How to Create a Menger Sponge

This article is about how to create a Menger Sponge. A Menger Sponge is known in mathematics as a Fractal curve.

Steps

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    Start with a three dimensional object which is commonly referred to as a 'cube' as the zero degree (origin) of the Menger Sponge.
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    Create a division on a single face in the cube with two horizontal and two vertical scores (scribed lines, not cut all the way through), so there are now nine squares shown on that face.
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    Repeat the above step with each of the remaining five faces of the cube. By the end of your process you will have a cube that has been sub-divided into 3 X 3 X 3 = 27 smaller cubes (That is the volume of the original cube at this first degree of divisions.).
    • Do not let the 6 X 9 = 54 visible "exterior" faces of the 27 smaller cubes in the larger cube confuse you.
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    Remove the center cube of each side of the original cube of the construct. That would be the middle "third" cube from each middle row and column, the same way - throughout the original cube - when all of x and y and z axis are taken into consideration; along with the cube contained (in the interior) at the centre of the original cube. This removes 7 smaller cubes all together, in total.
    • You now can see straight through the original cube from the point of view of each of the 6 faces.
      Image titled Create a Menger Sponge Step 4Bullet1
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    Count all the cubes as a check. Now you should have 20 cubes.
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    Repeat the above steps (except step 1) for all cubes that now remain.
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    Observe. You will have now constructed what may be called the second degree of that which mathematicians know as the "Menger Sponge". Such steps could be repeated indefinitely but it would be difficult to do as cubes become smaller and more numerous within the interior of the larger cubes. You would visualize them all taken apart and put back together to do the interiors, but imagine the minute sizes if you worked through to some nth degree, and how would you keep it together (pins?), as it would collapse if all cubes were separate pieces!

Tips

  • A study of Mathematics shall avoid a state of confusion. You may wish to think about this until it makes sense to you, if it does not. Study the pictures and imagine the cubes for all 6 faces of the original cube -- 4 lateral (side) faces and the top and bottom faces.
  • The correct Menger Sponge is that which consists of a number of cubes equal to 20xn where n is the number of times the repetition mentioned above has indeed been completed, and 20 is the number of smaller cubes after the 7 middle ones were removed form each divided cube.

Article Info

Categories: Mathematics