I really enjoyed Escher and the Droste effect project at Universiteit Leiden. They recreate Escher’s Prentententoonstelling.
The general technique is to start with a picture that contains itself inside it (the Droste effect). In this case you start with a man looking at a print of a town with a print gallery where a man is looking at a print of a town with a print gallery where a man …. This defines an image at every point except one fixed point. Consider the image as drawn on the complex plane. Pull this image back through a the exponential map. The resulting image is tiled vertically because the exponential map is periodic in that direction, but the image is also tiled horizontally because the original image repeats itself on a smaller scale. The smaller images end up on the left of pulled back image.
The interesting bit is to now rotate the pullback image by 45°. The rotated image is still tiled horizontally and vertically. You can imagine rotating a square tiled floor 45°, and the diamonds are still tiled vertically and horizontally. Now push this image back through the exponential map. This is sound because the rotated image is still tiled vertically. The result causes the angle to twist as the scale changes. All the transformations involved are conformal, so angles are preserved.
Escher developed his grid by hand, without this mathematical derivation. He did a remarkably good job, although his grid isn’t exactly the same.
I’m told that if you actually go to Universiteit Leiden then there is a video setup that creates this effect in real time, so you can view yourself in your own “Video Gallery”.
I actually prefer the image rotating the other way.
