2-D Maps of the Prime Numbers

Here are a few two-dimensional mappings of some prime numbers distributed across Quadrant I.  Mathematica was used for the algorithm and resulting graphics. 

i = the total amount of integer-coordinate points in Quadrant I.  Pi(x) = j = the amount of primes less than or equal to the amount x.  (i, i) is where the composite-number points are mapped to.

Not quite random, but not quite predictable.  A fractal has a specific generating relation (algebraic expression).  There is no such single relation for all of the primes numbers.  Is there a possible 'semi-fractal' pattern?  Is such a concept valid?  What would the be the criteria?  There are some ideas that begin to link fractals and the distribution of primes, but I have not found a clear connection as of yet.  There are algebraic expressions for some primes, there are probablistic suggestions, many modulo ideas, but nothing as clear and direct as I searched for.  The next idea to be revisited is the Jumping Champion model.  With the direct 2-D map, as i increases, pi(x) increases, and the quadrant becomes more filled.  The only clear characteristic that remains constant for each generation is the semi-random, semi-predictable nature of the distribution, and that as i increases, pi(x) increases.  This is reflected in the 2-D mapped prime vector field.  The field progresses across the entire Quadrant I from the origin outward.  The vector field is composed of the consecutive points from where (i,i) = (20,20), in the image just above, where i = 401, but the vector lengths were not made to remain inside the i x i area as of yet.

This 2-D Prime Pattern idea was first presented to me by my then math department chair back in 1997.  At the time my first thought was to generate a Markov Chain.  Using pi(x) as the theme for the transitional probabilities, the patterns converged to one single pattern as x was increased using the Mathematica program I used. 

My most recent effort is the consecutive connected prime points graphs.  The more prime points, the more the connecting line has a fractal top and bottom boundary.  This chracteristic contiunes as the amount of prime points increases.  The behavior of when higher or lower points appear stays the same no matter how many prime points are connected, much as like the stock market graphs.  There can be a jump or drop at any time, not quite predictably.  As the amount of prime points decreases, this could be a zoom-in/change in scale, and as the prime points increase, this could be a zoom-out/change in scale.  The jump - behavior does not change.  Fractal?

Click to enlarge the top two images.