  The level set of an elliptic function is a doubly periodic point set in C. To
obtain a wider spectrum of point sets, we consider, more generally, a Riemann
surface S immersed in C^2 and its sections (``cuts'') by C. We give S a
crystallographic isometry in C^2 by defining a fundamental surface element as a
conformal map of triangular domains and S as its extension by reflections in
the triangle edges. Our main result concerns the special case of maps of right
triangles, with the right angle being a regular point of the map. For this
class of maps we show that only seven Riemann surfaces, when cut, form point
sets that are discrete in C. Their isometry groups all have a rank-four lattice
subgroup, but only three of the corresponding point sets are doubly periodic in
C. The remaining surfaces form quasiperiodic point sets closely related to the
vertex sets of quasiperiodic tilings. In fact, vertex sets of familiar tilings
are recovered in all cases by applying the construction to a piecewise flat
approximation of the corresponding Riemann surface. The geometry of point sets
formed by cuts of Riemann surfaces is no less ``rigid'' than the geometry
determined by a tiling, and has the distinct advantage in having a regular
behavior with respect to the complex parameter which specifies the cut.
