The square is the 2dimensional analog of the cube. As can be seen on the right, a square can be triangulated in two ways. These two triangulations are obtained from each other by a flip. There are 74 ways to triangulate the usual (3dimensional) cube [1], i.e. to decompose it into a set of tetrahedra that intersect along common faces and whose vertices are also vertices of the cube. These 74 triangulations can be partitioned into 6 symmetry classes [1,2]. All these triangulations are regular and, as a consequence, each of them can be transformed into any other triangulation of the cube by performing a sequence of flips. What about the 4dimensional cube? Until recently, the number of its triangulations was unknown. Indeed, the only method fast enough to possibly allow for their enumeration consists in exploring the flipgraph of the 4dimensional cube, under the assumption that this graph is connected. I proved the connectedness of this graph in [3]. Part of this proof is computerassisted: Proposition 3 from [3] is obtained using an algorithm you will find several implementations of here. The number of triangulations of the 4dimensional cube is found as a consequence using TOPCOM: there are 92 487 256 such triangulations, partitioned into 247 451 symmetry classes. These numbers are reported within sequences A238820 and A238821 in the Online Encyclopedia of Integer Sequences:
These numbers are yet unknown for cubes of dimension larger than 4. It is an interesting problem to find them: after all, they may be thought of as distant relatives of Catalan numbers as they also count the triangulations of a structured set of points. The two triangulations of a square are related by a flip. The six triangulations, up to symmetry, of a cube [1]. [1]Nonregular triangulations of products of simplices
