Menger sponge

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

Contents

Construction

Image 3: A sculptural representation of iterations 0 (bottom) to 3 (top). Mengerova houba.jpg
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.
  2. Divide every face of the cube into nine squares, like Rubik's Cube. This sub-divides 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. This is a level-1 Menger sponge (resembling a void cube).
  4. Repeat steps two and three for each of the remaining smaller cubes, and continue to iterate ad infinitum .

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.

An illustration of the iterative construction of a Menger sponge up to M3, the third iteration Menger sponge (Level 0-3).jpg
An illustration of the iterative construction of a Menger sponge up to M3, the third iteration
Menger sponge animation through (4) recursion steps Mengersponge.gif
Menger sponge animation through (4) recursion steps

Properties

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

The th stage of the Menger sponge, , is made up of smaller cubes, each with a side length of (1/3)n. The total volume of is thus . The total surface area of is given by the expression . [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.

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]

Cubes with Menger fractal structures after shockwave loading. The color indicates the temperature rise associated with plastic deformation. 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.

Formal definition

360deg equirectangular projection from the middle of a level-5 Menger sponge
(view as a 360deg interactive panorama) Mengerschwamm aus Sicht Koordinatenursprung HQ 72MP Gleichwinklig zbuffer negative.jpg
360° equirectangular projection from the middle of a level-5 Menger sponge
( view as a 360° interactive panorama )

Formally, a Menger sponge can be defined as follows:

where is the unit cube and

MegaMenger

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 960000 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

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.
  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

See also

Related Research Articles

Fractal Self similar mathematical structures

In mathematics, a fractal is a subset of Euclidean space with a fractal dimension that strictly exceeds its topological dimension. Fractals appear the same at different scales, as illustrated in successive magnifications of the Mandelbrot set. Fractals often exhibit similar patterns at increasingly smaller scales, a property called self-similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge, it is called affine self-similar. Fractal geometry lies within the mathematical branch of measure theory.

Hausdorff dimension Invariant

In mathematics, Hausdorff dimension is a measure of roughness, or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a line segment is 1, of a square is 2, and of a cube is 3. That is, for sets of points that define a smooth shape or a shape that has a small number of corners—the shapes of traditional geometry and science—the Hausdorff dimension is an integer agreeing with the usual sense of dimension, also known as the topological dimension. However, formulas have also been developed that allow calculation of the dimension of other less simple objects, where, solely on the basis of their properties of scaling and self-similarity, one is led to the conclusion that particular objects—including fractals—have non-integer Hausdorff dimensions. Because of the significant technical advances made by Abram Samoilovitch Besicovitch allowing computation of dimensions for highly irregular or "rough" sets, this dimension is also commonly referred to as the Hausdorff–Besicovitch dimension.

Sierpiński carpet Plane fractal built from squares

The Sierpiński carpet is a plane fractal first described by Wacław Sierpiński in 1916. The carpet is a generalization of the Cantor set to two dimensions; another is Cantor dust.

Sierpiński triangle Fractal composed of triangles

The Sierpiński triangle, also called the Sierpiński gasket or Sierpiński sieve, is a fractal attractive fixed set with the overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles. Originally constructed as a curve, this is one of the basic examples of self-similar sets—that is, it is a mathematically generated pattern that is reproducible at any magnification or reduction. It is named after the Polish mathematician Wacław Sierpiński, but appeared as a decorative pattern many centuries before the work of Sierpiński.

Koch snowflake Fractal and mathematical curve

The Koch snowflake is a fractal curve and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Curve Without Tangents, Constructible from Elementary Geometry" by the Swedish mathematician Helge von Koch.

In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern changes with the scale at which it is measured. It has also been characterized as a measure of the space-filling capacity of a pattern that tells how a fractal scales differently from the space it is embedded in; a fractal dimension does not have to be an integer.

This is a list of fractal topics, by Wikipedia page, See also list of dynamical systems and differential equations topics.

Sierpiński curve

Sierpiński curves are a recursively defined sequence of continuous closed plane fractal curves discovered by Wacław Sierpiński, which in the limit completely fill the unit square: thus their limit curve, also called the Sierpiński curve, is an example of a space-filling curve.

In mathematics, the T-square is a two-dimensional fractal. It has a boundary of infinite length bounding a finite area. Its name comes from the drawing instrument known as a T-square.

Fractal curve Mathematical curve whose shape is a fractal, pathological irregularity, regardless of magnification. Each non-zero arc has infinite length

A fractal curve is, loosely, a mathematical curve whose shape retains the same general pattern of irregularity, regardless of how high it is magnified, that is, its graph takes the form of a fractal. In general, fractal curves are nowhere rectifiable curves — that is, they do not have finite length — and every subarc longer than a single point has infinite length.

Hexaflake

A hexaflake or Lindstrøm Snowflake is a fractal constructed by iteratively exchanging hexagons by a flake of seven hexagons; it is a special case of the n-flake.

In mathematics, a power of three is a number of the form 3n where n is an integer, that is, the result of exponentiation with number three as the base and integer n as the exponent.

Vicsek fractal

In mathematics the Vicsek fractal, also known as Vicsek snowflake or box fractal, is a fractal arising from a construction similar to that of the Sierpinski carpet, proposed by Tamás Vicsek. It has applications including as compact antennas, particularly in cellular phones.

In the mathematical field of point-set topology, a continuum is a nonempty compact connected metric space, or, less frequently, a compact connected Hausdorff space. Continuum theory is the branch of topology devoted to the study of continua.

Osgood curve Non-self-intersecting curve of positive area

In mathematics, an Osgood curve is a non-self-intersecting curve of positive area. More formally, these are curves in the Euclidean plane with positive two-dimensional Lebesgue measure.

An n-flake, polyflake, or Sierpinski n-gon, is a fractal constructed starting from an n-gon. This n-gon is replaced by a flake of smaller n-gons, such that the scaled polygons are placed at the vertices, and sometimes in the center. This process is repeated recursively to result in the fractal. Typically, there is also the restriction that the n-gons must touch yet not overlap.

Rep-tile

In the geometry of tessellations, a rep-tile or reptile is a shape that can be dissected into smaller copies of the same shape. The term was coined as a pun on animal reptiles by recreational mathematician Solomon W. Golomb and popularized by Martin Gardner in his "Mathematical Games" column in the May 1963 issue of Scientific American. In 2012 a generalization of rep-tiles called self-tiling tile sets was introduced by Lee Sallows in Mathematics Magazine.

Jeannine Mosely holds a Ph.D. in EECS from the Massachusetts Institute of Technology, and is known for her work as an origami artist. She is best known for her modular origami designs, especially her work using business cards. She has organized several crowd-sourced origami projects built from tens of thousands of business cards involving hundred of volunteers for each project. She is also known for her minimalist origami designs, curved crease models, and her invention of "or-egg-ami" models made from egg cartons.

Mosely snowflake

The Mosely snowflake is a Sierpiński–Menger type of fractal obtained in two variants either by the operation opposite to creating the Sierpiński-Menger snowflake or Cantor dust i.e. not by leaving but by removing eight of the smaller 1/3-scaled corner cubes and the central one from each cube left from the previous recursion (lighter) or by removing only corner cubes (heavier). In one dimension this operation is trivial and converges only to single point. It resembles the original water snowflake of snow. By the construction the Hausdorff dimension of the lighter snowflake is

The Fibonacci word fractal is a fractal curve defined on the plane from the Fibonacci word.

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. 1 2 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.

Further reading