  We construct infinite families of abstract regular polytopes of type
$\{4,p_1,\ldots,p_{n-1}\}$ from extensions of centrally symmetric spherical
abstract regular $n$-polytopes. In addition, by applying the halving operation,
we obtain infinite families of both locally spherical and locally toroidal
regular hypertopes of type $\left\{{p_1 \atop p_1},\ldots,p_{n-1}\right\}$.
