  This paper is concerned with the problem of approximating a homeomorphism by
piecewise affine homeomorphisms. The main result is as follows: every
homeomorphism from a planar domain with a polygonal boundary to R^2 that is
globally Holder continuous of exponent \alpha, and whose inverse is also
globally Holder continuous of exponent \alpha can be approximated in the Holder
norm of exponent \beta by piecewise affine homeomorphisms, for some \beta that
only depends on \alpha. The proof is constructive. We adapt the proof of
simplicial approximation in the supremum norm, and measure the side lengths and
angles of the triangulation over which the approximating homeomorphism is
piecewise affine. The approximation in the supremum norm, and a control on the
minimum angle and on the ratio between the maximum and minimum side lengths of
the triangulation suffice to obtain approximation in the Holder norm.
