (1926), "Allgemeine Räume und Cartesische Räume. I.", Communications to the Amsterdam Academy of Sciences. English translation reprinted in Edgar, Gerald A., ed. (2004), Classics on fractals, Studies in Nonlinearity, Westview Press. Advanced Book Program, Boulder, CO, ISBN 978-0-8133-4153-8, MR 2049443
  6. Wolfram Demonstrations Project, Volume and Surface Area of the Menger Sponge
  7. University of British Columbia Science and Mathematics Education Research Group, Mathematics Geometry: Menger Sponge
  8. Chang, Kenneth (27 June 2011). "The Mystery of the Menger Sponge". Retrieved 8 May 2017 – via NYTimes.com.
  9. "A299916 - OEIS". oeis.org. Retrieved 2018-08-02.
  10. 10.0 10.1 Dattelbaum, Dana M.; Ionita, Axinte; Patterson, Brian M.; Branch, Brittany A.; Kuettner, Lindsey (2020-07-01). "Shockwave dissipation by interface-dominated porous structures". AIP Advances. 10 (7): 075016. doi:10.1063/5.0015179.
  11. Tim Chartier. "A Million Business Cards Present a Math Challenge". Retrieved 2015-04-07.
  12. "MegaMenger". Retrieved 2015-02-15.
  13. Robert Dickau (2014-08-31). "Cross Menger (Jerusalem) Cube Fractal". Robert Dickau. Retrieved 2017-05-08.
  14. Eric Baird (2011-08-18). "The Jerusalem Cube". Alt.Fractals. Retrieved 2013-03-13., published in Magazine Tangente 150, "l'art fractal" (2013), p. 45.
  15. Wade, Lizzie. "Folding Fractal Art from 49,000 Business Cards". Retrieved 8 May 2017.
  16. W., Weisstein, Eric. "Tetrix". mathworld.wolfram.com. Retrieved 8 May 2017.

Category:Fractals

分类: 分形


This page was moved from wikipedia:en:Menger sponge. Its edit history can be viewed at 门格海绵/edithistory