  A team of anonymous mobile agents represented by points freely moving in the
plane have to gather at a single point and stop. Agents start at different
points of the plane and at possibly different times chosen by the adversary.
They are equipped with compasses, a common unit of distance and clocks. They
execute the same deterministic algorithm and travel at speed 1. When agents are
at distance at most $\epsilon$, for some positive constant $\epsilon$ unknown
to them, they can exchange all information. Due to the anonymity of the agents
and the symmetry of the plane, gathering is impossible, e.g., if agents start
simultaneously at distances larger than $\epsilon$. However, if some agents
start with a delay with respect to others, gathering may become possible. In
which situations such latecomers can enable gathering? To answer this question
we consider initial configurations formalized as sets of pairs $\{(p_1,t_1),
(p_2,t_2),\dots , (p_n,t_n)\}$, for $n\geq 2$ where $p_i$ is the starting point
of the $i$-th agent and $t_i$ is its starting time. An initial configuration is
gatherable if agents starting at it can be gathered by some algorithm, even
dedicated to this particular configuration. We characterize all gatherable
initial configurations. Is there a universal deterministic algorithm that can
gather all gatherable configurations of a given size. It turns out that the
answer is no. We show that all gatherable configurations can be partitioned
into two sets: bad and good configurations. We show that bad gatherable
configurations (even of size 2) cannot be gathered by a common gathering
algorithm, and we prove that there is a universal algorithm that gathers all
good configurations of a given size.
