  S.~Elnitsky (1997) gave an elegant bijection between rhombic tilings of
$2n$-gons and commutation classes of reduced words in the symmetric group on
$n$ letters. P.~Magyar (1998) found an important construction of the
Bott-Samelson varieties introduced by H.C.~Hansen (1973) and M.~Demazure
(1974). We explain a natural connection between S.~Elnitsky's and P.~Magyar's
results. This suggests using tilings to encapsulate Bott-Samelson data (in type
$A$). It also indicates a geometric perspective on S.~Elnitsky's combinatorics.
We also extend this construction by assigning desingularizations to the
zonotopal tilings considered by B.~Tenner (2006).
