One approach to gauge the complexity of the computational problem underlying haptic perception is to determine the number of dimensions needed to describe it. In vision, the number of dimensions can be estimated to be seven. This observation raises the question of what is the number of dimensions needed to describe touch. Only with certain simplified representations of mechanical interactions can this number be estimated, because it is in general infinite. Organisms must be sensitive to considerably reduced subsets of all possible measurements. These reductions are discussed by considering the sensing apparatuses of some animals and the underlying mechanisms of two haptic illusions.