In the classification of Hietarinta, three triangular 4 x 4 R-matrices lead
, via the FRT formalism, to matrix bialgebras which are not deformations of
the trivial one. In this paper, we find the bialgebras which are in dualit
y with these three exotic matrix bialgebras, We note that the L-T duality o
f FRT is not sufficient for the construction of the bialgebras in duality.
We find also the quantum planes corresponding to these bialgebras both by t
he Wess-Zumino R-matrix method and by Manin's method.