Infinity Series II: Zero Volume With Infinite Surface Area

Ascending and Descending (M. C. Escher, Lithograph, 1960).
This is the second entry of the “Infinity Series”, which will introduce you to the peculiar world of infinity. Last time we discussed the Koch Snowflake, which has finite area but infinite perimeter. We did not specify the size of the circle into which we fitted the triangles, so the circle could have been as small as you wanted and still have an object that sits inside with infinite perimeter. But that raises the question, “can you make it to be 0?”
In fact, you can – but you have to add a dimension. This is basically an extension of the last entry.

Initially start with a cube which is constructed by putting 6 square faces perpendicular to each other.

Then divide every face of the cube into nine equal smaller squares. This can be done nicely by dividing every edge into thirds. This will create 27 smaller cubes. Then, extract the smaller cube in the middle of each face as well as the cube in the very center; this leaves us with 20 cubes. The surface area has increased, since every removed cube on the faces actually created three more faces of the same size (i.e. removing a cube on the face deletes one of the nine divided squares of the face but creates four more of those squares).

Similarly, divide each remaining cubes into nine smaller cubes and extract the middle cube of each face as well as the central one. Since you’re essentially performing the same operation on the smaller cubes, the volume has again decreased and the surface area increased.

Performing this one more time will leave you something like this.

As you can imagine, this operation can be done an infinite number of times. After an infinite time of iteration, you are left with something that has an infinite surface area but 0 volume.

How can I be sure that I have 0 volume?

I challenge you to give me any positive number, say epsilon, and this number can be as small as you want it to be (but not zero  – positive numbers don’t include 0). I will win the challenge if I can perform the operation enough (finite) times to get a volume to be smaller than that epsilon. The truth is, I will always win and this can be seen from the fact that the volume decreases by 20/27 per iteration.

How can I be sure that I have infinite surface area?

Let’s play a similar challenge. Give me any number, say delta, and this can be the biggest number that you can possibly think of. But I have the power on mathematics to my side and I can assure you that by performing the above operations enough (finite) times, I will have a surface area larger than your delta (by a similar argument as above, but this time increasing the area at every iteration).

This particular object with infinite surface area but 0 volume is called the Menger Sponge.

If you had built a house of Menger sponge, it would have taken you an infinite amount of time to paint the walls but you would have just built yourself a house that no one (including yourself) can fit inside.

If you were to make a swimming pool of Menger sponge, then you would need an infinite amount of tiles but even a single drop of water would overflow the pool. Perhaps not so much fun.

By creating the Menger sponge, you are essentially creating something (as a matter of fact, something infinite) out of nothing.

NOTES: Images of Menger Sponges from Wikimedia Commons (Saperaud / Wikimedia Commons)
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