M. C. Escher’s art is renowned for its seemingly effortless depiction of some of the most abstract concepts in mathematics, logic and philosophy. His Print Gallery (below) is a lithograph from 1956 which depicts a man standing in a gallery viewing a painting of a harbour; yet, this painting appears to extend into his own reality as the buildings by the port contain the same gallery which he is visiting.

This is striking enough as an artistic feat, but what has caught the curiosity of mathematicians is the void at the centre of the picture. Can it be filled? Is it there on purpose? Two research teams almost 25 years apart investigated just this issue. In his 1979 book Gödel, Escher, Bach Douglas Hofstadter explains Escher’s Print Gallery through Incompleteness Theorems of the mathematician and philosopher Kurt Gödel. The First Incompleteness Theorem simply states that in a system S there exist statements of language which cannot be proved nor disproved in S – i.e. the system is “consistent’”. The Second Incompleteness Theorem elaborates to explain that the consistency (or “non-contradiction”) in S cannot be proved in S.

However, Hofstadter paraphrases the theorems eloquently, stating that “for any record player, there are records which it cannot play because they will cause its indirect self-destruction”. Using self-referential systems and different levels of reality, Hofstadter stipulates that as a result of the Incompleteness Theorems, the void at the lithograph’s centre could be arbitrarily small, but must exist – that is, filling the void is the record which would cause self-destruction to the record player.

From The Mathematical Structure of Escher’s Print Gallery by B. de Smit and H. W. Lenstra Jr. in the American Mathematical Society Journal Volume 50 Number 4.

In 2003, Dutch mathematicians Hendrik Lenstra and Bart de Smit analysed Print Gallery using conformal mappings, which preserve the angles between two very close points when a function is applied to them. The researchers found that it was possible to fill the empty centre if the picture were to be taken as elliptic curves over the complex plane, which houses imaginary numbers z=x+iy where i=the square root of -1. Indeed, Lenstra and de Smit found that by applying a number of transformations (essentially a combination of functions) to the grid Escher used as a sketching aid, it was possible to “unfold” and straighten Print Gallery to find an “original image” (i.e. a simple straight grid or the man looking at a painting without distortion).

Print Gallery also subscribes to the Droste Effect (a picture containing itself infinitely – mise en abyme in art) and the Leiden team found that the necessary transformations largely rest on multi-valued functions, such as the exponential and logarithm, as they enable this recursion since several points of the function return the same value. In the words of Bruno Ernst, Escher was attempting to create an “annular bulge, a cyclic expansion…without beginning or end”.  Early sketches reveal he first tried to create this illusion by using a straight grid, but this produced an image too distorted for the interplay between levels of reality to exist; thus, Escher decided against it in favour of curved lines.

Hofstadter also relates Print Gallery to strange loops, which give the viewer the impression of moving ever further from the origin while always returning to it, i.e. they are self-referential. The grid Escher used to sketch Print Gallery demonstrates this, since if one were to the path from ABCDA clockwise around the centre in the grid above, the squares which comprise it magnify by 4 in all four directions. In other words, as one moves from a point A cyclically around the origin, one would ultimately return to the starting point A, but will have undergone an expansion of factor 256.

My own work began with retracing Lenstra and de Smit’s steps by using computer programmes to reconstruct the grids they created; but I soon came to wonder whether the two theories would be reconcilable instead. Logarithms have a singularity at zero, i.e. a point at which the function is undefined. Loosely, one can liken it to the logarithmic function having a “void” at zero. As such, the logarithm’s singularity could be considered the “arbitrarily” small void that Gödel’s Incompleteness Theorems predict to exist.

The Droste Effect is a visual representation of a strange loop. The loop itself, as Hofstadter explains, forces a viewer to question the different levels of reality Print Gallery portrays. As Hofstadter rightly asks, is Print Gallery then “a picture of a picture which contains itself? Or is it a picture of a gallery which contains itself? Or of a town which contains itself? Or a young man who contains himself?” Escher is known for his strangely fitting constructions of the infinite. There are two different, explicit levels of reality: that of the picture and that of the man standing in the gallery. However, these two levels allow the viewer to consider herself as part of a third level of reality, which, in turn, seems to engulf her as part of Escher’s implicit chain of “reality levels”. For any position in this chain, there will exist a greater and lesser reality, one more and one less real than the “current position”. Although this is difficult to imagine, it is still a linear ordering akin to the simple ranking of numbers by magnitude. Escher’s Print Gallery, however, is not linearly conceived. Its difficulty then lies in determining what is real and what is not; does the young man truly exist but the port is a mere picture? Or is the harbour real, yet the man is fantastical?

Escher’s artwork appears to be perfectly explicable through the lens of both conformal mappings and Gödel’s Incompleteness Theorems. Strange loops are an important feature of Print Gallery that distort that which the viewer intuitively feels must be real and what she knows truly is real. Returning to the practical, Lenstra and de Smit work in ℂ*, the complex plane excluding (0,0) from the set of transformations that relate the distorted and straight image. Therefore, Lenstra and de Smit’s “filling” of the void is near-perfect – it excludes this one single point, which is humanly impossible to view and thereby fulfils the Incompleteness Theorems’ prediction that an arbitrarily small void must always exist. Therefore, it is possible to reconcile the two approaches and the complex logarithm’s singularity at zero can be seen as the epitome of Gödel’s Incompleteness Theorems.

Art, philosophy and mathematics overlap in countless ways, perhaps beginning with perspective, proportion and symmetry; in testing the limits of reality, the relationship between the three studies is one which has mystified generations, blurring the lines between what is possible and what is not. As such, Escher’s artistic genius lies in his ability to portray the incomprehensible.

Siobhán Fraile Ordóñez studied Mathematics and Economics. Her work focussed on mathematically analysing M. C. Escher’s art. In 2015, she founded a female student group to advance the understanding of investment strategies, the financial industry, and global markets.