= Weyl chamber geometry of C3
{c}
{title2=$x_1>x_2>x_3>0$}
The <C3 root system> consists of $\pm2e_i$ and $\pm e_i\pm e_j$ in $\mathbb R^3$. Its <Weyl group> is the <signed symmetric group> on three coordinates, of order $48$. The reflecting planes are $x_i=0$ and $x_i=\pm x_j$. The open <Weyl chamber> $x_1>x_2>x_3>0$ has closure generated by $(1,0,0)$, $(1,1,0)$ and $(1,1,1)$. Sign choices and absolute-coordinate orderings give its $48$ translates. On the unit sphere the chamber is a spherical triangle with angles $\pi/2$, $\pi/3$, $\pi/4$, and hence area $\pi/12$; the areas of all $48$ triangles sum to $4\pi$.
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