Peter Doyle's Amazing Spirals

The previous section was concerned with transformations obtained by composing a series of inversions in circles. These transformations are now known as Möbius transformations, and thanks to Poincaré, we know they have a particularly simple algebraic formula (which I won't tell you). Instead, I will list the basic types of Möbius transformations that can occur

• Reflections in lines and inversions in circles
• Translations by a fixed distance in a fixed direction
• Rotations by a given angle around a given center
• Glide reflections: translations followed by reflection in the line of translation.
• Dilations by a given scale factor
• Elliptic transformations: inversion in a circle followed by a rotation (not necessarily the center of the circle) followed by inversion in the original circle.
• Parabolic transformations (same thing for translations)
• Glide parabolics (same thing for glide reflections)
• Hyperbolic (same thing for dilations)
• Loxodromic or spiral transformations (combination of a dilation relative to a given point and a rotation around that point, possibly conjugated by an inversion.)

Some of these transformations may be familiar. We shall limit ourselves to an illustration of the last and most beautiful transformation, the spiralling one, with an intrepid walker in the movie loxowalkB.mpg. We have also put in a few `guardrails' for his safety.

Here we are already showing a more general idea of `tiling'; where the tile is Dr. Stickler and the group is generated by a single loxodromic transformation.

In general, the study of kleinian groups begins with the selection of the generating transformations and then examines their collective action on geometric objects. Peter Doyle discovered (in a slightly different form) an amazing symmetry of loxodromic elements acting on a simple configuration of tangent circles. We shall phrase it in the form of a visual theorem in Figure 23.

Figure 23: Start with a Möbius transform $a$ that maps a pair of tangent circles to a second pair, which are also tangent to the first as shown. Then there is a unique Möbius transform $b$ that pairs them in the other possible way, and $ab=ba$.

The theorem means that any four such circles that comes with one pairing transformation $a$ automatically has another $b$. The group generated by these transformations can be applied to the original starting circles to develop a beautiful packing of circles in the plane. If the four circles are all the same size, we simply get the usual hexagonal packing of the plane. But if the circles are of different sizes, we get something a lot more beautiful: a Doyle Spiral. The transformations $a$ and $b$ will be loxodromic spiralling element whirling the circles all around and through each other.

In general, the circles will overlap; but if we solve delicate algebraic equations for certain special crystalline algebraic numbers, we will achieve what Ken Stephenson dubbed a coherent Doyle spiral (see [#!BeardonDubejkoetal1994!#]). In fact, the example in the theorem is just such a coherent packing, as we see by applying $a$ repeatedly in Figure 24. There is a hidden structure in that 16 applications of $a$ is the same as one application of $b$. Thus, the group presentation has relations $ab=ba$ and . Beardon, Stephenson, and Dubejko proved there is exactly one such Doyle spiral for any similar such relation .

Figure 24: The $(1,16)$ coherent spiral Doyle packing with group presentation .

This is a relatively uncomplicated group, and yet if we have the technique of finite state automata it's convenient for enumerating precisely the group elements. When we do so, the complete Doyle spiral looks like Figure 25. Here we change colors each time we apply $a$, and we leave the color alone when we apply $b$. The color emphasizes both spiral transformations.

Figure 25: The $(1,16)$ coherent spiral Doyle packing.

It is a delicate calculation to find the transformations for a given integer pair that produce the Doyle Spiral. The next figures show the (17,43) spiral and the (10,101) spiral.

Figure 26: The $(17,43)$ coherent spiral Doyle packing.

Figure 27: The $(10,101)$ coherent spiral Doyle packing.

If you've seen some color visualizations of functions of a complex variable, these may seem familiar. The spirit of using circle packings to approximate conformal functions is explained in Ken Stephenson's book Introduction to Circle Packing, and indeed the Doyle packings converge to a certain well known function. But that theorem is just a little off in the future.

The idea of interlocking families of spirals appears in nature as the phenomenon of `phyllotaxis', visible in pine cones, sunflower seed arrangements, shells, etc., although in nature the integers are almost always consecutive Fibonacci numbers.

We'll end this section with some of Jos Leys' striking visualizations of the coherent Doyle packings.

Figure 28: Doyle packings rendered by Jos Leys in Ultrafractal.