Graphic artist M.C. Escher’s mind-twisting prints are famous for their repeating geometric patterns, spatial illusions, and impossible structures. Some of these depictions pose intriguing mathematical puzzles.

In his April 4 lecture “Escher and the Droste Effect,” Bart de Smit explained to a capacity crowd at the MAA Carriage House how he and Leiden University colleague Hendrik Lenstra used mathematics to fill a mysterious hole in Escher’s 1956 lithograph Print Gallery.

The Droste effect, de Smit explained, refers to the repetition of a picture within itself. Think of the Land O’Lakes logo, in which a kneeling Indian maiden holds a butter box on which her likeness appears. The Dutch name for this sort of recursive image derives from a brand of cocoa. A box of Droste cocoa powder depicts a nun carrying a tray bearing a cup and—surprise!—a Droste box.

“There are really an infinite number of nuns in here,” de Smit said, gesturing toward an animated zoom of the Droste packaging. “They’re just getting smaller and smaller.”

A French foodstuff provided another example. The laughing cow that adorns each wheel of La Vache Qui Rit cheese wears as earrings wheels of the same. To transition from the circular wheel of cheese to the angled and therefore oblong earrings, though, it is necessary to both zoom in and rotate.

It’s no mystery that there’s rotation afoot in Print Gallery. In Escher’s lithograph, a man stands in a gallery looking at a picture of the city in which the gallery is located. Arches warp, picture frames curve. And at the center of this contorted, self-referential representation of the Maltese town of Senglea sits a circular void, which prompted Lenstra and de Smit to ask: “Can you continue this picture? Can you keep going?”

De Smit spent the balance of his lecture demonstrating that the answer is “yes.”

He showed four of Escher’s preliminary sketches, all drawn in normal perspective, each a blow-up of the previous, zoomed in by a factor of four.

De Smit presenting the La Vache Qui Rit logo during his MAA Distinguished Lecture.

He flashed a slide of the odd curved grid paper Escher used when constructing Print Gallery and explained that, according to Escher popularizer Bruno Ernst, the artist divided his rectilinear sketches into squares and then transferred them piece by piece onto the curved grid.

De Smit described how reversing Escher’s transformation yields a “straight world” cut through with a white spiral corresponding to the hole in the finished print. It’s easy enough to produce a seamless picture by filling in the spiral, de Smit noted, but less clear that the resulting image can be twisted into an imperforate version of Print Gallery.

But it can be done, as de Smit and his colleagues discovered. By investigating how paths in the straight world correspond to paths in the curved one, they determined that in creating Print Gallery Escher used a scaling factor of 18 and a rotation of 160 degrees.

The mathematicians then figured out that in his quest to accomplish cyclic expansion in an aesthetically pleasing way, Escher stumbled upon what mathematicians call conformal transformations. These functions have the artistically useful property of preserving angles.

Bart de Smit illustrates how Escher took advantage of conformal

transformations to create his lithograph Print Gallery.

“If you use a conformal transformation, you can transform anything, and you will get something where you can still recognize the details,” de Smit said.

Lenstra and de Smit determined the precise scaling factor and degrees of rotation required to make the transformation between Escher’s straight and curved worlds perfectly conformal. Their results were published in the Notices of the AMS (pdf). They found that, as Escher tweaked his grid paper, he approached the “mathematically ideal solution” to the problem posed by his artistically motivated vision for the piece.

Bruno Ernst doesn’t think the artist knew it, but it turns out that the way to fill the central hole in Print Gallery follows directly from Escher’s construction of the scene. Lenstra and de Smit were able to finish what Escher started by enlisting illustrators to bridge the gaps in the straight sketches of the Senglea gallery and then applying to these pictures the conformal transformation Lenstra and de Smit identified.

Interested readers can see the product—look closely at the inverted miniature of the gallery and you’ll spot the anachronistic Möbius Strip II, which postdates Print Gallery—at the project website maintained by de Smit.—Katharine Merow

Listen to the full lecture (mp3)

Photographs and audio by Laura McHugh

This MAA Distinguished Lecture was funded by the National Security Agency.