  We describe the structure of quasiflats in two-dimensio\-nal Artin groups. We
rely on the notion of metric systolicity developed in our previous work. Using
this weak form of non-positive curvature and analyzing in details the
combinatorics of tilings of the plane we describe precisely the building blocks
for quasiflats in all two-dimensional Artin groups -- atomic sectors. This
allows us to provide useful quasi-isometry invariants for such groups --
completions of atomic sectors, stable lines, and the intersection pattern of
certain abelian subgroups. These are described combinatorially, in terms of the
structure of the graph defining an Artin group. As an important tool, we
introduce an analogue of the curve complex in the context of two-dimensional
Artin groups -- the intersection graph. We show quasi-isometric invariance of
the intersection graph under natural assumptions.
  As immediate consequences we present a number of results concerning
quasi-isometric rigidity for the subclass of CLTTF Artin groups. We give a
necessary and sufficient condition for such groups to be strongly rigid (self
quasi-isometries are close to automorphisms), we describe quasi-isometry
groups, we indicate when quasi-isometries imply isomorphisms for such groups.
In particular, there exist many strongly rigid large-type Artin groups. In
contrast, none of the right-angled Artin groups are strongly rigid by a
previous work of Bestvina, Kleiner and Sageev.
