Regular 3-polytopes of type $\{n,n\}$

For each integer $n \geq 3$, we construct a self-dual regular 3-polytope $\mathcal{P}$ of type $\{n, n\}$ with $2^n n$ flags, resolving two foundamental open questions on the existence of regular polytopes with certain Schl\"afli types. The automorphism group $\operatorname{Aut}(\mathcal{P})$ is explicitly realized as the semidirect product $\mathbb{F}_2^{n-1} \rtimes D_{2n}$, where $D_{2n}$ is the dihedral group of order $2n$, with a complete presentation for $\operatorname{Aut}(\mathcal{P})$ is provided. This advances the systematic construction of regular polytopes with prescribed symmetries.