The plane figure formed by a sequence of circles 
, 
, 
, ... that are all tangent to each other at the same point
and such that the sequence of radii converges to zero. In the figure above, 
is chosen to be the circle
with center 
and radius 
.
The topology of this set and of its generalizations to higher dimensions has been intensively studied in recent years (Eda 2000, Eda and Kawamura 2002ab). This research
was motivated by the following striking discovery: although the fundamental
group of the circle is 
, the fundamental group
of the figure eight is 
, where 
denotes the free product,
and in general the fundamental group of the
-petalled rose
curve is 
(Massey 1989, pp. 123-125), the fundamental
group of the Hawaiian ring is not a free group
(Higman 1952, de Smit 1992, Black 1996).
-Dimensional Hawaiian Earring." Fund. Math. 165,
17-28, 2000a.Eda, K. and Kawamura, K. "The Singular Homology of
the Hawaiian Earring." J. London Math. Soc. 62, 305-310, 2000b.Higman,
G. "Unrestricted Free Products and Varieties of Topological Groups." J.
London Math. Soc. 27, 73-81, 1952.Massey, W. S.