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→Removing Points with Rigid Body Motions
Make sure you understand the content of <math>W \ </math>. If the first few values of <math>A = \left\{{6,2\sqrt{3},\frac{e}{\pi}, ... }\right\} \ </math>, then <math>W \ </math> contains, among other things, <math>6-2\sqrt{3}, (2\sqrt{3}-\frac{e}{\pi})/653, \ </math> etc. Now, since <math>W \ </math> is countable, and the angles of <math>\mathbb{S}^1 \ </math> are uncountable, there is an angle in <math>\mathbb{S}^1 \ </math> which is not in <math>W \ </math>. Pick one such angle, and call it <math>\theta \ </math>. This is our magic rotation angle.
Make sure you understand the content of <math>W \ </math>. If the first few values of <math>A = \left\{{6,2\sqrt{3},\frac{e}{\pi}, ... }\right\} \ </math>, then <math>W \ </math> contains, among other things, <math>6-2\sqrt{3}, (2\sqrt{3}-\frac{e}{\pi})/653, \ </math> etc. Now, since <math>W \ </math> is countable, and the angles of <math>\mathbb{S}^1 \ </math> are uncountable, there is an angle in <math>\mathbb{S}^1 \ </math> which is not in <math>W \ </math>. Pick one such angle, and call it <math>\theta \ </math>. This is our magic rotation angle.
Note that if we rotate <math>A \ </math> by an angle not in <math>W \ </math>, it is guaranteed to not carry any point in <math>A \ </math> to any other point in <math>A \ </math> because of our clever definition. Rotation by <math>\theta \ </math>, for example, cannot carry <math>a_7 \ </math> to <math>a_11 \ </math>, because then <math>\theta = a_{11}-a_7 \in W \ </math>, but we specifically said <math>\theta \notin W \ </math>. Furthermore, 12 rotations of <math>\theta \ </math> could never carry <math>a_7 \ </math> to <math>a_{11} \ </math>, because then <math>\theta = \frac{a_{11}-a_7}{12} \ </math>, but that's in <math>W \ </math> too, since we constructed it to contain not just all the differences of angles in <math>A \ </math>, but all the fractions of those differences as well.
So we construct the infinite super-necklace <math>T = A \cup R_\theta(A) \cup R_{2\theta}(A) ... \ </math> and finally, observe that <math>R_\theta(T) = R_\theta(A) \cup R_{2\theta}(A) \cup ... = T-A \ </math>.
We form the partition <math>\mathbb{S}^1 = T \cup (\mathbb{S}^1-T) \ </math> and then note <math>R_\theta(T) \cup (\mathbb{S}^1 - T) = \mathbb{S}^1 - A \ </math> and we have shown that <math>\mathbb{S}^1 \cong \mathbb{S}^1 - A \ </math>
===The Free Group===
===The Free Group===