An interactive display of the mathematics behind Escher's Print Gallery

This applet is meant to complement the information found at the University of Leiden on Escher and the Droste effect.
(Requires Java 1.4 or higher.)

Below is a representation of Esher's Print Gallery. In understanding the map used to go from the normal "flat" perspective to the curved one shown here one may click on a point on one side to see where the corresponding point lies in the other world. Try dragging the red dot along the frame or window pane dividers. What happens as you go around the frame multiple times? Note that there are multiple copies of this single point. Why is that?

To help facilitate visualizing this relation, we have colorized the person in the images below. Try zooming in or out by right (or command-) clicking on the image and you can see that this image does indeed contain a full copy of itself.

You may also view the grid that is used to do the transformation. Try tracing along lines in the grid. (This may take a moment to generate the first time.)

Although this lets us see where the map sends specific points, it isn't very helpful in explaining how the periodicity works. We note that the image contains scaled and rotated copies of itself. Recall that logarithmic function on the complex plane turns scaling into horizontal translation and rotation into vertical translation. We not take both images as mapped under the logarithmic function and map them above their respective images. Play around with the red dot as before. Note that although this logarithmic image looks even more twisted, it is still conformal--in other words although the global distortion is great locally things still have about the same shape.

It may be useful to see what horizontal and vertical movement in the logarithmic plane corresponds to in the other planes.

There is something else that becomes clearer in this representation. What relation do you seen in the two top pictures?

The logarithm of the "escherized" picture is no more than a rotation and scaling of the logarithm of the "flat" picture! In the complex plane, this is the equivalent to multiplication by a constant. This leads us to the function mapping from the flat world to that of Print Gallery: take the logarithm, rotate and scale, and then drop back down with the exponential function (the inverse or opposite of the logarithm). Although the rotation and scaling must preserve the vertical periosity of 2pii, many different choices of rotation are possible and all yield different pictures.

But what about the droste effect? Try zooming out on the logarithmic pictures in the applet above to see if you can figure out where the smaller and smaller copies come from.

If you zoom out enough on the logarithmic pictures, you see that it is actually a grid of many pictures placed next to each other as if tiling a floor with square tiles. The ones on the right are all the "outer" copies, and the ones on the left are all the "inner" copies. As this tiling goes on forever, this is what yields the infinite self-containment of each of the lower pictures.

Although this applet only shows the process as applied to the Print Gallery, any self-containing image is possible, say, a window on your computer screen containing the entire screen. Although security technicalities prevent this from being done in an applet, you can look at an example or download a jar file that does these transformations here.

Robert Bradshaw
MSRI 2003 Summer Graduate Program, Mathematical Graphics
Reed College, Portland, Oregon