It was discussed previously that the projective line in the projective geometry is the set of lengths. Then R (i.e. the projective line) is the set of lengths too. The continuous line of R can be turned into a discrete line by setting any interval smaller than some value U to be equal to zero. However, the previous algorithm (where R was turned into a set of discrete intervals and at the limit point intervals would tend to ep) is a mechanism to turn the projective line (i.e. a set of lengths) into a set of points (that’s because |ep|=0). But in doing so, the linear structure of R turns into a point. In other words, R as a set of points is presented as one point, where “distance” between consecutive elements is ep.
ep objects are multiples of natural numbers. It means the line of real numbers tends to the lattice of natural numbers at inf (lattice is constructed by points rather than lengths). So instead of trying to establish a 1-1 mapping between R and N, an evolutionary connection between them can be established. R evolves into a lattice with a fixed step size. This should be reflected in the paper model. In fact, the c-plane is constructed for this purpose and is defined to be the set of discrete elements of N.
The inverse of this process (i.e. when discrete elements turn into a continuous line) should also be considered. Since the continuous line of R is defined as the projective line, discrete elements can be turned into a continuous line based on a type of lens. This means the c-plane should include a type of lens where the unit length (or the step size) is the image produced by the lens and can be blurred to produce other lengths other than the unit length. The process of blurring the unit length involves changing the parameters of the lens.
So the question this article tries to answer is no longer, how to establish a 1-1 mapping between R and N, instead, the question is how to evolve one into another.