Assignment 1 Generating and viewing the Menger sponge Solution

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Description

The Menger sponge is an important object in mathematics and the theory of fractals. The figures below show how it can be constructed.

2) Remove the cube in the middle.

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3) Create next layer of the sponge by adding 4 cubes in the corners, as shown.

1. Add the third layer, which has the same structure as the first layer shown in Step 2. If you wish, you may think if this object as a cube subdivided into 17 sub-cubes, 7 of which have been removed (one in the bottom layer, 5 in the middle, and 1 in the top layer.)

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4’) Remove gaps between the cubes (actually, they were never needed for the construction, but were shown in Fgures 1-4 to clearly illustrate the arrangement). This is stage-1 Menger sponge.

1. Now, arrange 20 stage-1 Menger sponges in the same manner as the initial cubes have been arranged. This is stage-2 sponge. If you prefer, you can also alternatively think of the stage-2 sponge as resulting from the replacement of each cube in Figure 4 with a stage-1 sponge.

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1. Repeat Step 5 of the construction recursively a least one more time. Don’t go too far! The number of sub-cubes and their faces grows very fast (exponentially)!

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6’) Again, gaps in Figure 6 have only been added for illustrative purposes. The actual Menger sponge does not have them.

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