# 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...!!!!

## No comments:

## Post a Comment