Menger sponge

Image 1: An illustration of M4, the sponge after four iterations of the construction process.

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]

Construction

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

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

  1. Begin with a cube (Image 2 - first from left).
  2. Divide every face of the cube into 9 squares, like a Rubik's Cube. This will sub-divide the cube into 27 smaller cubes.
  3. 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 (Image 2 - second from left). This is a level-1 Menger sponge (resembling a Void Cube).
  4. Repeat steps 2 and 3 for each of the remaining smaller cubes, and continue to iterate ad infinitum.

The second iteration gives a level-2 sponge (Image 2 - third from left), the third iteration gives a level-3 sponge (Image 2 - 4th from left), and so on. The Menger sponge itself is the limit of this process after an infinite number of iterations.

Image 2: An illustration of the iterative construction of a Menger sponge up to M3, the third iteration.
Menger sponge animation through (4) recursion steps.

Properties

True view of the cross-section of a level-4 Menger sponge through its centroid and perpendicular to a space diagonal. In this interactive SVG, the cross-sections are true-view and to scale.

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.

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 , with [9].

The sponge's Hausdorff dimension is log 20/log 3 ≅ 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 curve 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 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.

Formal definition

Formally, a Menger sponge can be defined as follows:

where M0 is the unit cube and

MegaMenger

A model of a tetrix viewed through the centre of the Cambridge Level-3 MegaMenger at the 2015 Cambridge Science Festival
One of the MegaMengers, at the University of Bath

MegaMenger is 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 6 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.[10] In 2014, twenty level-three Menger sponges were constructed, which combined would form a distributed level-four Menger sponge.[11]

Similar fractals

Jerusalem cube

Third iteration Jerusalem cube
3D-printed model Jerusalem cube

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.[12][13] 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.
  2. 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.
  3. Repeat the process on the cubes of rank 1 and 2.

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

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. log 9/log 3=2
  • A Mosely snowflake is a cube-based fractal with corners recursively removed.[14]
  • A tetrix is a tetrahedron-based fractal made from four smaller copies, arranged in a tetrahedron.[15]

See also

References

  1. ↑ Beck, Christian; Schögl, Friedrich (1995). Thermodynamics of Chaotic Systems: An Introduction. Cambridge University Press. p. 97. ISBN 9780521484510.
  2. ↑ Bunde, Armin; Havlin, Shlomo (2013). Fractals in Science. Springer. p. 7. ISBN 9783642779534.
  3. ↑ Menger, Karl (2013). Reminiscences of the Vienna Circle and the Mathematical Colloquium. Springer Science & Business Media. p. 11. ISBN 9789401111027.
  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. ↑ Tim Chartier. "A Million Business Cards Present a Math Challenge". Retrieved 2015-04-07.
  11. ↑ "MegaMenger". Retrieved 2015-02-15.
  12. ↑ Robert Dickau (2014-08-31). "Cross Menger (Jerusalem) Cube Fractal". Robert Dickau. Retrieved 2017-05-08.
  13. ↑ 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.
  14. ↑ Wade, Lizzie. "Folding Fractal Art from 49,000 Business Cards". Retrieved 8 May 2017.
  15. ↑ W., Weisstein, Eric. "Tetrix". mathworld.wolfram.com. Retrieved 8 May 2017.

Further reading

  • Iwaniec, Tadeusz; Martin, Gaven (2001), Geometric function theory and non-linear analysis, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, ISBN 978-0-19-850929-5, MR 1859913 .
  • Zhou, Li (2007), "Problem 11208: Chromatic numbers of the Menger sponges", American Mathematical Monthly, 114 (9): 842, JSTOR 27642353
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