The Mass Gap of the Space-time and its Shape

Snyder’s quantum space-time which is Lorentz invariant is investigated. It is found that the quanta of space-time have a positive mass that is interpreted as a positive real mass gap of space-time. This mass gap is related to the minimal length of measurement which is provided by Snyder’s algebra. Several reasons to consider the space-time quanta as a 24-cell are discussed. Geometric reasons include its self-duality property and its 24 vertices that may represent the standard model of elementary particles. The 24-cell symmetry group is the Weyl/Coxeter group of the $F_4$ group which was found recently to generate the gauge group of the standard model. It is found that 24-cell may provide a geometric interpretation of the mass generation, Avogadro number, color confinement, and the flatness of the observable universe. The phenomenology and consistency with measurements is discussed.