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  "body": "\n## Definition\n\nLet $W$ be a finite group of isometries of $\\R^n$ generated by [[reflections|06BRDF3F0E]] in hyperplanes through the origin, a finite [[Coxeter group|G6FND7X69J]].\nAssume that no nonzero vector is fixed by all of $W$.\nLet $C$ be a fundamental chamber of $W$: the closure of one connected component of the complement of all the reflecting hyperplanes of $W$.\nThen $C$ is a simplicial cone bounded by $n$ hyperplanes $H_1, \\dots, H_n$, and the reflections $s_1, \\dots, s_n$ in them generate $W$.\nChoose a point $v \\in C$ with $v \\ne 0$.\nThe *Wythoff construction* produces the polytope\n$$P(v) = \\operatorname{conv}(W v),$$\nthe [[convex hull|0FQSDC2EQ2]] of the [[orbit|0YQPZB5Q1G]] $W v = \\set{w(v) : w \\in W}$.\n\n## Remarks\n\nAll points of $W v$ lie on the sphere of radius $\\norm{v}$, so each of them is a vertex of $P(v)$, and $W$ acts transitively on the vertices of $P(v)$.\nThe polytope depends, up to similarity, mainly on the set of mirrors $H_i$ that do not contain $v$; in a Coxeter diagram one marks the corresponding nodes with rings.\nIf $v$ does not lie on the mirror $H_i$, then $v$ and $s_i(v)$ are joined by an edge of length twice the distance from $v$ to $H_i$.\nWhen $v$ has the same distance to every mirror that does not contain it, all edges of $P(v)$ have the same length, and $P(v)$ is a [[uniform polytope|BZXHGNGQAX]].\nLet $W$ be the symmetry group of a regular polytope.\nIf $v$ lies on all mirrors except the one at an end of the Coxeter diagram, then $P(v)$ is that regular polytope or its dual, up to similarity.\n\nExample: let $W$ be the symmetry group of the square, of order $8$, generated by the reflections in the lines $y = 0$ and $y = x$, with chamber $C = \\set{(x, y) : 0 \\le y \\le x}$.\nFor $v = (1, 0)$, on the mirror $y = 0$, the orbit is $\\set{(\\pm1, 0), (0, \\pm1)}$, and $P(v)$ is a square.\nFor $v = (1, 1)$, on the mirror $y = x$, the orbit is $\\set{(\\pm1, \\pm1)}$, again a square.\nFor $v = (1, \\tan(\\pi/8))$, which is at equal distance from both mirrors, the orbit has $8$ points spaced by angles $\\pi/4$, and $P(v)$ is a regular octagon.\n\nNon-example: the choice $v = 0$ is excluded; its orbit is the single point $0$, and the construction gives no polytope of dimension $n$.\n",
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