We explain the BETA rewrite, as a limit of the SHUFFLE rewrite, which appeared for the first time in Pure See, Emergent rewrites.
The SHUFFLE rewrite in hexagon form is:
which can be drawn as a graph rewrite:
The nodes depicted in red depend on a parameter which can be made to converge to infinity, as explained in Pure See, Reduction by passing to the limit. Then the hexagon form under this limit becomes:
which can be drawn as a BETA graph rewrite:
But then we can represent the same rewrite in the following form:
and we remark that the blue arrow with the BETA name can be completely replaced by the graph of the node "5"
After all this tells us that the BETA rewrite itself has this component which sits somewhere in between the two frames (before and after, left hand side and right hand side of the rewrite).
Thus the rewrite is embodied by this graph...