Here, we review our progress on a geometrical approach of quantum physics and topological crystals starting from nature, electrodynamics of planets and linking with Dirac magnetic monopoles and gauge fields. The Bloch sphere of a quantum spin-$\frac{1}{2}$ particle can also acquire an integer topological charge in the presence of a radial magnetic field. We show that the global topological properties are revealed from the poles of the surface allowing a correspondence between smooth fields, metric and quantum distance. The information is transported from each pole to the equatorial plane on a thin Dirac string. We develop the theory, "the quantum topometry" in space and time, and present applications on transport from a Newtonian approach, on a quantized photo-electric effect from circular dichroism of light towards topological band structures of crystals. The occurrence of robust edge modes related to the topological lattice models are revealed analytically when deforming the sphere or ellipse onto a cylinder. The topological properties of the quantum Hall effect, the quantum anomalous Hall effect and the quantum spin Hall effect on the honeycomb lattice can be measured locally in the Brillouin zone from the light-matter coupling. The formalism allows us to include interaction effects from the momentum space. Interactions may also result in fractional entangled geometry within the curved space. We develop a relation between entangled wavefunction in quantum mechanics, coherent superposition of geometries, a way to one-half topological numbers and Majorana fermions. We show realizations in topological matter. We present a relation between axion electrodynamics, topological insulators on a surface of a cube and the two-spheres' model via the meron.