Minecraft Menger sponge (Menger Universal Curve) :
The Menger sponge is a 3 dimensional cube that models the Sierpinski carpet. The Sierpinski carpet is a fractal generalization of the Cantor set.
The Sierpinski carpet is a 2 dimensional figure starting with a square and then subdividing into 9 sections. You then remove the center section.
The next step is to take the remaining 8 sections and repeat the process for each one.
After that you are left with an image where the middle section is missing from each of the 8 subsections of the original figure. You then take each of those subsections and repeat the process again. This will leave you with this figure:
You can then repeat the process again....
An amazing property of the Sierpinski carpet is that as the number of iterations of this process approaches infinity the area will approach 0
When this process is moved up into 3 dimensions you construct a Sierpinski sponge. The Menger cube is slight alteration of the Sierpinski sponge.
The Menger sponge has a similar amazing property that as the iterations approach infinity the Surface area approaches infinity and the Volume approaches 0..
I created a 4 iteration Menger sponge in Minecraft, starting with a 3X3X3cube.
Then I removed the center block from each of the 6 faces.
I Then cloned this cube using the Minecraft /clone command to get to 20 cubes arranged to show the next iteration of the Menger sponge.
I then cloned that cube to make 20 more copies to construct the next iteration.
At this point I will note that for the Menger sponge the number of cubes (or in this case blocks) is given by the formula (Number of blocks) N = 20ⁿ where n is the number of iterations and the above cube is iteration 3. which is N = 20³ = 8000 blocks.
For the final project I decided to move over into my realms server and construct a 4 iteration Menger cube with N = 20⁴ = 160,000 blocks...!!!!