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In this article, we study the correspondence between the geometry of del Pezzo surfaces 
${{s}_{r}}$ and the geometry of the 
$r$ -dimensional Gosset polytopes ( 
${{(r-4)}_{21}}$ . We construct Gosset polytopes ${{(r-4)}_{21}}$ in Pic 
${{S}_{r}}\,\otimes \,\mathbb{Q}$ whose vertices are lines, and we identify divisor classes in Pic ${{s}_{r}}$ corresponding to 
$(a-1)$ -simplexes 
$(a\le r)$ , 
$(r-1)$ -simplexes and $(r-1)$ -crosspolytopes of the polytope ${{(r-4)}_{21}}$ . Then we explain how these classes correspond to skew 
$a$ -lines $(a\le r)$ , exceptional systems, and rulings, respectively.
As an application, we work on the monoidal transform for lines to study the local geometry of the polytope ${{(r-4)}_{21}}$ . And we show that the Gieser transformation and the Bertini transformation induce a symmetry of polytopes 
${{3}_{21}}$ and 
${{4}_{21}}$ , respectively.
