  It is shown that the fundamental group of the Griffiths double cone space is
isomorphic to that of the triple cone. More generally if $\kappa$ is a cardinal
such that $2 \leq \kappa \leq 2^{\aleph_0}$ then the $\kappa$-fold cone has the
same fundamental group as the double cone. The isomorphisms produced are
non-constructive, and no isomorphism between the fundamental group of the $2$-
and of the $\kappa$-fold cones, with $2 < \kappa$, can be realized via
continuous mappings. We also prove a conjecture of James W. Cannon and Gregory
R. Conner which states that the fundamental group of the Griffiths double cone
space is isomorphic to that of the harmonic archipelago.
