A Slice of Interdimensional Sponge Cake

The Menger sponge is a delightfully pathological shape to soak up. Start with a cube of side-length 1, and “hollow it out” by slicing it into 27 sub-cubes (like a Rubik’s Cube) and removing the 7 central sub-cubes as illustrated below. Call this the level 1 sponge. To make the 2nd level, perform the same “hollowing out” operation on each of the 20 cubes in level 1. Keep going to make levels 3, 4, and so on. The end result, i.e., level \(\infty\), is a fractal known as the Menger Sponge. It has a crazy[1] network of tunnels and passageways running through it:

Stages in the construction of the Menger sponge
Stages in the construction of the Menger sponge: Levels 0, 1, 2, and 3. Click for larger image.

What is its volume? We made the Menger Sponge by cutting out smaller and smaller chunks from the original unit cube, so how much volume did we leave behind? Each “hollowing out” step reduces the volume by a factor of 20/27, so level k has volume \((20/27)^k\) and the final Menger Sponge has volume 0. We removed all the volume!

Since the Menger Sponge doesn’t have enough “3D stuff” to be considered a truly 3-dimensional shape, maybe it behaves more like a 2-dimensional surface? This isn’t quite right either. The Menger Sponge has “too much 2D stuff”: its surface area is infinite.[2] In fact, this shape falls squarely (hah!) in the middle: it has dimension 2.7268….

Wait, what does it mean to have non-integer dimension? One interpretation, formalized in the notion of Hausdorff dimension, is to look at how the “stuff” it is made from behaves under scaling. If we take a square (which is made of “2-dimensional stuff”) and scale it by a factor of 1/3, its area changes by a factor of \((1/3)^2\). Similarly, scaling a cube (“3D stuff”) by 1/3 scales its volume by \((1/3)^3\). In general, scaling “d-dimensional stuff” by a factor of 1/3 scales its “d-dimensional volume” by \((1/3)^d\).

How a plane, Menger sponge, and cube behave under scaling
How different dimensional "stuff" behaves when scaled by a factor of 1/3. Left: A 2D plane is split into 9=3^2 smaller copies. Right: A 3D cube is split into 27=3^3 copies. Middle: A Menger sponge is split into 20 smaller copies.

As shown in this image, a Menger Sponge can be chopped into 20 smaller Menger Sponges, each scaled by 1/3. So if the Menger Sponge is made of m-dimensional stuff, its m-dimensional volume scales by 1/20 when the shape is scaled by 1/3, so we should have \((1/3)^m=1/20\), i.e., \(m=\log_3(20) = 2.7268\ldots\).

As one more example of the quirkiness that can be squeezed out of the Menger Sponge, what happens when we slice this shape along one of its primary diagonal planes[3]? The Menger Sponge’s crazy network of tunnels creates a stellar array of 6-pointed stars cut out of a regular hexagon:

A slice through a Menger sponge
Slicing a Menger sponge along a primary diagonal plane produces a dazzling fractal (living in the slicing plane) made from 6-pointed stars.

What is the Hausdorff dimension of this hexagonal fractal? It turns out to be $$\log_3\left(\tfrac{9+\sqrt{33}}{2}\right) = 1.8184\ldots.$$ To find out where this ridiculous number comes from, you’ll have to come back next week!


  1. This is a technical term, of course. []
  2. The surface area of level k can be computed to be \(2\cdot(20/9)^k+4\cdot(8/9)^k\), and this approaches infinity as k grows. []
  3. Specifically, slice it along the perpendicular-bisecting plane of one of the cube’s diagonals []

4 thoughts on “A Slice of Interdimensional Sponge Cake

    1. Thanks again, Sid!

      I made most of the images on this site (including the ones in this post) with Mathematica 8, sometimes with minimal post-processing (e.g., cropping, resizing, joining) in GIMP. My other graphics program of choice is the Asymptote language, which I use when I need vector formats like PDF and/or embedded LaTeX labels/text. (In particular, I usually prefer Asymptote for academic publishing.)

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