Menger sponge

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The Menger sponge is a fractal solid. It is also known as the Menger-Sierpinski sponge or, incorrectly, the Sierpinski sponge. It is a three-dimensional extension of the Cantor set and Sierpinski carpet, with Hausdorff dimension (ln 20) / (ln 3) (approx. 2,726833). It was first described by Austrian mathematician Karl Menger in 1926.

Construction

Construction of a Menger sponge can be visualized as follows:
  1. Begin with a cube, (first image).
  2. Shrink the cube to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/3} of its original size and make 20 copies of it.
  3. Place the copies so they will form a new cube of the same size as the original one but lacking the centerparts, (next image).
  4. Repeat the process from step 2 for each of the remaining smaller cubes.

After an infinite number of iterations, a Menger sponge will remain.

The number of cubes increases by : Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 20^n} . Where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} is the number of iterations performed on the first cube:

Iters Cubes Sum
0 1 1
1 20 21
2 400 421
3 8,000 8,421
4 160,000 168,421
5 3,200,000 3,368,421
6 64,000,000 67,368,421

At the first level, no iterations are performed, (20 n=0 = 1).

Properties

File:Gasket14.png
An illustration of M3, the third iteration of the construction process. Image © Paul Bourke, used by permission

Each face of the Menger sponge is a Sierpinski carpet; furthermore, any intersection of the Menger sponge with a diagonal or medium of the initial cube M0 is a Cantor set. The Menger sponge is a closed set; since it is also bounded, the Heine-Borel theorem yields that it is compact. Furthermore, the Menger sponge is uncountable and has Lebesgue measure 0.

The topological dimension of the Menger sponge is one; indeed, the sponge was first constructed by Menger in 1926 while exploring the concept of topological dimension. Note that the topological dimension of any curve is one; that is, curves are topologically one-dimensional. Menger showed, in the 1926 construction, that the sponge is a universal curve, in that any possible one-dimensional curve is homeomorphic to a subset of the Menger sponge. Note that by curve we mean any object 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 gasket 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 flat, and might be embedded in any number of dimensions. Thus, as a humorous example, any geometry of quantum loop gravity can be embedded in a Menger sponge.


Formal definition

Formally, a Menger sponge can be defined as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M := \bigcap_{n\in\mathbb{N}} M_n}

where M0 is the unit cube and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_{n+1} := \left\{\begin{matrix} (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} \end{matrix}\right\}}

See also

References

  • Karl Menger, General Spaces and Cartesian Spaces, (1926) Communications to the Amsterdam Academy of Sciences. English translation reprinted in Classics on Fractals, Gerald A.Edgar, editor, Addison-Wesley (1993) ISBN 0-201-58701-7
  • Karl Menger, Dimensionstheorie, (1928) B.G Teubner Publishers, Leipzig.

External links

de:Menger-Schwamm pl:Kostka Mengera sv:Mengers tvättsvamp