Menger Cube – Super Facts

Super fact 113 : There are shapes that have a limited volume but an infinite surface area. Two examples are Gabriel’s Horn and the Menger Sponge.

A trumpet looking shape with a decreasing thickness as you move right along the trumpet. It stretches out infinitely far. — 3D illustration of Gabriel’s horn. RokerHRO, Public domain, via Wikimedia Commons.

You get Gabriel’s horn by rotating the curve y = 1/x around the x-axis in a coordinate system with the horn starting at x = 1, and y = 1, as in the picture below. Well, you can choose other start values too. A Gabriel’s horn with a beginning radius 1, or widest radius 1, (as in this example) will have a volume equal to pi. Pi is a popular constant. However, it will have an infinite surface area.

This seems like a paradox. The amount of paint you need to fill up Gabriel’s Horn (with widest radius = 1) is pi. However, the surface area of the outside and inside is infinite. So, wouldn’t you need an infinite amount of paint to paint the inside of Gabriel’s Horn? The amount of paint you need to fill up the horn should be more than if you just paint the inside, shouldn’t it? What’s going on? The solution to the paradox is to realize that the radius of Gabriel’s Horn will become increasingly small as it stretches out to the right, and for a coat of paint to take up volume it must have thickness. This is explained well here.

A coordinate system with the curve y = 1/x beginning at x = 1 and y =1. — If you rotate this curve around the x-axis, you get a trumpet shape. That is Gabriel’s Horn. I was lazy and drew this using ChatGPT instead of drawing it myself.

To understand why it is possible for the surface area of Gabriel’s Horn to become infinite you can imagine two cylinders of equal volume, one short (and thick), and one long (and thin). The longer and thinner cylinder will have a larger surface area as shown in the picture below. As Gabriel’s Horn is stretched out and getting thinner and thinner you get an infinite surface area, as you go towards infinity, while the volume does not become infinite. This is analogous to the infinite series in my previous post where adding an infinite number of subsequently smaller addends results in a finite number (corresponding to the volume being finite in this case).

The Menger Sponge

The picture shows a Menger Cube with square holes in its surface. There are also square holes inside the cube. — An illustration of M4, the sponge after four iterations of the construction process. Niabot, CC BY 3.0 https://creativecommons.org/licenses/by/3.0, via Wikimedia Commons

The way you construct a Menger sponge or a Menger cube is by starting with a cube.
Then divide every face of the cube into nine squares in a similar manner to a Rubik’s Cube, dividing the cube into 27 smaller cubes.
Then remove the smaller cube in the middle of each face and remove the smaller cube in the center of the larger cube, leaving 20 smaller cubes. This is a level 1 Menger sponge.
Repeat steps two and three for each of the remaining smaller cubes and continue to iterate infinitely many times.

As you are repeating this process over and over the volume of the Menger sponge will decrease a little bit in every step whilst the area will grow towards infinity.