Step 4
Finally, connect every point on S2 with a ray to the origin; the paradoxical decomposition of S2 then yields a paradoxical decomposition of the solid unit ball minus the point at the ball's centre (this center point needs a bit more care, see below).
N.B. This sketch glosses over some details. One has to be careful about the set of points on the sphere which happen to lie on the axis of some rotation in H. However, there are only countably many such points, and like the point at the centre of the ball, it is possible to patch the proof to account for them all (see below).
Read more about this topic: Banach–Tarski Paradox, A Sketch of The Proof
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“The machine unmakes the man. Now that the machine is perfect, the engineer is nobody. Every new step in improving the engine restricts one more act of the engineer,—unteaches him.”
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