Klein's Polyhedra for the Forth Extremal Cubic Form

Davenport and Swinnerton-Dyer had found the first 19 extremal thernar cubic forms g_i, the meaning of which is the same as the meaning of the Markov forms for binary quadratic forms. The Klein's polyhedra for the forms g₁,g₂,g₃ were recently computed by Bruno and Parusnikov. They computed convergents of continued fractions for some matrix generalizations of the continued fractions algorithm as well. In this paper the analogious problems for the form g₄ are studied. Namely, the Klein polyhedra for the form g₄ and its conjugated one ^ˆg^*₄ are computed. It appears that they are essentially different. Their periods and fundamental domains were found. The matrix algorithm's expansions of the vectors of these forms are computed as well.