Generalizations of Schöbi's Tetrahedral Dissection

Let v1; : : : ; vn be unit vectors in R such that vi ¢ vj = ¡w for i 6= j where ¡1 < w < 1 n¡1 . The points Pn i=1 ‚ivi (1 ‚ ‚1 ‚ ¢ ¢ ¢ ‚ ‚n ‚ 0) form a “Hill-simplex of the first type”, denoted by Qn(w). It was shown by Hadwiger in 1951 that Qn(w) is equidissectable with a cube. In 1985, Schöbi gave a three-piece dissection of Q3(w) into a triangular prism cQ2(2) £ I, where I denotes an interval and c = p 2(w + 1)=3. The present paper generalizes Schöbi’s dissection to an n-piece dissection of Qn(w) into a prism cQn¡1( 1 n¡1)£I, where c = p (n ¡ 1)(w + 1)=n. Iterating this process leads to a dissection of Qn(w) into an n-dimensional rectangular parallelepiped (or “brick”) using at most n! pieces. The complexity of computing the map from Qn(w) to the brick is O(n2). A second generalization of Schöbi’s dissection is given which applies specifically in R4. The results have applications to source coding and to constant-weight binary codes.