  We show that there exists a quasi-isometric embedding of the product of $n$
copies of $\mathbb{H}_{\mathbb{R}}^2$ into any symmetric space of non-compact
type of rank $n$, and there exists a bi-Lipschitz embedding of the product of
$n$ copies of the $3$-regular tree $T_3$ into any thick Euclidean building of
rank $n$ with co-compact affine Weyl group. This extends a previous result of
Fisher--Whyte. The proof is purely geometrical, and the result also applies to
the non Bruhat--Tits buildings.
