About
Mengermania tracks the progress of building a level four Menger sponge. The sponge I am building will be a five foot cube comprising 1,296,384 units. The only material used in the structure will be ordinary index cards. No glue, tape, or scissors will be used during the construction or assembly processes.
Learn how to build a Menger sponge.
What is a Menger sponge?
A Menger sponge is a three-dimensional fractal named for its creator, Karl Menger.
To make a Menger sponge, do the following:
- Picture a cube.
- Divide the cube into 27 smaller cubes arrannged in a 3×3×3 setup like a Rubik's Cube.
- Remove the center cube from each face and the one in the middle of the arrangement.
- Repeat steps one through three for each of the remaining cubes to create more detailed sponages.
Steps for creating a level one Menger sponge
Steps for creating a level two Menger sponge
Each time the process of dividing and removing cubes is completed, a new "level" of sponge is created. For example, following steps one through three above for a single cube creates a level one sponge (20 cubes). If they're done a second time, a level two sponge is created (400 cubes). This process can be done an infinite number of times resulting in a sponge with an infinite number of cubes and surface area.
Level three sponge with counting guide
To determine the level of a sponge, count the number of gaps that are different sizes. The figure to the right displays a portion of a level three sponge with the three different gaps highlighted in orange.