  In any attempt to build a quantum theory of gravity, a central issue is to
unravel the structure of space-time at the smallest scale. Of particular
relevance is the possible definition of coordinate functions within the theory
and the study of their algebraic properties, such as non-commutativity. Here we
approach this issue from the perspective of loop quantum gravity and the
picture of quantum geometry that the formalism offers. In particular, as we
argue here, this emerging picture has two main elements: i) The nature of the
quantum geometry at Planck scale is one-dimensional, polymeric with quantized
geometrical quantities and; ii) Appropriately defined operators corresponding
to coordinates by means of intrinsic, relational, constructions become
non-commuting. This particular feature of the operators, that operationally
localize points on space, gives rise to an emerging geometry that is also, in a
precise sense, fuzzy.
