Being the nerdy geeky sort I had a special fondness while growing up for the paradoxical nature, mathematical precision and infinite recursions present in MC Eschers lithographs, but recently I’ve grown to have an even more profound respect and admiration for his work from a fractal/loop perspective.

It started off with reading the article The Mathematical Structure of Escher’s Print Gallery by B. de Smit and H. W. Lenstra Jr. in which they analyze the print gallery as you see above where a young man is standing in a gallery looking at a painting that against all logic warps out and encompasses the gallery in which he is standing. There is a white spot in the center where the drawing stops. Why? Was it too complex to draw? Is there a mathematical paradox hinted at? The analysis of the original paper is pretty heady for those without a mathematics background and I hope the mathematicians don’t think of this article as blasphemy because of my attempt to make it simpler to grasp with my flimsy grasp on math. For those who have a solid grasp on math please don’t read my page (you’ll make me feel ashamed) and read the actual article instead.

The effect known as the Droste effect is named after a well known dutch chocolate in which the graphic on the box depicts a lady standing holding a tray with a box of Droste chocolate depicting a lady standing holding a tray with Droste chocolate… ad infinitum. This is known as recursion but how does this apply to Eschers piece?

The Print Gallery is actually a loop which contains a smaller version of itself, you start with a man in an art gallery looking at a print. This print hes looking at happens to contain the gallery in which he is standing but only 256 times smaller. Below is a video illustrating this part.

 

Why 256 times smaller?  First some math, but dont worry, it’s simple: 2 to the power of 8 = 256 (2x2x2x2x2x2x2x2=256) This means if you zoom in at a factor of 2X – 8 times to a certain point in to the image where there is a copy of the image 256 times smaller you will eventually end up back where you started. A seamless loop. As quoted from Bruno Ernst’s book  The Magic Mirror of MC Escher Escher started "from the idea that it must…be possible to make an annular (ringlike) bulge," "a cyclic expansion…without beginning or end." The realization of this idea caused him "some almighty headaches." At first, he "tried to put his idea into practice using straight lines [left], but then he intuitively adopted the curved lines shown in the image on the right. In this way the original small squares could better retain their square appearance."

  

What does this mean in laymans terms? Basically it means the small picture seamless warps into the larger one by using a grid to transform the image. Even cooler is that it creates a never ending seamless loop since it’s done with a zoom that ends up back where it started if you zoom in 256 times.

Below is one of the four original  studies for The Print shop.

Below are 8 images each one zooming into the previous picture by a factor of 2 (2x2x2x2x2x2x2x2=256) The rough areas are the white spot in the center of the drawing. By the time you get to the end you can see you are back where you started (the next zoom level is #1)

B. de Smit and H. W. Lenstra Jr of Leiden University in the Netherlands together with a team applied a 4 step process of reverse engineering this piece of Escher and reconstructing it

1) Decontrstruct – apply the reverse of the transformations to arrive at the original unaltered image.

2) Redraw the image

3) Recolor the image and apply the first logarithmic tiling transformation

4) apply the final transformation and the center is filled in!

While this is really cool and all I think what is the most amazing part of this is that different parameters can be used to create a number of variations of pieces escher might have come up with had he used different values. This was done by using a different transformation grid in the final step

By doing this the team created a large number of seamlessly looping zooming animations of what eschers print shop might have been – This page has all of them The animation below is the opposite values of what escher chose so it is in essence the yin to the print shops yang.

 One of my favorite visual mathematicians, Jos Leys has an amazing page explaining how to create the droste effect and use it in practical applications which will come in handy for those of us nerdy enough to learn how to put it into processing and code.

This page will be updated as I do more research on the subject.

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