  Let W be an infinite Coxeter group. We initiate the study of the set E of
limit points of "normalized" positive roots (representing the directions of the
roots) of W. We show that E is contained in the isotropic cone of the bilinear
form B associated to a geometric representation, and illustrate this property
with numerous examples and pictures in rank 3 and 4. We also define a natural
geometric action of W on E, and then we exhibit a countable subset of E, formed
by limit points for the dihedral reflection subgroups of W. We explain that
this subset is built from the intersection with Q of the lines passing through
two positive roots, and finally we establish that it is dense in E.
