Generalizations of Schöbi’s Tetrahedral Dissection

Abstract

Let v 1,…,v n be unit vectors in ℝ n such that v i ⋅v j =−w for i≠j, where $-1<w<\frac{1}{n-1}$ . The points ∑ =1 λ i v i (1≥λ 1≥⋅⋅⋅≥λ n ≥0) form a “Hill-simplex of the first type,” denoted by $\mathcal {Q}_{n}(w)$ . It was shown by Hadwiger in 1951 that $\mathcal {Q}_{n}(w)$ is equidissectable with a cube. In 1985, Schöbi gave a three-piece dissection of $\mathcal {Q}_{3}(w)$ into a triangular prism $c\mathcal {Q}_{2}(\frac{1}{2})\times I$ , where I denotes an interval and $c=\sqrt{2(w+1)/3}$ . In this paper, we generalize Schöbi’s dissection to an n-piece dissection of $\mathcal {Q}_{n}(w)$ into a prism $c\mathcal {Q}_{n-1}(\frac{1}{n-1})\times I$ , where $c=\sqrt{(n-1)(w+1)/n}$ . Iterating this process leads to a dissection of $\mathcal {Q}_{n}(w)$ into an n-dimensional rectangular parallelepiped (or “brick”) using at most n! pieces. The complexity of computing the map from $\mathcal {Q}_{n}(w)$ to the brick is O(n 2). 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.