In this paper we examine the correlation between the information content and the spatial realization of range measurements taken by a mapping robot. To do so, we consider the task of constructing an occupancy grid map with a binary Bayesian filter. Using a narrow beam-based sensor model (versus an additive white Gaussian noise model), we prove that any controller tasked to maximize a mutual information reward function is eventually attracted to unexplored space. This intuitive behavior is derived solely from the geometric dependencies of the occupancy grid mapping algorithm and the monotonie properties of mutual information. Since it is a function of both the robot's position and the uncertainty of the surrounding cells, mutual information encodes geometric relationships that are fundamental to robot control, thus yielding geometrically relevant reward surfaces on which the robot can navigate. Lastly, we present the results of two experiments employing an omnidirectional ground robot equipped with a laser rangefinder.