Extensions of regular polytopes with preassigned Schlafli symbol

We say that a (d + 1)-polytope P is an extension of a polytope K if the facets or the vertex figures of P are isomorphic to K. The Schlafli symbol of any regular extension of a regular polytope is determined except for its first or last entry. For any regular polytope K we construct regular extensions with any even number as first entry of the Schlafli symbol. These extensions are lattices if K is a lattice. Moreover, using the so-called CPR graphs we provide a more general way of constructing extensions of polytopes. (C) 2008 Elsevier Inc. All rights reserved.