Consider any graph with trivalent nodes from the eight ones of the permutation cube (A, L, D, FOX, FI, FOE, S, N), as well as 1-valent nodes like I, K, T, and with free half edges which will receive 1 valent FRIN or FROUT nodes.
Denote such a graph by G.
Equivalently, thing about G as being a mol file. Or, following the conventions of Pure See, think about G as being a list of Pure See commands, because each of the trivalent nodes A, L, D, FOX, FI, FOE are in correspondence with a permutation of (from, see, as) and S is a fanout and N is a fanin.
Now, let’s make all recursive, or perhaps fractal?
Suppose that the free half edges of G are decorated with 3 tags: from, see, as.
So, from far away, G is like a trivalent node, provided we group the free edges by tag.
The next thing we do is to put a mask on the graph G with decorated free edges.
Each such mask is a permutation of three elements (from, see, as).
We write:
G.from a G.see b G.as c
or simply
G a b c
to describe a node which is G, whose “from” decorated half edges are the 1st port, which is connected with the rest of the world with the edge with label “a”, whose “see” decorated half edges are the 2nd port, connected with the edge with label “b”, etc
As in Pure See, we may have
G.see b G.from a G.as c
which is denoted by
(1-G) b a c
Why?
Because now we have a node G whose 1st port is the “see” decorated half edges and the second port is the “from” decorated half edges, otherwise all the rest is the same.
So we permuted the ports by using the permutation (213) or if you wish (see, as, from).
This corresponds by the isomorphism of the group of permutations of 3 elements with the anharmonic group, to the element of the anharmonic group
1-z
hence the notation
1-G
Read Space, combinators and life for a more detailed explanation of the correspondence.
All in all we thus may have
G, 1-G, 1/G, (G-1)/G, G/(G-1), 1/(1-G)
which are our numbers! Along with 0, 1 and infinity, which are related to termination nodes (with masks) as explained in Pure See.
Then we may multiply two numbers G H, take differences G – H, or other algebraic operations of these, as described here.
We can generate integers (which are not quite the Church integers, but alike) and fractions, like 2 and 1/2. And -1.
This is basically what is described in em-convex, except that here we use a graph G instead of a generic “z” or “epsilon”.
Mind that we don’t use the rewrites of Pure See, only the notation!
So this is the description of numbers which I use. It is, relative to Pure See, fractal or recursive, because we start from a pure graph which is a D node, then we mask it and we produce the other 5 nodes, then we produce larger and larger graphs by this number mechanism.
All are nested from… see.. as.. like statements, or mol files with numbers like notation for the type of the node.
chorasimilarity 