Volume 9, Issue 3
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The Modular Group
The code that constructs the illustrations in this notebook lies in the package file ModularGroup.m.
Construction of the animations. The action of modular group elements map the tesselation onto itself, that is, before and after pictures of the action will appear identical. The intermediate frames of the animations, which imply continuous motion rather than the abrupt jump from one configuration to another, are produced as "fractional powers" of the group elements. They accomplish a portion of the same action on fundamental regions. The facility to produce these transformations is due to the fact that in order to produce frames for the animations, it is sufficient to compute compositions of extremely simple functions such as
If, for example, an animation for the action of a function were to have six frames from beginning to end, the points for each frame could be computed as follows.
where transforms the first fixed point of to the point at and the second fixed point, if there is one, to 0. transforms the points at and 0 to the fixed points of g, and is one of the following: , , or ; . A specific example is given in the following section.
Conjugation and Canonical Forms
We may view the conjugation we speak of in this section as algebraic only if we broaden the group within which we see the conjugation taking place. Here we conjugate by elements outside the modular group, but still within the group of all Möbius transformations.
It turns out that every Möbius transformation, including every element of the modular group, is conjugate equivalent to a Euclidean rotation, translation, or dilation. Thus, for example, is conjugate equivalent to a Euclidean rotation of about the origin. The conjugating transformations used in our computations are those that move fixed points to the point at infinity and to the origin. If there is a single fixed point for a group element, that fixed point is moved to the point at infinity. For a methodical discussion about Möbius transformations moving the fixed points of other transformations, see [1, 98].
Figure 13. A set of fundamental regions (far left illustration) and the set's three images under repeated action by .
Figure 13 depicts the images of three fundamental regions under the iterated action by . An animation can be produced from these figures alone by using the integer 1 as input for the function g3Animation. However, a more revealing illustration is produced by using an integer larger than 1 that inserts intermediate steps for the action so that the animation can suggest a continuous motion. The key to this process is the conjugate equivalence of each modular group element to a Euclidean rotation, translation, or dilation.
If we permit conjugation by Möbius transformations not necessarily in , we can relate the 2-fixed point transformations in to "canonical" transformations--those that have fixed points at the origin and at infinity.
Figure 14. Conjugation with rotation of 120°.
The animation on the left shows the fixed point of , that we have already seen, being translated to the origin. The second fixed point of , which has remained out of our view because it lies outside of the hyperbolic plane, is simultaneously moved off to the point at infinity. Following this transformation, all circles that passed through the original fixed point become straight lines passing through the origin. A Euclidean rotation about the origin accomplishes the desired rearrangement of the regions. Finally, translating the fixed points back to their original positions maps the fundamental regions to their proper, final positions. The animation on the right shows a hyperbolic rotation about the fixed point in the upper half-plane model of the hyperbolic plane. Each frame in the right-hand animation was computed by composing the functions that are explicitly portrayed in the left-hand animation.
If we know that is an element of the modular group, then there is a Möbius transformation , such that , where is a rotation of 180°, a rotation of 120°, a right shift by units, or a dilation about the origin.
Because , the right shift by one unit, is itself an element of the modular group, the conjugation for these types of modular group elements can be accomplished within the modular group. We have seen this winding action in the previous section.
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