This project is based on this one: Escher and the Droste effect.

Sometime in 2002, Paul-Olivier and I went to a lecture given by Hendrik Lenstra (Universiteit Leiden and previously UC Berkeley) about his investigations into Escher's Printgallery at the Stanford math department. He initiated the project (with, and currently run by Bart de Smit, also Universiteit Leiden) to recreate Escher's image and 'fill in the hole'. See their website for their results, and the theory behind both what they and we did. Namely, the obvious extension to their work was to do the same with photographs rather than line drawings. The same twisting technique works with any image that contains a 'picture frame' within it that the whole picture can be put inside of. When we heard that Lenstra would give a public lecture at Stanford on May 12 we decided to try to make a photograph version similar to Escher's image (but with a Stanford theme) for that talk.

In order to get the necessary difference in scales, we needed to be able to take multiple pictures at different levels of zoom. We also wanted interesting, recognisable scenery. We went for photographs from the top of Hoover Tower looking down into the Quad near Memorial Church. This map shows Main Quad to the left and Hoover Tower to the right. Somehow we managed to get an intersection between everyone's schedules and took the photographs on March 27th 2003, when Hendrik Lenstra was in Palo Alto for a conference at AIM.

Downstairs, earlier that morning.

Our photographer, Ernie also did some sharpening of the raw images, and 'cloned out' some people we didn't want in the shot. It helped that we took the photos during the Easter holiday, outside of termtime, and the weather was good to us.

Bart de Smit suggested we use a plugin for the graphics program GIMP, called MathMap. We did all the calculations to the log picture and back again using this package. It has a few quirks but generally works well.

The view from the top. If you look very closely you can just make out the stand with the whiteboard we used as a picture frame.

Ernie Buckels, our photographer. He used his 6 mega pixel Canon D60, and we rented a large zoom lens.

These are some of the images we used, ready to be converted.

The highest zoom factor pictures aren't really as good resolution as we'd want. This was only a real problem with Lenstra's face, so we cheated. Paul took another photograph of Lenstra the next day, trying to get the angle and lighting similar (standing on a table, less than 10 meters away). After some Photoshop work we get the highest zoom image below.

This image is the result of combining half a dozen pictures of different zooms after taking logs, using MathMap to make the log images and Photoshop to combine them. All the following images come from this image. The maps to here are not conformal, this gets corrected later on.

The end result. We chose the setting to try to have some features for the eye to follow - the edge of the garden island, the tree, Memorial Church all work well.

Twisting the other way... there are almost a ZxZ worth of possible ways to put the picture back together again.

The zoom ratio turned out to be approx 224.5, very close to Escher's 256.
Untwisted movie:
High quality (14100kb)
Low Quality (2000kb)

These movies made by Bart de Smit from our log image.
Twisted movie:
High quality (mpeg, 10600kb)
High quality (mov, 7700kb, thanks to Dick Palais)
Low Quality (1500kb)


We had a lot of help from many people, but in particular we especially thank: