  We consider the moduli space M_r of polygons with fixed side lengths in
five-dimensional eucledian space. We analyze the local structure of its
singularities and exhibit a real-analytic equivalence between M_r and a
weighted quotient of the n-fold product of the quaternionic projective line
HP^1 by the diagonal PSL(2,H)-action. We explore the relation between M_r and
the fixed point set of an anti-symplectic involution on a GIT quotient
Gr(2,4)^n/SL(4,C). We generalize the Gel'fand-MacPherson correspondence to more
general complex Grassmannians and to the quaternionic context, and realize our
space M_r as a quotient of a subspace in the quaternionic Grassmannian
Gr_H(2,n) by the action of the group Sp(1)^n. We also give analogues of the
Gel'fand-Tsetlin coordinates on the space of quaternionic Hermitean matrices
and briefly describe generalized action-angle coordinates on M_r.
