Particle Swarms in N-Dimensional Simplex Conformations Quantum Mechanical and Topological Problems

N-dimensional simplices, apart of their highly symmetric properties, exhibit interesting properties, both in the structure of the coordinates of their vertices and in the uniform distances between pairs of them. Relationships of simplices with the vertices of dimensional hypercubes are also of interest. Discussing these facts in depth as an introduction, the present study examines the implications of the coordinate structure of simplices for the Theoretical and Quantum Mechanical formalism, applied to swarms of particles located at the vertices of a simplex. Also, the properties of the topological matrices that represent such a distance-conformation will be studied. Some interesting results arise from this analysis. Among others, a paradox about the existence of swarms of particles with dimensional simplex-vertex distance structure and the possibility of overcoming it, allowing the description of an dimensional description of the particles’ location. A brief discussion on the time-dependence of simplex coordinates is also provided.