Menger sponge
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.
The Menger sponge itself is the limit of this process after an infinite number of iterations.
smaller cubes, each with a side length of (1/3)n. The total volume of
[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 the following recurrence relation:
[10] 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 2024, Broden, Nazareth, and Voth proved that all knots can also be found within a Menger sponge.
[11] The Menger sponge is a closed set; since it is also bounded, the Heine–Borel theorem implies that it is compact.
Experiments also showed that cubes with a Menger sponge-like structure could dissipate shocks five times better for the same material than cubes without any pores.
[12] Formally, a Menger sponge can be defined as follows (using set intersection): where
is the unit cube and 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.
A Jerusalem cube is a fractal object first described by Eric Baird in 2011.
It is created by recursively drilling Greek 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 a Jerusalem cross pattern.
which means the fractal cannot be constructed using points on a rational lattice.
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.
