Rolling Ball Argument - Scale-dependent Topology

Scale-dependent Topology

If the surface being probed contains connections whose scale is a smaller than the ball diameter, then these connections may not appear in the ball's map. If the surface contains a wormhole whose throat narrows to slightly less than the ball's diameter, the ball may be able to enter and explore each wormhole mouth, but will not be able to pass through the throat, and will produce a map in which the narrowing mouth walls each terminate in a sharp geometrical spike.

The smooth and multiply connected surface will be mapped by the physics of a "large" particle as being singly connected and including geometrical singularities.