The answer to all questions is No. In fact, even the right reaction to the first sentence – that the Planck scale is a “discrete measure” – is No.
The Planck length is a particular value of distance which is as important as 2π times the distance or any other multiple. The fact that we can speak about the Planck scale doesn’t mean that the distance becomes discrete in any way. We may also talk about the radius of the Earth which doesn’t mean that all distances have to be its multiples.
In quantum gravity, geometry with the usual rules doesn’t work if the (proper) distances are thought of as being shorter than the Planck scale. But this invalidity of classical geometry doesn’t mean that anything about the geometry has to become discrete (although it’s a favorite meme promoted by popular books). There are lots of other effects that make the sharp, point-based geometry we know invalid – and indeed, we know that in the real world, the geometry collapses near the Planck scale because of other reasons than discreteness.
— Is reality discrete at the quantum level?
— Lubos Motl
2012.01.31 Tuesday ACHK