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→Paradoxical Decomposition of the Free Group
===Paradoxical Decomposition of the Free Group===
===Paradoxical Decomposition of the Free Group===
Why are we talking about the free group? It is important for two reasons, the first of which is that it admits a "paradoxical decomposition", if we allow such a notion to refer to the graphical representation of it above. We begin by partioning the group as <math>F_2 = U \cup D \cup L \cup R \ </math>, where the sets <math>U,D,L,R \ </math> are those described below.
[[Image:Free group decomp.jpg|right|200px|thumb|U,L, and D are within the ovals. R is what is left over.]]
Why are we talking about the free group? It is important for two reasons, the first of which is that it admits a "paradoxical decomposition", if we allow such a notion to refer to the graphical representation of it above. We begin by partioning the group as <math>F_2 = U \cup D \cup L \cup R \ </math>, where the sets <math>U,D,L,R \ </math> are those described in the image to the right.
===A Geometric Model of the Free Group===
===A Geometric Model of the Free Group===