Zigzag Structure of Thin Chamber Complexes.

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 zigzags in the simplex αn<inline-graphic></inline-graphic>, the cross-polytope βn<inline-graphic></inline-graphic>, the 24-cell, the icosahedron and the 600-cell are equal to the Coxeter numbers of An<inline-graphic></inline-graphic>, Bn=Cn<inline-graphic></inline-graphic>, F4<inline-graphic></inline-graphic> and Hi<inline-graphic></inline-graphic>, i=3,4<inline-graphic></inline-graphic>, respectively. We also discuss in which cases two faces in a thin chamber complex can be connected by a zigzag. [ABSTRACT FROM AUTHOR]