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  "body": "\n## Definition\n\nLet $d \\ge 1$ and let $G = (V, E)$ be a finite [[graph|00C0BR7D7H]].\nA *framework* in $\\R^d$ is a pair $(G, p)$ where $p \\colon V \\to \\R^d$ is a map.\nThe framework $(G, p)$ is *rigid* if there is $\\varepsilon \u003e 0$ with the following property: every map $q \\colon V \\to \\R^d$ such that\n\n1. $\\norm{q(u) - q(v)} = \\norm{p(u) - p(v)}$ for every edge $uv \\in E$, and\n2. $\\norm{q(v) - p(v)} \u003c \\varepsilon$ for every $v \\in V$\n\nalso satisfies $\\norm{q(u) - q(v)} = \\norm{p(u) - p(v)}$ for all $u, v \\in V$.\nA framework that is not rigid is *flexible*.\n\n## Remarks\n\nExample: the triangle $K_3$ placed at three non-collinear points of $\\R^2$ is rigid.\nClause 1 already fixes all three pairwise distances, so the conclusion holds for every $\\varepsilon$.\n\nNon-example: the $4$-cycle placed at the corners of the unit square in $\\R^2$ is flexible.\nFor every $\\varepsilon \u003e 0$ there is a rhombus with sides $1$ close to the square, which satisfies clauses 1 and 2, but its diagonals have lengths different from $\\sqrt{2}$.\n\nRigidity is equivalent to the absence of a nontrivial flex.\nA flex is a continuous motion of the points, starting at $p$, that keeps all bar lengths; it is nontrivial if it changes some distance between two vertices.\n",
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