  The main difficulty in solving the Helmholtz equation within polygons is due
to non-analytic vertices. By using a method nearly identical to that used by
Fox, Henrici, and Moler in their 1967 paper; it is demonstrated that such
eigenvalue calculations can be extended to unprecedented precision, very often
to well over a hundred digits, and sometimes to over a thousand digits.
  A curious observation is that as one increases the number of terms in the
eigenfunction expansion, the approximate eigenvalue may be made to alternate
above and below the exact eigenvalue. This alternation provides a new method to
bound eigenvalues, by inspection.
  Symmetry must be exploited to simplify the geometry, reduce the number of
non-analytic vertices and disentangle degeneracies. The symmetry-reduced
polygons considered here have at most one non-analytic vertex from which all
edges can be seen. Dirichlet, Neumann, and periodic-type edge conditions, are
independently imposed on each polygon edge.
  The full shapes include the regular polygons and some with re-entrant angles
(cut-square, L-shape, 5-point star). Thousand-digit results are obtained for
the lowest Dirichlet eigenvalue of the L-shape, and regular pentagon and
hexagon.
