Generalizations of Sch\"{o}bi's Tetrahedral Dissection

TL;DR

This paper generalizes Schöbi's tetrahedral dissection to higher dimensions, providing explicit n-piece dissections of Hill-simplices into bricks with applications in coding theory.

Contribution

It introduces an n-piece dissection method for Hill-simplices into rectangular parallelepipeds, extending Schöbi's 3D dissection to arbitrary dimensions with efficient computation.

Findings

Dissection of Q_n(w) into a brick with at most n! pieces

Efficient O(n^2) computation of the dissection map

Applications to source coding and binary codes

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 < 1/(n-1). The points Sum_{i=1..n} lambda_i v_i, where 1 >= lambda_1 >= ... >= lambda_n >= 0, form a ``Hill-simplex of the first type'', denoted by Q_n(w). It was shown by Hadwiger in 1951 that Q_n(w) is equidissectable with a cube. In 1985, Sch\"{o}bi gave a three-piece dissection of Q_3(w) into a triangular prism c Q_2(1/2) X I, where I denotes an interval and c = sqrt{2(w+1)/3}. The present paper generalizes Sch\"{o}bi's dissection to an n-piece dissection of Q_n(w) into a prism c Q_{n-1}(1/(n-1)) X I, where c = sqrt{(n-1)(w+1)/n}. Iterating this process leads to a dissection of Q_n(w) into an n-dimensional rectangular parallelepiped (or ``brick'') using at most n! pieces. The complexity of computing the map from Q_n(w) to the brick is O(n^2). A second generalization of Sch\"{o}bi's dissection is given which applies specifically in R^4. The results have applications to source coding and to constant-weight binary codes.