I packed up my office recently, so some mathematical objects came home with me: a house with two rooms, an origami counterexample, some British objects of constant width. And then there were the leftovers from that time we built a Menger sponge.
Thanks to Megamenger, a worldwide effort to build a huge fractal distributed around the globe, the Menger sponge will always be one of my favorite fractals. You can make a Menger sponge by taking a solid cube and drilling a square hole through the center of one face through to the other side. Now do this twice more for the remaining faces. (Worried about drilling a square hole? A Reuleaux triangle will come in handy.) You’ve now completed one level of the Menger sponge, and you’re left with something that looks like a Rubik’s cube with a bunch of cubes removed.
To keep Menger-izing, you drill square holes into all the remaining cubes the same way you did the first one. Then drill square holes into the remaining cubes, and so on. Like the Cantor set or Wallis sieve, which I’ve written about earlier, the only barrier to actually making this object in the real world is that you have to repeat the construction process forever.
In Megamenger, instead of making the object by drilling and removing, each group built it from the ground up. We made 20 small cubes out of six business cards each, assembled them into one holey cube, and took 20 of those to make a larger cube. Assembly details are available on the Megamenger site.
The reason I wanted to write about the Menger sponge right now is because it is 2.7268(ish)-dimensional. Earlier this month I wrote about dimensions. You can read my post about it, but the basic idea was to link dimension to the number of coordinates used to measure or label something. That kind of dimension is always a whole number. Fractals do have dimensionality in the traditional sense, but they are such puzzling creatures that people had to come up with a new definition of dimension to fully describe them.
Here’s the problem: the Menger sponge has infinite surface area but zero volume. It’s too big if we measure it in two dimensions but too small if we measure it in three dimensions. In contrast, let’s think about a 1×1 solid square in the two-dimensional plane. Its length is infinite: there are infinitely many 1-unit line segments that are part of this square. Its area is 1 square unit, and its volume is 0 because it has no depth. In this example, there’s one dimension in which it has infinite size, one dimension in which it has zero size, and one dimension in which it has a measurable size. It’s like Goldilocks. The two-dimensional plane is “just right” for measuring a solid square.
In the case of the Menger sponge, two-dimensional space is too small, and three-dimensional space is too large. Could there be some number in between there that is “just right” for the Menger sponge?
There is! To figure out how to find it, we need to be creative about how we think about dimension. What features other than the number of coordinates define dimension? One is scaling. At the risk of being Captain Obvious, let’s start small. In one-dimensional space, if you triple an object’s length, the object becomes three times as long. In two-dimensional space, if you triple an object’s length in every direction—for example, tripling the length and width of a rectangle—the object becomes nine times as large. Equivalently, it takes nine small squares to make one square with triple the side length. In three dimensions, if you triple the length, width, and height of a cube, it becomes 27 times as large. It takes 27 small cubes to make one cube with triple the side length. We can see a pattern here. The numbers 3, 9, 27 can also be written 31, 32, 33—the dimension is the power to which we raise the scale factor to get the new size.
Now we can see why the Menger sponge doesn’t quite work as either a two-dimensional or a three-dimensional object. When we stretch each side of the Menger sponge by a factor of three, we get 20 copies of the Menger sponge back, so we have multiplied its size by 20. Another way to think about it is that it only takes 20 cubes to build a level-one Menger sponge and 20 level-ones to build a level-two. But the level-two Menger sponge’s sides are three times as long as the level-one’s.
Because we can raise numbers to non-integer powers, we can use scaling to understand what dimension might mean for objects like the Menger sponge that don’t scale the way whole number-dimensional objects like a solid cube do. The dimension is the power to which we raise the scale factor to get the new size. For the Menger sponge, its dimension d is the number that solves the equation 3d=20. The number 2.7268 just about does it. (The exact answer is log320 or log(20)/log(3).)
It would be a shame to leave a post about the Menger sponge and fractal dimension without a rabbit-hole for you to dive down, so here you go: the Wikipedia list of fractals by dimension. You’re welcome!
Read about more of my favorite spaces:
The Cantor Set
Fat Cantor Sets
The Topologist’s Sine Curve
Cantor's Leaky Tent
The Infinite Earring
The Line with Two Origins
The House with Two Rooms
The Fano Plane
The Möbius Strip
The Long Line
The Wallis Sieve
Two Tori Glued along a Slit
The Empty Set