The relationship between abstract geometrical concepts and reality? In pure geometry points have no parts, lines no breadth and surfaces no thickness, geometrical figures can be exactly congruent. But how do these concepts relate to the physical world. What is the relationship between the laws of geometry and the laws of our physical world?
 A: Geometry started as an observatianal discipline, fitting data to algebraic functions. It developed into a paradigm for a mathematical theory. A mathematical theory has some axioms, and from them, with logic , algebra, calculus... theorems are developed . It is a rigorous mathematical construct, closed and precise.
Nature is neither closed nor precise. Geometry is a model for physics. A model means that in addition to the mathematical axioms, laws and correspondences of physical observables  to the mathematics  have to be assigned. This introduces errors , to start with, because all measurements have an error assigned. A point has to be defined, as well as a line and a plane in the physical space. When errors appear between the mathematical predictions and the physical observations, the mathematical model is revised, to fit the data. For example plane geometry had to be changed to spherical geometry, where two parallel lines do meet, when the globe was modelled.
The relationship between the accurate mahematical model and reality is that reality tests the validity of the model and discards it if there are deviations not excusable by measurement errors.
