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.

The Menger sponge is a fractal which is the three-dimensional analog of the Sierpiński carpet. Menger (1926) proved that the Menger sponge is universal for all compact 1-dimensional topological spaces, meaning that any compact topological space of dimension 1 has a homeomorphic copy as a subspace of the Menger sponge (Peitgen et al. 1992 ...

There is a simpler way of deducing the area of a Menger Sponge: Start with the area of the previous iteration (A n-1) Remove the middle of all the faces ((8/9)A n-1) Add the internal surface area, which is 20n-1(smaller cubes)×6(holes)×4(sides)×(1/9n)(scaling down) (A n =(8/9)A n-1 +(4×6×20n-1)/9n)

Mar 1, 2014 · Poke holes through it, removing 7 of these small cubes. Repeat this process for each remaining small cube. Do this forever! The result is the Menger sponge. What’s the volume of the Menger sponge? At each stage we remove 7/27 of the volume, so only 20/27 of the volume is left. As we repeat this forever, the volume drops to zero.

A Menger sponge is a square which is similar to itself. To make a Menger sponge, take a square, and cut a square hole in the middle. Repeat the process around the edge of the square 9 times. Do the same thing aforementioned infinitely.

Feb 9, 2018 · The Menger sponge is almost always represented as being constructed from Cantor sets using the “middle third” rule. The Menger sponge consists of all points ( x , y , z ) such that ( x , y ) , ( y , z ) , and ( x , z ) are all in Sierpinski carpets.

Mar 29, 2009 · In mathematics, the Menger sponge is a fractal curve. It is the universal curve, in that it has topological dimension one, and any other curve (more precisely: any compact metric space of topological dimension 1) is homeomorphic to some subset of it. It is sometimes called the Menger-Sierpinski sponge or the Sierpinski sponge.

Nov 26, 2024 · Three high schoolers and their mentor revisited a century-old theorem to prove that all knots can be found in a fractal called the Menger sponge.

Austrian mathematician Karl Menger first described what became known as the Menger Sponge in 1926. He was working in the field of topology and was trying to develop a definition of dimension. But the Menger Sponge is in fact an instance of what is nowadays termed a fractal.

Hexahedron Fractal (the Menger Sponge) Nice thing about the Menger Sponge is that it is all over the web, easy to get different views of. One of the more famous fractal structures, it is a cube made of cubes.