  We consider Holder smoothness classes of surfaces for which we construct
piecewise polynomial approximation networks, which are graphs with polynomial
pieces as nodes and edges between polynomial pieces that are in `good
continuation' of each other. Little known to the community, a similar
construction was used by Kolmogorov and Tikhomirov in their proof of their
celebrated entropy results for Holder classes.
  We show how to use such networks in the context of detecting geometric
objects buried in noise to approximate the scan statistic, yielding an
optimization problem akin to the Traveling Salesman. In the same context, we
describe an alternative approach based on computing the longest path in the
network after appropriate thresholding.
  For the special case of curves, we also formalize the notion of `good
continuation' between beamlets in any dimension, obtaining more economical
piecewise linear approximation networks for curves.
  We include some numerical experiments illustrating the use of the beamlet
network in characterizing the filamentarity content of 3D datasets, and show
that even a rudimentary notion of good continuity may bring substantial
improvement.
