Is every point on a Menger Sponge visible from the outside?

Choose an arbitrary point on the surface of a Menger Sponge. Can you find a straight line starting at that point and extending beyond the sponge that doesn't intersect the sponge anywhere else? That is, is there a position and angle 'outside' the sponge from which an observer could see that point?

In one sense it seems the answer should be 'no', because a point on the sponge can be inside arbitrarily many twisting tunnels. But then again the construction of the shape means that every point is somehow 'near' the outside.

1 Answer

Yes, the whole surface is visible.

You can make the sponge by starting with a cube and then drilling out a square segment form all three sides. Then you take a smaller drill and repeat the process 3*8 times and so on.

Since all you do is drilling holes, the question simplifies to "can you see the entire inner surface of a straight pipe when looking at it from one end?"
The answer to it is obviously 'yes' even though the viewing angles will become infinitesimally small.

Edit:
We can probably generalize this for other shapes and higher dimensions like this:
"For a set $S$ Every point on its surface $\partial S$ can be viewed from the outside of the convex hull of $S$ if the set complementary to $S$ can be constructed from the union of straight lines (of infinite length and of which there are infinitely many)"

This still does not cover all possible shapes, like a hourglass, but it works for the Menger Sponge, Cantor Dust and similar fractals

• Somehow your explanation doesn't satisfy me, because it presumes that the set of all points contained in the Menger sponge consists only of those points that are located on a boundary; i.e., $M = \partial M$. But even for a 1-dimensional Cantor set, this is false: there exist points of the Cantor set that are not endpoints, so it should be easy to see that there also must exist some point $x \in M$ but $x \not\in \partial M$: for instance, because $1/4$ is in the Cantor set on $[0,1]$ but not an endpoint, $(1/4,1/4,1/4)$ is a non-boundary point of the Menger sponge. Commented Oct 21, 2014 at 23:00
• not only the cantor set, this does not work for any non-connected set in 1d. But that wasnt the question, the question was, in fact, about the surface = $\partial M$ of the sponge Commented Oct 21, 2014 at 23:20
• Indeed, I had missed that part of the original question! Commented Oct 21, 2014 at 23:33
• This depends in essential fashion on the fact that all of the 'holes' in the sponge go all the way through from one end to the other (or, essentially, that the sponge is the intersection of three generalized cylinders); this is true but maybe not so self-evident as it would seem. Commented Oct 21, 2014 at 23:38
• It's at least intuitively clear from looking at a picture that any hole through a cubelet actually extends all the way through the sponge, and that observation indeed makes the question trivial Commented Oct 22, 2014 at 9:50