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  "body": "\n## Example\n\nTake three [[ellipses|0VKCCEXBJ1]] in the three coordinate planes of $\\R^3$:\n$$C_1 = \\set{(x, y, 0) : \\tfrac{x^2}{4} + y^2 = 1}, \\quad C_2 = \\set{(0, y, z) : \\tfrac{y^2}{4} + z^2 = 1}, \\quad C_3 = \\set{(x, 0, z) : x^2 + \\tfrac{z^2}{4} = 1}.$$\nThey are pairwise disjoint.\nFor instance, a common point of $C_1$ and $C_2$ has $x = 0$ and $z = 0$.\nThen $y = \\pm 1$ from $C_1$ and $y = \\pm 2$ from $C_2$, which is impossible.\nThe other pairs are similar.\nSo $C_1 \\cup C_2 \\cup C_3$ is a [[link|1A5WRCNTM3]] with $3$ components, the *Borromean rings*.\nAny two of the components form the unlink of two components.\nFor example, $C_2$ meets the plane $z = 0$ only at $(0, \\pm 2, 0)$, outside the region bounded by $C_1$.\nSo $C_1$ bounds a flat disk disjoint from $C_2$, and it can be shrunk in that disk to a tiny circle far from $C_2$.\nBut the three components together cannot be pulled apart: the link is not equivalent to the unlink of three components.\n\n## Remarks\n\nAll three pairwise [[linking numbers|SFJ329YB9F]] are $0$, so linking numbers do not detect the Borromean rings.\nTheir nontriviality is detected by Milnor's triple linking number, or by the fact that their link group is not free.\nA nontrivial link in which removing any one component leaves an unlink is called Brunnian, and the Borromean rings are the simplest example.\nThey have a diagram with $6$ crossings that is alternating.\nTheir complement in $S^3$ carries a complete hyperbolic metric of finite volume, and it can be built from two regular ideal octahedra.\nMichael Freedman and Richard Skora showed that the Borromean rings cannot be built from three round circles.\n\n## Explanations\n\n### L1 human\n\nThe Borromean rings are three rings that hold together as a group, although no two of them are linked.\nIf any one ring is cut and removed, the other two fall apart.\nThey appear in old coats of arms, among them that of the Borromeo family of Italy, which gives the name.\n",
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