Find the area!

Geometry Level 5

\[S=\{(x,y,z) \in \mathbb{R}^3: x^2+y^2+z^2=1\}\]

Let \(S\) be the surface of a unit sphere centered at the origin.

Let \(F\) be a subset of \(S\). Suppose that there exist three subsets \(F_1,F_2,F_3 \) of \(S\), each of which is obtained by rigidly rotating the set \(F\) around the origin such that:

  1. These three sets are pairwise disjoint, i.e., \[F_i\cap F_j = \phi, i\neq j\]
  2. Their union spans the entire surface of the sphere, i.e.,
    \[F_1\cup F_2 \cup F_3=S\]

Find the area of any such set \(F\) satisfying the above conditions.

P.S. Rotation is defined in the usual way: \[F_i=\{(x,y,z): \exists (a,b,c) \in F \text{ s.t. } (x,y,z)'=Q_i (a,b,c)'\} \] for some orthonormal matrix \(Q_i,i=1,2,3\).


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