An illustration of the iterative construction of a Menger sponge up toM3, the third iteration
The construction of a Menger sponge can be described as follows:
Begin with a cube.
Divide every face of the cube into nine squares in a similar manner to aRubik's Cube. This sub-divides the cube into 27 smaller cubes.
Remove the smaller cube in the middle of each face and remove the smaller cube in the center of the larger cube, leaving 20 smaller cubes. This is a level 1 Menger sponge (resembling a void cube).
Repeat steps two and three for each of the remaining smaller cubes and continue to iteratead 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.
Hexagonal cross-section of a level-4 Menger sponge. (Part of aseries of cuts perpendicular to the space diagonal.)
Theth 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 aSierpinski carpet, and the intersection of the sponge with any diagonal of the cube or any midline of the faces is aCantor set. The cross-section of the sponge through itscentroid and perpendicular to aspace diagonal is a regular hexagon punctured withhexagrams arranged in six-fold symmetry.[8] The number of these hexagrams, in descending size, is given by the followingrecurrence relation:, with.[9]
The sponge'sHausdorff dimension islog 20/log 3 ≅ 2.727.[10] TheLebesgue covering dimension of the Menger sponge is one, the same as anycurve. Menger showed, in the 1926 construction, that the sponge is auniversal curve, in that every curve ishomeomorphic to a subset of the Menger sponge, where acurve means anycompactmetric space of Lebesgue covering dimension one; this includestrees andgraphs with an arbitrarycountable number of edges, vertices and closed loops, connected in arbitrary ways. Similarly, theSierpinski 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 notplanar and might be embedded in any number of dimensions.
In 2024, Broden, Nazareth, and Voth proved that all knots can also be found within a Menger sponge.[11]
Experiments also showed that cubes with a Menger sponge-like structure could dissipateshocks five times better for the same material than cubes without any pores.[12]
MegaMenger was a project aiming to build the largest fractal model, pioneered byMatt Parker ofQueen Mary University of London andLaura Taalman ofJames 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 moreaesthetically pleasing.[13] In 2014, twenty level three Menger sponges were constructed, which combined would form a distributed level four Menger sponge.[14]
AJerusalem cube is afractal object first described by Eric Baird in 2011. It is created by recursively drillingGreek cross-shaped holes into a cube.[15][16] The construction is similar to the Menger sponge but with two different-sized cubes. The name comes from the face of the cube resembling aJerusalem cross pattern.[17]
The construction of the Jerusalem cube can be described as follows:
Start with a cube.
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.
Repeat the process on the cubes of ranks 1 and 2.
Iterating an infinite number of times results in the Jerusalem cube.
Since the edge length of a cube of rank N is equal to that of 2 cubes of rank N+1 and a cube of rank N+2, it follows that the scaling factor must satisfy, therefore which means the fractal cannot be constructed using points on arationallattice.
Since a cube of rank N gets subdivided into 8 cubes of rank N+1 and 12 of rank N+2, the Hausdorff dimension must therefore satisfy. The exact solution is
which is approximately 2.529
As with the Menger sponge, the faces of a Jerusalem cube are fractals[17] with the same scaling factor. In this case, the Hausdorff dimension must satisfy. The exact solution is
Atetrix is a tetrahedron-based fractal made from four smaller copies, arranged in a tetrahedron.[19]
A Sierpinski–Menger snowflake is a cube-based fractal in which eight corner cubes and 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
^Menger, Karl (1928),Dimensionstheorie, B.G Teubner,OCLC371071
^Menger, Karl (1926), "Allgemeine Räume und Cartesische Räume. I.",Communications to the Amsterdam Academy of Sciences. English translation reprinted inEdgar, Gerald A., ed. (2004),Classics on fractals, Studies in Nonlinearity, Westview Press. Advanced Book Program, Boulder, CO,ISBN978-0-8133-4153-8,MR2049443
^Quinn, John R. (2013). "Applications of the contraction mapping principle". In Carfì, David; Lapidus, Michel L.; Pearse, Erin P. J.; van Frankenhuijsen, Machiel (eds.).Fractal geometry and dynamical systems in pure and applied mathematics. II. Fractals in applied mathematics. Contemporary Mathematics. Vol. 601. Providence, Rhode Island: American Mathematical Society. pp. 345–358.doi:10.1090/conm/601/11957.ISBN978-0-8218-9148-3.MR3203870.. SeeExample 2, p. 351.
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