How to achieve persistence of a 2D coordinate system representing connexions? This post explains it in a bottom-up approach. post 
Let the binary string 0000000010… be an object description, a description of a coordinate system with two dimensions. Obviously there is some information ‚in‘ the binary string. At position 9 we perceive a difference between this position and the positions before (and after): the binary digit 1 instead of all the other 0’s.
If we hand over this (marked by the binary digit 1 at position #9) to an inverse pairing function called split, we get two coordinates: split 9 = (0, 3).
Before proceeding, please find the Haskell code for the pairing function and its inverse:
{{{ -- fusion f :: (Fractional a) => a -> a -> a f x y = (1/2) * (x + y) * (x + y + 1) + y -- split s :: (RealFrac a, Floating a) => a -> (a, a) s z = (x, y) where j = getJ z y = z - (auxF j) x = j - y -- auxiliary functions auxF :: (Fractional a) => a -> a auxF j = (1/2) * j * ( j + 1 ) getJ :: (Num a1, RealFrac a, Floating a) => a -> a1 getJ z = w where q = (-1/2) + sqrt(1/4 + 2 * z) w = fromInteger (truncate q ) -- End of Haskell script }}}
Let’s visualize the configuration which we’ve got so far at position #9:
y = 3 ‚is‘ the z of another such configuration and x = 0 looks like a dead end, cause split 0 = (0, 0) yields no difference anymore.
Moving from the configuration at position #9 to another such configuration (along the orange arrow), the z of the second configuration is now 3 as well: z = 3. The value 3 ‚is‘ the same as the connexion (x, y) — beneath the dotted line — i.e. the relation between x and y (at another position than #9).
Any such configuration consists of 4 elements: [x, y, f x y, z] with the pairing function f x y, called fusion, to give us the position which we mark by means of the binary digit 1 in the persistent binary string above.
Once more the configuration [x, y, z] represents a value f x y which is all of the values x, y, and z at once.
(Cf. *Monads as containers* page for the list monad and such nondeterministic computations)
Okay?! But… where does the z come from? To be continued.