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  "body": "\n## Statement\n\nAssume ZFC, that is, ZF with the [[axiom of choice|0HTGB13FE2]].\nLet $B = \\set{x \\in \\R^3 : \\norm{x} \\le 1}$ be the closed unit ball in [[Euclidean space|0XPR1E85W0]] $\\R^3$, and let $B'$ be a translate of $B$ that is disjoint from $B$.\nThen there are an integer $n \\ge 1$ and a partition of $B$ into disjoint sets $A_1, \\ldots, A_n$ with the following property.\nThere are [[isometries|0878DN3WCS]] $g_1, \\ldots, g_n$ of $\\R^3$ such that $g_1(A_1), \\ldots, g_n(A_n)$ are pairwise disjoint with union $B \\cup B'$.\n\n## Remarks\n\nThe proof starts with two rotations that generate a [[free group|020Z5SEN27]] $F_2$ of rank $2$ inside the [[special orthogonal group|0CGYNDCWRR]] $SO(3)$.\nApart from the identity, the group $F_2$ splits into four pieces.\nLeft multiplication by the generators turns two of them into a copy of $F_2$, and the other two into another copy.\nThe axiom of choice picks one point from each orbit of $F_2$ on the sphere.\nThis transfers the splitting from the group to the sphere, outside a countable set of fixed points.\nRadial segments carry it to the ball minus its centre, and the centre and the countable set are absorbed by a further rotation argument.\nThe pieces cannot all be Lebesgue measurable, since isometries preserve volume.\nFive pieces suffice, and no fewer than five work.\nThe analogue fails in $\\R^1$ and $\\R^2$, because their isometry groups are [[amenable|PG8Z6F6DV9]].\nSo there are finitely additive isometry-invariant measures on all bounded subsets that give the unit ball a positive value.\nA stronger form holds in $\\R^n$ for each $n \\ge 3$: any two bounded subsets with nonempty interior can be cut into finitely many pieces that match by isometries.\n\n## Explanations\n\n### L1 human\n\nYou can cut a solid ball into a few pieces.\nMoving the pieces only by rotations and translations, you can put them back together as two balls of the same size as the first.\nThis seems to break the conservation of volume.\nThere is no contradiction, because the pieces are so scattered that they have no volume at all.\nSuch pieces can only be shown to exist with the axiom of choice; nobody can draw them.\n\n### L2 human\n\nThe theorem shows that there is no finitely additive measure on all subsets of $\\R^3$ that is invariant under isometries and gives the unit ball a positive finite value.\nThis is one reason why measure theory works with a restricted family of measurable sets.\nThe key algebraic input is a free subgroup of $SO(3)$.\nThe general idea behind the paradox is the failure of amenability.\nA group that contains a free subgroup of rank $2$ is not amenable, and its actions can be paradoxical.\n",
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