In this paper we study the art gallery problem, which is one of the
fundamental problems in computational geometry. The objective is to place a
minimum number of guards inside a simple polygon such that the guards together
can see the whole polygon. We say that a guard at position x sees a point y
if the line segment xy is fully contained in the polygon.
Despite an extensive study of the art gallery problem, it remained an open
question whether there are polygons given by integer coordinates that require
guard positions with irrational coordinates in any optimal solution. We give a
positive answer to this question by constructing a monotone polygon with
integer coordinates that can be guarded by three guards only when we allow to
place the guards at points with irrational coordinates. Otherwise, four guards
are needed. By extending this example, we show that for every n, there is
polygon which can be guarded by 3n guards with irrational coordinates but
need 4n guards if the coordinates have to be rational. Subsequently, we show
that there are rectilinear polygons given by integer coordinates that require
guards with irrational coordinates in any optimal solution.Comment: 18 pages 10 Figure