The point of showing people things like the Hilbert curve is probably to give a nice example of recursion. However, it's a good opportunity to do some proselytizing for studying mathematics.

For instance, anyone who has ever taught the Hilbert curve example has probably told the class that the process generates a "space-filling curve" (and probably left it there--that's what I've done when I've used this example). If you're lucky you'll have someone in the class (every class should have one) who asks, "But what practical use is it?" (This same person probably asked you that question when you showed the "Towers of Hanoi" puzzle, the 8-Queens problem, the Knight's tour problem, etc.). Now I know that I can hit them (metaphorically, of course) with a practical application of space-filling curves: image retrieval.

"Using Hilbert Curve in Image Storing and Retrieving" by Zhexuan Song and Nick Roussopoulos.

Here's my understanding of the basic idea: extremely large images (images too large to fit in memory all at once) must sometimes be accessed to retrieve subimages. To keep things simple, assume we need to retrieve a rectangular subarray of pixels from the image. The "obvious" way to store the image pixels is in row-major order, which introduces discontinuities at the edges of the image (two consecutive pixels in storage might represent two very widely separated pixels in the actual image if those pixels straddle a "line break" in the image). If, instead, the pixels are stored in a pattern that imitates the meandering of a Hilbert curve throughout the space of pixels, we have always have nearby pixels in storage corresponding to nearby pixels in the image. This, in turn, reduces the number of non-contiguous memory accesses needed to retrieve the rectangle, reducing retrieval time.

I haven't pursued the references or even carefully read the paper; I probably got the details wrong. Probably this idea is older than the paper cited above and the authors are merely extending it. (Project for a future day.)

Would someone taking only the minimum mathematics needed to get by be likely to stumble across something like this (unless, of course, they had an instructor who liked to show pretty pictures to illustrate recursion)?

There's a list of links about applications of Hilbert curves to data retrieval at http://citeseer.nj.nec.com/338420.html.

Please send suggestions to rroos@allegheny.edu