Quantum Geometry I : Basics of Loop Quantum Gravity. The Quantum Polyhedra

General Relativity describes gravity in geometrical terms. This suggests that quantizing such theory is the same as quantizing geometry. The subject can therefore be called quantum geometry and one may think that mathematicians are responsible of this subject. Unfortunately, most mathematicians are not aware of this beautiful area of study. Here we give a basic introduction to what quantum geometry means to a community working in a theory known as loop quantum gravity. It is directed towards graduate or upper students of physics and mathematics. We do it from a point of view of a mathematician.