The warped space of M C Escher

Grid for Escher's Print Gallery (Lenstra and de Smit)

The Dutch graphic artist M C Escher is famous for his impossible objects, infinitely repeating patterns and surreal worlds.

Thumbnail of Escher's Print GalleryIn my opinion, his 1956 lithograph “Print Gallery” is his greatest masterpiece, combining an infinitely repeating image with a precise mathematical warping of space to create a truly mind-bending piece of surreal art.

In the bottom-left corner, a man is standing in an art gallery, looking at a painting of the Maltese city of Senglea.  Following the picture clockwise to the top-right corner, the painting expands, revealing an increasing amount of detail in the city’s buildings.  Continuing round to the bottom-right corner, we see that one of the buildings is an art gallery.  Returning to the bottom-left corner, we discover that this is the original art gallery in which the man is standing!

Thumbnail of Escher's woodcut Senglea, MaltaCompare this with Escher’s 1935 woodcut of Senglea, and observe the similarity of the ship in the harbour and the buildings in the top-left corner.  According to this website, Escher did indeed use this woodcut as the basis for his “Print Gallery”.  You may also notice that the gallery contains references to some of Escher’s other works, including Three Spheres I, Horseman, Sky and Water I and Rind.

Thumbnail of Escher's original Print Gallery gridThe foundation of Escher’s image is a warped grid, which he described as “a cyclic expansion or bulge without beginning or end”. [1]  His original grid can be seen here.  “Although fascinated by visual mathematical concepts, Escher had only a high school education in mathematics and little interest in its formalities.” [1]  It’s believed that Escher constructed his grid through visual experimentation and successive improvements.

Lenstra and de Smit’s project

In April 2003, Hendrik Lenstra and Bart de Smit (two mathematicians at Leiden University) published an article in which they analysed the mathematics of the image, and created their own reconstruction of it, assisted by a small group of artists and programmers.  They also set up an excellent website to present their work.

Lenstra and de Smit’s team determined the exact mathematical structure of Escher’s grid, and produced their own version of it (see page 5 of their article), which is shown at the top of this post.  Steve Witham has made another rendering of the grid, which is shown below:

Grid for Escher's Print Gallery (Witham)

There are two important properties of this grid which Escher designed.  Firstly, all the lines cross at right angles (90°), ensuring that when you look closely at any small part of the picture, it looks normal and undistorted (although it may be rotated).  In mathematical terms, the grid is said to be conformal.

Secondly, as you travel clockwise around the grid, the squares continuously expand.  Moving from one quadrant to the next, the squares expand by a factor of 4.  This is highlighted in the grid below:

Grid for Escher's Print Gallery - Warped squares

The small green square in the top-left corner becomes 4 times larger in the top-right corner.  Similarly, the small pink square in the top-right corner becomes 4 times larger in the bottom-right corner, and so on.

For one complete revolution travelling clockwise through the four quadrants of the grid, the squares expand by a factor of 4×4×4×4 = 256.

Having made the grid, Lenstra and de Smit’s team then created their own reconstruction of “Print Gallery”:

Lenstra and de Smit's version of Escher's Print Gallery

By looking at the structure of the grid, they were able to answer the question: “What details are hidden by the white hole at the centre of Escher’s original image?”

Centre of Lenstra and de Smit's version of Escher's Print Gallery

In the illustration above, the blue circle corresponds to Escher’s white hole.  Inside the circle is an identical copy of the original image, “rotated clockwise by 157.6 degrees and scaled down by a factor of 22.58”. [2]  This is indicated by the small rotated square removed from the blue circle.

In order to create a complete picture, Lenstra and de Smit’s team had to invent suitable details to fill the blue area.  They could then untwist the picture to reveal the original image:

Escher's Print Gallery untwisted

The picture contains a copy of itself, 256 times smaller.  In the illustration below, the four untwisted images show that if you continually zoom into the picture (travelling clockwise around the grid), you eventually arrive back where you started:

Four untwisted views of Escher's Print Gallery

Each untwisted image is magnified 4 times more than the previous one, so one complete revolution of the grid will zoom in by a factor of 256.

A box of Droste cocoa powderLenstra and de Smit’s team have created some very nice animations which continually zoom into the picture (both twisted and untwisted).

When a picture contains a smaller identical copy of itself like this, the effect is known as the Droste effect.  “The effect is named after the image on the tins and boxes of Droste cocoa powder, one of the main Dutch brands, which displayed a nurse carrying a serving tray with a cup of hot chocolate and a box with the same image.” [3]

Thumbnail of photo of Bart de Smit's lectureIn April 2011, de Smit gave a lecture in Washington DC to the Mathematical Association of America, in which he spoke in detail about his “Print Gallery” project, as well as discussing Escher’s interest in mathematics, and other examples of the Droste effect.  Details of the lecture are available here, as well as a complete MP3 recording which is well worth listening to.

Derivative works

Thumbnail of the Stanford Print GalleryIn 2003, Henry Segerman and Paul-Olivier Dehaye created a photographic equivalent of “Print Gallery” at Stanford University, featuring Lenstra standing next to a picture frame.  The frame contains an aerial photo of the Stanford campus, and following the image around in a clockwise direction, the view warps and expands in exactly the same manner as “Print Gallery”, until it arrives back at Lenstra again.

Thumbnail of a transformed image by Jos LeysIn March 2006, Jos Leys wrote an article in which he showed “how Lenstra’s method can be applied more generally for the transformation of images, and the generation of endless zoom animations.”  After describing the mathematics of his image transformation, he then implemented it in Ultra Fractal, and used it to generate some example images and an animation.  He has created a gallery of his work, and a user guide for creating the Droste effect in Ultra Fractal.

Thumbnail of Lloyd Burchill's OrganolepticLater in 2006, Lloyd Burchill used Lenstra and de Smit’s mathematics to create an image of an endlessly-repeating nose and mouth, and posted it on Flickr.  This image would turn out to be a very powerful inspiration for others, triggering a chain reaction of amazing creativity across the internet, eventually leading to the creation of a plug-in for Adobe Photoshop / After Effects and an iPhone app!

Thumbnail of Sebastien Perez-Duarte's Spiral PianoLloyd’s image inspired Sébastien Pérez-Duarte to start creating his own images in this style, using GIMP (an image editor) and MathMap (an image-processing plug-in for GIMP).  This spiral piano was his first creation, and he has more similar images in this set.  He experimented with images whose infinitely-repeating area is either rectangular or circular, and he also experimented with different values for the parameters which control the image transformation.

Thumbnail of David Swart's First Droste Post with the MostIn August 2006, David Swart, inspired by Sébastien’s work, started creating his own images in this style, writing his own image-transformation code in CThis was his first creation, and he has more similar images in this set.  He has also shared his example of a C/C++ implementation of the Droste effect here.

Thumbnail of Alexandre Duret-Lutz's Neverending StoryIn September 2006, Alexandre Duret-Lutz, inspired by this image of Sébastien’s, started creating his own images in this style, also using GIMP and MathMap.  This endless bookcase was his first creation, and he has more similar images in this set.  Like Sébastien, he has also experimented with different values for the transformation parameters, and has created this overview using a chessboard as the source image (similar to Lenstra and de Smit’s grid of Escher-like images).

To be continued …

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