Asymptotically efficient triangulations of the d-cube

Triangulating the regular d-cube I = [0, 1] in a “simple” way has many applications, like solving differential equations by finite element methods or calculating fixed points. See, for example, [7]. Determining the smallest number of simplices needed has brought special attention both from a theoretical point of view and from an applied one (see [6, Section 14.5.2] for a recent survey). Before going on, let us clarify that when we speak about triangulations of a polytope P of dimension d we mean decompositions of P into d-simplices (i) using as vertices only the vertices of P , and (ii) intersecting face to face (i.e., forming a geometric simplicial complex). If the second condition is not fulfilled, we call them simplicial dissections of P . The number of d-simplices of a triangulation or dissection T will be called its size and denoted |T |. A general method to obtain the smallest triangulation of a polytope P of dimension d as the optimal integer solution of a linear program is described in [2]. For the d-cube, the direct application of this method is impossible in practice beyond dimension 4 or 5. With a somewhat similar method but simplifying the equations using simmetries of the cube, Anderson and Hughes [1] have calculated the smallest size of a triangulation of the 6-cube and the 7-cube, in a computational tour-deforce which involved a problem with 1,456,318 variables and ad hoc ways of decomposing the system into smaller subsystems. In order to compare sizes of triangulations of cubes in different dimensions, Todd [7] defines the efficiency of a triangulation T of the d-cube to be the number (|T |/d!) 1