Let $\Sigma_{\Delta^{n}}$ be the fan in Example 3.70. The toric variety $X_{\Sigma_{\Delta^{n}}}$ is the projective space $\mathbb{P}^{n}_{K}$. More generally, to a polytope $\Delta\subset M_{\mathbb{R}}$ of maximal dimension we can associate a complete toric variety $X_{\Sigma_{\Delta}}$, where $\Sigma_{\Delta}$ is the fan of Example 3.71.