Escher's Three Balls Paradox

Despite lacking any formal background in the subject, Escher's drawings undoubtedly have mathematical flavor. Paradox, in particular, is a commonly explored theme. In many of his lithographs, Escher makes use of carefully crafted optical illusions to depict geometrically impossible toys, buildings, stairs, etc. Many of these paradoxes are well known and much has been written about them. However, I recently stumbled across a beautiful paradox in Three Balls, a piece belonging to his reflection collection, which to my knowledge is not widely recognized. Take a moment to ponder the image above and see if you can spot it.

Did you find it? If not, no worries, I'll walk you through it.

Let's start from the beginning. Escher's lithograph is of three balls: a translucent one, a reflective one, and an opaque one. The outer two are not particularly interesting, so let's ignore those. On the reflective ball (defined as R), Escher draws the reflection of himself drawing the three balls in his studio. We are able to see what Escher is drawing in the reflection given by R. Let's focus on reflective ball in this reflection (defined as S).

Already, there's an interesting question here. Is the image on S a depiction of the final drawing, or is it a depiction of the drawing process? If it's the final image then there should be a completed picture of Escher drawing the three balls on S (with another reflective ball T whose relation to S is the same as the relation of S to R). If it's the image in progress, we should be seeing a sketch of this picture on S. The latter alternative seems more likely, but it's still quite unconvincing. Based on the attention to detail in his other work, I posit it would be uncharacteristically lazy of Escher to so sloppily draw S, and he is lucidly drawing something else entirely. What is it? Well, it's reflection of the bottom of Escher's face.

Imagine you are sitting across a table from Escher. The drawing with the three balls in the reflection from R is on the table between the two of you and under Escher's chin. If S were a mirror, then you would expect to see the bottom of Escher's face. This is the paradox! In S, Escher isn't depicting what is actually being drawn as R. Instead, S behaves as an actual reflective surface from the perspective of the viewer of the lithograph.

One might argue this is a convoluted explanation for what is in actuality a simple exploration of reflection. I disagree. Escher's work is famous for its complexity, and my explanation aligns perfectly with his generally understood fondness for paradox. But, even if this wasn't his intention, who cares? Art is founded on the interplay of intention and interpretation anyway.