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  "body": "\n## Statement\n\nLet $G$ be a finite simple [[planar graph|0NC9R64TXP]].\n\n1. There are closed discs $D_v$ in the plane, one for each vertex $v$ of $G$, with pairwise disjoint interiors. For distinct vertices $u$ and $v$, the discs $D_u$ and $D_v$ are tangent if and only if $u$ and $v$ are adjacent.\n2. Let $G$ be maximal planar with at least three vertices. Then such a family of discs is unique up to [[Möbius transformations|0SG74WBFA2]] and reflections in lines.\n\n## Remarks\n\nThe theorem is also called the Koebe–Andreev–Thurston theorem.\nOne proof first adds vertices to make $G$ a triangulation.\nFor a triangulation, the angles of the triangles of centres at each interior vertex must sum to $2\\pi$.\nThis is a system of equations in the radii, and an iterative or variational argument shows that it has a unique normalized solution.\nExample: the complete graph $K_4$ is realized by three mutually tangent equal discs together with a fourth small disc in the gap between them, tangent to all three.\nThe simplicity hypothesis is needed.\nTwo discs with disjoint interiors touch in at most one point, so a double edge cannot be shown by tangency, and a loop has no meaning.\nConversely, the tangency graph of such a family of discs is always planar, so the theorem characterizes planar graphs.\n\n## Explanations\n\n### L1 human\n\nDraw a planar graph, that is, a network of dots and lines without crossings.\nThe theorem says that you can replace the dots by coins lying flat on a table so that two coins touch exactly when the dots are joined.\nFor a triangulation, the picture is essentially unique.\nThis gives a rigid, geometric way to draw any planar graph.\n\n### L2 human\n\nThurston suggested that circle packings of fine hexagonal triangulations approximate the [[Riemann mapping|02WHMEG5ZR]] of a domain, and Rodin and Sullivan proved this convergence.\nSo circle packings give a discrete version of [[conformal maps|0YYT36R4YD]], the subject of discrete conformal geometry.\nThe theorem also gives a geometric proof of planar separator theorems and of Fáry's theorem that planar graphs can be drawn with straight edges, by joining centres of tangent discs.\n",
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