Math is everywhere! – fractals on the Franz Josef glacier

One of the stops on our New Zealand adventure was at the Franz Josef glacier on the West coast. We went on the full day hike which gave us plenty of time to explore the various ice formations on the glacier under the careful eye of our guide. Along the way up the glacier, I took the following series of pictures:

All of these were taken on the way up the glacier. Can you tell in what order I took them? If you’re like my students (and a few others I have shown these to), you will likely be incorrect.

I realized as I was walking that this might be because of the idea of self-similarity, a characteristic of fractals in which small parts are similar to the whole. When I showed this set of pictures to my geometry class, I then showed them a great video video zooming in on the Mandelbrot fractal to show them what this meant.

The formations in the ice and the sizes of the rocks broken off my the glacier contributed to the overall effect. Here is another shot looking down the face of the glacier in which you can see four different groups of people for a size comparison:

The cooler thing than seeing this in the first place was discovering that it’s a real phenomenon! There are some papers out there discussing the fact that the grain size distribution of glacial till (the soil, sand, and rocks broken off by the glacier) is consistent throughout a striking range of magnitudes. The following chart is from Principles of Glacier Mechanics by Roger Leb. Hooke:

In case you are interested in exploring these pictures more, here are the full size ones in the same A-B-C-D order from above:

Oh, and in case you are wondering, the correct order is B-C-A-D.