On the enumeration of subcells within hypercubes and its application to the Borsuk-Ulam theorem

The conventional triangulation of 2-spheres and subdivision of tetrahedrons in R³ are difficult to generalize to higher dimensions. The challenge lies in finding a systematic way to characterize each of subcells after the division. This work discusses the dissection of high-dimensional hypercubes and presents a way where all subsequent subcells and their symmetries can be systematically enumerated. Of particular interest is a generic coordinate system that is employed to construct all cells through suitable homeomorphisms. By repeatedly applying this generic coordinate to all cells, multitasking in parallel is possible. On the other hand, the Borsuk-Ulam theorem asserts that every continuous function from an n-sphere into the Euclidian n-space maps at least one pair of antipodal points on the sphere with the same function value. The exquisiteness lies in that only the continuity is assumed in the theorem with yet such profound applications. As an application, this enumeration scheme can be employed to find the Borsuk-Ulam antipodal pair guaranteed without evoking any derivative information for the task. Numerical experiments manifest the effectiveness and potential of this enumeration scheme.