Two-dimensional crystallography introduced by the sprinkler watering problem
2012 - J.A. De Toro, G.F. Calvo, P. Muñiz
European Journal of Physics 33, 167-177 (2012)
Fernández Calvo, Gabriel.
he problem of optimizing the number of circular sprinklers watering large fields is used to introduce, from a purely elementary geometrical perspective, some basic concepts in crystallography and comment on a few size effects in condensed matter physics. We examine square and hexagonal lattices to build a function describing the, so-called, dry area fraction of the field to be irrigated, providing a simple concept for first-year undergraduate students to understand the similarities between the arrangement of atoms/molecules in two dimensions and that of sprinklers. It is shown that the search for the most efficient (least-overlap) sprinkler configuration is equivalent to the search for the Wigner–Seitz cell covering the largest area of its own circumcircle.