Famous Fractals

Megner Sponge

The Megner Sponge is a 3-dimensional version of a fractal cluster, which is formed by continuously cutting out pieces of a figure. The Megner Sponge uses a cube for the initial figure (base). To form it, you have to divide a cube into 27 smaller cubes and then cut out the center cube and cubes at the centers of every side:

You then do the same for every one of the 20 cubes left and continue the process infinitely (see iteration). At the end, the fractal formed looks like this:

FRACTAL DIMENSION

After cutting out some cubes from the base, you are left with 20 identical cubes the side of each of which is 1/3 the size of the original cube. Using the similarity method, you can find the fractal dimension to be log 20 / log 3, which is approximately 2.73.

LENGTH

After an infinite number of iterations, the fractal will be an infinitely complex network of lines. The overall length of the lines will be infinite. This is true for all fractals whose fractal dimension is greater than 1.

SURFACE AREA

As you can see from the picture, the figure formed at the surface of every face of the cube is the Sierpinski Carpet. The area of the Sierpinski Carpet is equal to 0. Because of this, the surface area of the Megner Sponge also has to be 0.

VOLUME

After every iteration, we take away 7/27 of the volume of the entire figure. Only 20/27 of the volume is left. To find the volume after some number of iterations, we would have to raise 20/27 to the power of that number. After an infinite number of iterations, the volume will become 0.