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David Mumford, Caroline Series, David Wright.
Indra's Pearls: the vision of Felix Klein.
CUP. 2002

rating : 3.5 : worth reading
review : 1 December 2010

Fractal fun with four circles.

[figure 8.1]

This is one long investigation into the fascinating geometric patterns that occur when a group of transformations called Möbius maps are iteratively applied to some initial special starting circles (known as Schottky disks). The underlying mathematics is explained, starting simple, and gradually building up (see for example figure 8.1 reproduced on the right; click to embiggen). It all started with Felix Klein's extension of the concept of geometry:

pp1-2. Klein proposed viewing geometry as 'the study of the properties of a space which are invariant under a given group of transformations'. To study geometry, he said, one needed not only objects (triangles, circles, icosahedra, or much wilder things like the fractal pictures in this book), but also movements. In the classical Euclidean regime which had been around for over two millennia, these movements had always been rigid motions: pick up a figure and place an identical copy down in a new place. Klein's radical idea was that other movements, which might stretch or twist the objects quite drastically, could be thought of geometrical movements too.

What's different, and nice, here is that algorithms and pseudocode for drawing the figures are also given and explained, including their limitations and when they fail. I don't have the time to play with these, but I wish I did, particularly to watch the serpentine tracing out of some of the fractals as continuous curves. Given the greater compute power available today than when this was written, I expect quite spectacular results can be achieved quite readily.

The progression is typical of mathematics: start with a concept, then generalise, then generalise again. The fractals start out relatively simple, but by the end are quite stunning in their intricate complexity, while clearly all being members of a family. And along the way, some surprising relationships with other fields appear, mainly because [p62] Möbius maps ... do for the Riemann sphere what the affine maps ... do for the complex plane (and we are used to the effect of iterating affine maps from Barnsley's Iterated Function Systems). In particular, a connection between the (convergent) limit sets generated here and the (divergent) behaviour of the chaotic IFS algorithm is demonstrated:

p133. Independence of starting point is actually the flip side of the famous butterfly effect in chaos theory ... The point of this analogy is to explain the feature of chaos called sensitive dependence on initial conditions (alias starting points). In our context, we can explain the butterfly effect like this. If P and Q are nearby limit points, then their infinite words begin with a long common string W = aBBaBABABaaa say. Now apply in order the maps A, b, b, steadily unravelling W. The pairs of points A(P), A(Q); bA(P), bA(Q); and so on will get steadily further apart (because their common strings are shorter and shorter), until eventually we reach the pair W-1(P) and W-1(Q) which must be in different initial Schottky disks ..., because they have no common string at all. In other words, no matter how close P is to Q, if we run the above process for long enough it will be completely unpredictable where we end up. Limit sets, it seems, are closely connected to chaos.

p140. the limit set is what is sometimes known as a 'strange attractor' for the group's dynamics, because as we continue to iterate our maps, then ... the orbit of any point will get sucked in ever closer to the limit set. A point in the limit set, on the other hand, hops around chaotically under the dynamics of the group

Not easy reading, but not dry, and certainly worth the effort, with plenty of projects to try out.