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

Abstract: 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.