Packings of a triangle and compactification of multiplication in $PGL_r$

The theme is a compactification of $PGL_r^3/PGL_r$ whose geometry is described in terms of the set of packings of an equilateral triangle with side $r$. This compactification, equipped with three morphisms $PGL_r^3/PGL_r\to PGL_r^2/PGL_r$ according to the De Concini-Procesi compactification, realizes the compactification of multiplication in $PGL_r$.