*Originally posted by drnihili*

Conversely, if AB and CD are macro distances composed of the same number of quanta, do they have precisely the same length?

Yes, this is the only way both distances can be equal.

As much it concerns other questions, I think that there is a common mistake that guys here makes when it comes to the quantized space. There is two kinds of spaces (and distances!), one is pure abstraction - the geometrical continuous space. This space is scale-invariant, it is not discreet. The other space is the real physical space, which had to be discreet at some level. Geometrical space can approximate physical space correctly only in macro scales. At micro scales continious geometry becom irrelevant.

I will demonstrate this to you in this way. Imagine 2D discreet space from the kind:

.................M.........................................................

.............................................................................

........A........B........C...............................................

...............................................................................

N......D........E........F......P...........................................

...............................................................................

........G........K.........L...............................................

..............................................................................

..................O.........................................................

A,B,C and so on are the discreet points from the space. The space exist only in this points. Point E is the central point, its distance to all periphery points- M,A,N.........., expressed in el. steps is the same. Because of this in the geometrical continious space the above figure is mapped as circle and in this points of view all periphery points lies on a circle. Inthis way a romb in the physical space is mapped as a sphere in the geometrical space.

I am interested from your proposal. Can I send you e-mail?