  A dilatation structure on a metric space, arXiv:math/0608536v4, is a notion
in between a group and a differential structure, accounting for the approximate
self-similarity of the metric space.
  The basic objects of a dilatation structure are dilatations (or
contractions). The axioms of a dilatation structure set the rules of
interaction between different dilatations.
  Linearity is also a property which can be explained with the help of a
dilatation structure. In this paper we show that we can speak about two kinds
of linearity: the linearity of a function between two dilatation structures and
the linearity of the dilatation structure itself.
  Our main result here is a characterization of contractible groups in terms of
dilatation structures. To a normed conical group (normed contractible group) we
can naturally associate a linear dilatation structure. Conversely, any linear
and strong dilatation structure comes from the dilatation structure of a normed
contractible group.
