Theoretical Physicist Carlo Rovelli tells us to ged rid of current idea about geometry:
We will continue to use geometry as a useful branch of mathematics, but is time to abandon the longstanding idea of geometry as the description of physical space. The idea that geometry is the description of physical space is engrained in us, and might sound hard to get rid of it, but it is unavoidable; it is just a matter of time. Better get rid of it soon.
Geometry developed at first as a description of the properties of parcels of agricultural land. In the hands of ancient Greeks it became a powerful tool for dealing with abstract triangles, lines, circles, and similar, and was applied to describe paths of light and movements of celestial bodies with very great efficacy. In the modern age, with Newton, it became the mathematics of physical space. This geometrization of physical space appeared to be further vindicated by Einstein, who described space (actually, spacetime) in terms of the curved geometry of Riemann. But in fact this was the beginning of the end. Einstein discovered that the Newtonian space described by geometry is in fact a field like the electromagnetic field, and fields are nicely continuous and smooth only if measured at large scales. In reality, they are quantum entities that are discrete and fluctuating. Therefore the physical space in which we are immersed is in reality a quantum dynamical entity, which shares very little with what we call “geometry”. It is a pullulating process of finite interacting quanta. We can still use expressions like “quantum geometry” to describe it, but reality is that a quantum geometry is not much of a geometry anymore.
Better getting soon rid of the idea that our spacial intuition is always reliable. The world is far more complicated (and beautiful) than a “geometrical space” and things moving in it.
Carlo Rovelli is Theoretical Physicist; Aix-Marseille University, in the Centre de Physique Théorique, Marseille, France; Author, The First Scientist: Anaximander and His Legacy