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→A Geometric Model of the Free Group
===A Geometric Model of the Free Group===
===A Geometric Model of the Free Group===
You may be unimpressed by this, since the free group isn't really a geometric object. If you recall, we said earlier that the free group was important for two reasons, and one of them was that it had a paradoxical decomposition. We now discuss the second reason.
Consider the unit sphere in ''xyz'' space. Pick a point on the sphere ''P'', and let <math>a \ </math> mean rotation of this point counterclockwise around the ''x''-axis by <math>\theta \ </math>, and <math>b \ </math> mean rotation around the ''z''-axis counterclockwise by that same angle <math>\theta \ </math>. If we define <math>\theta \ </math> so that <math>\cos(\theta) = 1/3 \ </math>, it turns out that this is precisely the free group <math>F_2 \ </math> if we define <math>ab \ </math> to mean first doing one rotation, then the other, and <math>a',b' \ </math> to be clockwise rotations respectively.
The isomorphism between this group, with <math>\theta = \arccos(1/3) \ </math> and so on, is not easy, and requires some hard linear algebra. It will not be given here.
From now on, we will use <math>\mathbb{F} \ </math> to refer to this group of rotations on the sphere.
===The Axiom of Choice===
===The Axiom of Choice===