Menger sponge


In mathematics, the Menger 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.

Construction

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.
  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.

Properties

The nth stage of the Menger sponge, Mn, is made up of 20n smaller cubes, each with a side length of n. The total volume of Mn is thus n. The total surface area of Mn is given by the expression 2n + 4n. 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. The number of these hexagrams, in descending size, is given by, with.
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.
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

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. In 2014, twenty level-three Menger sponges were constructed, which combined would form a distributed level-four Menger sponge.

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. 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 at the corners of the original cube, as well as twelve smaller cubes 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. Iterating an infinite number of times results in the Jerusalem cube.

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