Simplexity of the cube Original Research Article

A computer-assisted linear programming approach is used to study minimum-cardinality decompositions of the cube. A triangulation of the 7-cube into 1493 simplices is given and it is shown that this, and a previously given triangulation of the 6-cube into 308 simplices, are the smallest possible for these dimensions. A characterization is given for the numbers of the various types of simplices used in all minimum-cardinality triangulations of the d-cube for d = 5, 6, 7. It is shown that the minimum of the cardinalities of all corner-slicing triangulations of I7 is 1820. For decompositions of the cube more general than triangulations, it is shown for dimension 5 that the minimum cardinality is 67, and for dimension 6 it is at least 270.