Zigzag structure of thin chamber complexes

Zigzags and generalized zigzags in thin chamber complexes are investigated, in particular, all zigzags in the Coxeter complexes are described. Using this description, we show that the lengths of all generalized zigzags in the simplex $\alpha_{n}$, the cross-polytope $\beta_{n}$, the $24$-cell, the icosahedron and the $600$-cell are equal to the Coxeter numbers of $A_{n}$, $B_{n}=C_{n}$, $F_{4}$ and $H_{i}$, $i=3,4$, respectively. Also, we discuss the following problem: in which cases two faces in a thin chamber complex can be connected by a zigzag?