The square is the 2-dimensional 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 (3-dimensional) cube [1], i.e. to decompose the cube into a collection of tetrahedra that share their vertices with the cube and intersect along common faces. These 74 triangulations can be partitioned into 6 symmetry classes [1,2]. All of 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. Indeed, the graph induced by the regular triangulations in the flip graph of a set of points is the edge-graph of the corresponding secondary polytope.

What about the tesseract, the 4-dimensional analog of the cube? The only method fast enough to enumerate them explores its flip-graph, under the assumption that this graph is connected. I proved the connectedness of this graph in [3]. Part of this proof is computer-assisted: Proposition 3 from [3] is obtained using an algorithm you will find several implementations of here.

The triangulations of the tesseract can then be enumerated using TOPCOM: there are 92 487 256 of them, partitioned into 247 451 symmetry classes. These numbers are reported within sequences A238820 and A238821 in the Online Encyclopedia of Integer Sequences:

dimension	0	1	2	3	4
number of triangulations (A238820)	1	1	2	74	92 487 256
number of symmetry classes (A238821)	1	1	1	6	247 451

The corresponding numbers are as yet unknown for hypercubes of dimension greater than 4.