  Inspired by Coxeter's notion of Petrie polygon for $d$-polytopes (see
\cite{Cox73}), we consider a generalization of the notion of zigzag circuits on
complexes and compute the zigzag structure for several interesting families of
$d$-polytopes, including semiregular, regular-faced, Wythoff Archimedean ones,
Conway's 4-polytopes, half-cubes, folded cubes.
  Also considered are regular maps and Lins triality relations on maps.

Combinatorics