[Editor's notes: Escher's "Print Gallery" can be found at
Mathematical explanation of Escher and the Droste effect can be found at
I also learned that not everyone mentally factors street addresses so as to
remember them better. -- jdcc]
Mathematician Fills in a Blank for a Fresh Insight on Art
By SARA ROBINSON
n a flight to the Netherlands, Dr. Hendrik Lenstra, a mathematician, was
leafing through an airline magazine when a picture of a lithograph by the
Dutch artist M. C. Escher caught his eye.
Titled "Print Gallery," it provides a glimpse through a row of arching
windows into an art gallery, where a man is gazing at a picture on the wall.
The picture depicts a row of Mediterranean-style buildings with turrets and
balconies, fronting a quay on the island of Malta.
As the viewer's eye follows the line of buildings to the right, it begins to
bulge outward and twist downward, until it sweeps around to include the art
gallery itself. In the center of the dizzying whorl of buildings, ships and
sky, is a large, circular patch that Escher left blank. His signature is
scrawled across it.
As Dr. Lenstra studied the print he found his attention returning again and
again to that central patch, puzzling over the reason Escher had not filled
it in. "I wondered whether if you continue the lines inward, if there's a
mathematical problem that cannot be solved," he said. "More generally, I
also wondered what the structure is behind the picture: how would I, as a
mathematician, make a picture like that?"
Most people, having thought this far, might have turned the page, content to
leave the puzzle unsolved. But to Dr. Lenstra, a professor at the University
of California at Berkeley and the University of Leiden in the Netherlands,
solving mathematical puzzles is as natural as breathing. He has been known,
when walking to a friend's house, to factor the street address into prime
numbers in order to better fix it in his mind.
So Dr. Lenstra continued to mull over the mystery and, within a few days of
his arrival, was able to answer the questions he had posed. Then, with
students and colleagues in Leiden, he began a two-year side project,
resulting in a precise mathematical version of the concept Escher seemed to
be intuitively expressing in his picture.
Maurits Cornelis Escher, who died in 1972, had only a high school education
in mathematics and little interest in its formalities. Still, he was
fascinated by visual mathematical concepts and often featured them in his
One well-known print, for instance, shows a line of ants, crawling around a
Moebius strip, a mathematical object with only one side. Another shows
people marching around a circle of stairs that manage, through a trick of
geometry, to always go up. The goal of his art, Escher once wrote in a
letter, is not to create something beautiful, but to inspire wonder in his
Seeking insight into Escher's creative process, Dr. Lenstra turned to "The
Magic Mirror of M. C. Escher," a book written (under the pen name of Bruno
Ernst) by Hans de Rijk, a friend of Escher's, who visited the artist as he
created "Print Gallery."
Escher's goal, wrote Mr. de Rijk, was to create a cyclic bulge "having
neither beginning nor end." To achieve this, Escher first created the
desired distortion with a grid of crisscrossing lines, arranging them so
that, moving clockwise around the center, they gradually spread farther
apart. But the trick didn't quite work with straight lines, so he curved
Then, starting with an undistorted rendition of the quayside scene, he used
this curved grid to distort the scene one tiny square at a time.
After examining the grid, Dr. Lenstra realized that carried to its logical
extent, the process would have generated an image that continually repeats
itself, a picture inside a picture and so on, like a set of nested Russian
Thus, the logical extension of the undistorted picture Escher started with
would have shown a man in an art gallery looking at print on the wall of a
quayside scene containing a smaller copy of the art gallery with the man
looking at a print on the wall, and so on. The logical extension of "Print
Gallery," too, would repeat itself, but in a more complicated way. As the
viewer zooms in, the picture bulges outward and twists around onto itself
before it repeats.
Once Dr. Lenstra understood this basic structure, the task was clear: If he
could find an exact mathematical formula for the repetitive pattern, he
would have a recipe for making such a picture with the missing spot filled
Measuring with a ruler and protractor, he was able to estimate the bulging
and twisting. But to compute the distortion exactly, he resorted to elliptic
curves, the hot topic of mathematical research that was behind the proof of
Fermat's last theorem.
Dr. Lenstra knew he could apply elliptic curve theory only after reading a
crucial sentence in Mr. de Rijk's book. For esthetic reasons, Mr. de Rijk
explains, Escher fashioned his grid in such a way that "the original small
squares could better retain their square appearance." Otherwise, the
distortion of the picture would become too extreme, smearing individual
elements like windows and people to the point that they were no longer
"At first, I followed many false leads, but that sentence was the key," Dr.
Lenstra said. "After I read that, I knew exactly what was happening."
Escher was creating a distortion with a well-known mathematical property: if
you look at small regions of the distorted picture, the angles between lines
have been preserved. "Conformal maps," as such distortions are known, have
been extensively studied by mathematicians.
In practice, they are used in Mercator projection maps, which spread the
rounded surface of the earth onto a piece of paper in such a way that
although land masses are enlarged near the poles, compass directions are
preserved. Conformal principles are also used to map the surface of the
human brain with all the folds flattened out.
Knowing that Escher's distortion followed this principle, Dr. Lenstra was
able to use elliptic curves to convert his rough approximation of the
distortion into an exact mathematical recipe. He then enlisted a Leiden
colleague, Bart de Smit, to manage the project and several students to help
First, the mathematicians had to unravel Escher's distortion to obtain the
picture he started with. A student, Joost Batenburg, wrote a computer
program that took Escher's picture and grid as input and reversed Escher's
Once the distortion was undone, the resulting picture was incomplete. Some
of the blank patch in the center of "Print Gallery" translated into a
blurred swath spiraling across the top of the picture. So, the researchers
hired an artist to fill in the swath with buildings, pavement and water in
the spirit of Escher.
Starting with this completed picture, Dr. de Smit and Mr. Batenburg then
used their computer program in a different way, to apply Dr. Lenstra's
formula for generating the distortion.
Finally, they achieved their goal: a completed, idealized version of
Escher's "Print Gallery."
In the center of the mathematician's version, the mysterious blank patch is
filled with another, smaller copy of the distorted quayside scene, turned
almost upside-down. Within that is a still smaller copy of the scene, and so
on, with the remaining infinity of tiny copies disappearing into the center.
Since Escher's distortion was not perfectly conformal, the mathematician's
rendition differs slightly from his in other ways as well. Away from the
center, for example, the lines of some of the buildings curve the opposite
The researchers also used their program to create variations on Escher's
idea: one in which the center bulges in the opposite direction, and even an
animated version that corkscrews outward as the viewer seemingly falls into
the center. After a recent talk Dr. Lenstra gave at Berkeley, the audience
remained seated for several minutes, mesmerized by the spiraling scene.
While Dr. Lenstra has solved the mystery of the blank patch and more, one
question remains. Did Escher know what belonged in the center and choose not
to represent it, or did he leave it blank because he didn't know what to put
As a man of science, Dr. Lenstra said he found it impossible to put himself
inside Escher's mind. "I find it most useful to identify Escher with
nature," he said, "and myself with a physicist that tries to model nature."
Mr. de Rijk, now in his 70's, said he believed Escher knew his picture could
continue toward the center, but did not understand precisely what should go
there. "He would be astonished to experience that his print was still much
more interesting than was his intention," Mr. de Rijk said. He added that
while he knew of another effort to fill in Escher's picture, it was not
based on an understanding of the mathematics behind it.
"He was always interested when somebody used his prints as a base for
further study and applications," Mr. de Rijk said. "When they were too
mathematical, he didn't understand them, but he was always proud when
mathematicians did something with his work."