The warped space of M C Escher | Michael Hogg

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The warped space of M C Escher

Posted on 30 Jul 2011 | Leave a comment

Hendrik Lenstra and Bart de Smit&#8217;s rendering of the grid on which M C Escher&#8217;s &#8220;Print Gallery&#8221; is constructed.<br>Source: The Mathematical Structure of Escher&#8217;s Print Gallery, page 5.<br>Image inverted by me.<br>" data-image-caption="" data-large-file="https://michaelhogguk.wordpress.com/wp-content/uploads/2011/07/escher_grid_1.gif?w=500" class="aligncenter size-full wp-image-273" title="Grid for Escher's Print Gallery (Lenstra and de Smit)" src="https://michaelhogguk.wordpress.com/wp-content/uploads/2011/07/escher_grid_1.gif?w=500&h=504" alt="Grid for Escher's Print Gallery (Lenstra and de Smit)" width="500" height="504" srcset="https://michaelhogguk.wordpress.com/wp-content/uploads/2011/07/escher_grid_1.gif 500w, https://michaelhogguk.wordpress.com/wp-content/uploads/2011/07/escher_grid_1.gif?w=150&h=150 150w, https://michaelhogguk.wordpress.com/wp-content/uploads/2011/07/escher_grid_1.gif?w=298&h=300 298w" sizes="(max-width: 500px) 100vw, 500px" />

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

A thumbnail of M C Escher&#8217;s &#8220;Print Gallery&#8221;.<br>" data-image-caption="" data-large-file="https://michaelhogguk.wordpress.com/wp-content/uploads/2011/07/escher_print_gallery.gif?w=120" class="alignright size-full wp-image-278" title="Thumbnail of Escher's Print Gallery" src="https://michaelhogguk.wordpress.com/wp-content/uploads/2011/07/escher_print_gallery.gif?w=500" alt="Thumbnail of Escher's Print Gallery" />In 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!

A thumbnail of M C Escher&#8217;s woodcut &#8220;Senglea, Malta&#8221;.<br>" data-image-caption="" data-large-file="https://michaelhogguk.wordpress.com/wp-content/uploads/2011/07/escher_senglea_malta.jpg?w=120" class="alignright size-full wp-image-296" title="Thumbnail of Escher's woodcut Senglea, Malta" src="https://michaelhogguk.wordpress.com/wp-content/uploads/2011/07/escher_senglea_malta.jpg?w=500" alt="Thumbnail of Escher's woodcut Senglea, Malta" />Compare 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.

A thumbnail of M C Escher&#8217;s original grid for his &#8220;Print Gallery&#8221;.<br>Full-size image: Escher and the Droste effect &#8211; Grid.<br>" data-image-caption="" data-large-file="https://michaelhogguk.wordpress.com/wp-content/uploads/2011/07/escher_original_grid.gif?w=120" loading="lazy" class="alignright size-full wp-image-305" title="Thumbnail of Escher's original Print Gallery grid" src="https://michaelhogguk.wordpress.com/wp-content/uploads/2011/07/escher_original_grid.gif?w=500" alt="Thumbnail of Escher's original Print Gallery grid" />The 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:

Steve...