Changes

===Theorem===

===Theorem===

All sets discussed here are point sets in three dimensional, flat (ie, normal, everyday) space.  A point set is just that - a set of points.

All sets discussed here are point sets in three dimensional, flat (i.e., normal, everyday) space.  A point set is just that - a set of points.

A '''rigid motion''' of a set is a movement of the whole set one one direction, or a rotation, or a composition of those two kinds of motion.  That is to say, the set doesn't bend or get broken up.  A rigid motion of a set <math>A \ </math> is often written <math>g(A) \ </math>.  

A '''rigid motion''' of a set is a movement of the whole set one one direction, or a rotation, or a composition of those two kinds of motion.  That is to say, the set doesn't bend or get broken up.  A rigid motion of a set <math>A \ </math> is often written <math>g(A) \ </math>.  

Let <math>a,a',b,b' \ </math> be "letters."  We will form "words" by stringing finitely many of these letters together, subject to just a few simplification rules: <math>aa' = a'a = bb' = b'b= e \ </math>, where <math>e \ </math> is the identity, shorthand for the "word" of zero letters.

Let <math>a,a',b,b' \ </math> be "letters."  We will form "words" by stringing finitely many of these letters together, subject to just a few simplification rules: <math>aa' = a'a = bb' = b'b= e \ </math>, where <math>e \ </math> is the identity, shorthand for the "word" of zero letters.

We state without proof that such words form a group under "concatenation," or stringing back to back, ie, <math>aab' * bab' = aab'bab' = aaab' \ </math>.  By using exponents, ie, <math>aaa = a^3 \ </math>, each word has a unique spelling.   

We state without proof that such words form a group under "concatenation," or stringing back to back, i.e., <math>aab' * bab' = aab'bab' = aaab' \ </math>.  By using exponents, i.e., <math>aaa = a^3 \ </math>, each word has a unique spelling.   

This is called <math>F_2 \ </math>, the free group on two generators.  Any group structured like this, in which ''a'' and ''b'' have infinite order and are independent, (ie, no combination of ''a'' and ''b'' form ''e'') is this group.  We can picture this group graphically, with ''a'' and ''b'' axes:

This is called <math>F_2 \ </math>, the free group on two generators.  Any group structured like this, in which ''a'' and ''b'' have infinite order and are independent, (i.e., no combination of ''a'' and ''b'' form ''e'') is this group.  We can picture this group graphically, with ''a'' and ''b'' axes:

[[Image:free group.jpg|center|thumb|500px|This depiction demonstrates words of up to four letters; a graphical depiction is [[fractal]] in a fairly obvious fashion.]]

[[Image:free group.jpg|center|thumb|500px|This depiction demonstrates words of up to four letters; a graphical depiction is [[fractal]] in a fairly obvious fashion.]]

Since B is countable, and we can remove any countable set of points from the sphere, remove this set: <math>\mathbb{S}^2 - B \ </math>.

Since B is countable, and we can remove any countable set of points from the sphere, remove this set: <math>\mathbb{S}^2 - B \ </math>.

Now we define an orbit equivalence relation:  for two points <math>x,y \in \mathbb{S}^2-B \ </math>, define <math>x \sim y \ </math> whenever <math>x \ </math> and <math>y \ </math> are in the same orbit, ie, <math>\exists r \in \mathbb{F}:r(x)=y \ </math>.  We state without proof that this ''is'' an equivalence relation and thus partitions <math>\mathbb{S}^2-B \ </math> into equivalence classes, which will be disjoint.  Note there will be uncountably many orbits, since each only has countably many points and the union of all orbits covers the uncountable set <math>\mathbb{S}^2 - B \ </math>.

Now we define an orbit equivalence relation:  for two points <math>x,y \in \mathbb{S}^2-B \ </math>, define <math>x \sim y \ </math> whenever <math>x \ </math> and <math>y \ </math> are in the same orbit, i.e., <math>\exists r \in \mathbb{F}:r(x)=y \ </math>.  We state without proof that this ''is'' an equivalence relation and thus partitions <math>\mathbb{S}^2-B \ </math> into equivalence classes, which will be disjoint.  Note there will be uncountably many orbits, since each only has countably many points and the union of all orbits covers the uncountable set <math>\mathbb{S}^2 - B \ </math>.

Using the axiom of choice, we pick a single point from each orbit and call this set <math>W \ </math>. Note that <math>\mathbb{F}(W) = \mathbb{S}^2 - B \ </math>.

Using the axiom of choice, we pick a single point from each orbit and call this set <math>W \ </math>. Note that <math>\mathbb{F}(W) = \mathbb{S}^2 - B \ </math>.