  In this paper we study the extreme and the periodic $L_2$ discrepancy of
plane point sets. The extreme discrepancy is based on arbitrary rectangles as
test sets whereas the periodic discrepancy uses "periodic intervals", which can
be seen as intervals on the torus. The periodic $L_2$ discrepancy is, up to a
multiplicative factor, also known as diaphony. The main results are exact
formulas for these kinds of discrepancies for the Hammersley point set and for
rational lattices.
  In order to value the obtained results we also prove a general lower bound on
the extreme $L_2$ discrepancy for arbitrary point sets in dimension $d$, which
is of order of magnitude $(\log N)^{(d-1)/2}$, like the standard and periodic
$L_2$ discrepancies, respectively. Our results confirm that the extreme and
periodic $L_2$ discrepancies of the Hammersley point set are of best possible
asymptotic order of magnitude. This is in contrast to the standard $L_2$
discrepancy of the Hammersley point set. Furthermore our exact formulas show
that also the $L_2$ discrepancies of the Fibonacci lattice are of the optimal
order.
  We also prove that the extreme $L_2$ discrepancy is always dominated by the
standard $L_2$ discrepancy, a result that was already conjectured by Morokoff
and Caflisch when they introduced the notion of extreme $L_2$ discrepancy in
the year 1994.
