  We consider smooth isotropic immersions from the 2-dimensional torus into
$R^{2n}$, for $n \geq 2$. When $n = 2$ the image of such map is an immersed
Lagrangian torus of $R^4$. We prove that such isotropic immersions can be
approximated by arbitrarily $C^0$-close piecewise linear isotropic maps. If $n
\geq 3$ the piecewise linear isotropic maps can be chosen so that they are
piecewise linear isotropic immersions as well. The proofs are obtained using
analogies with an infinite dimensional moment map geometry due to Donaldson. As
a byproduct of these considerations, we introduce a numerical flow in finite
dimension, whose limit provide, from an experimental perspective, many examples
of piecewise linear Lagrangian tori in $R^4$. The DMMF program, which is freely
available, is based on the Euler method and shows the evolution equation of
discrete surfaces in real time, as a movie.
