It is our day-to-day experience that rigid objects, e.g. two books, can be placed right next to each other, in a way that they touch but don't overlap. However, if we think of space as composed of points, the concept of touching becomes different: if two geometrical objects touch each other, they always overlap. For example, when a cube touches a sphere, they have one point in common. This is inconsistent with our intuition that a given piece of space cannot be occupied by multiple rigid objects at the same time.