This work is the first half of a diptych presenting an informal visual proof of the nonplanarity of the complete graph on 5 points (K5) as a metaphor for the inevitable emergence of Adversary within society.
First we form a new graph by adjoining a point to the complete graph on 4 points (K4), then demonstrate that it is impossible to complete that graph “in the plane”.
We shall present two separate proofs because K4 can be rendered two different ways without loss of generality, so we must cover each case.
Here, we begin by adjoining a single point to this first planar representation of the complete graph on 4 points (K4), and it quickly becomes apparent why it is impossible to complete this newly formed graph in the plane.
While every other line necessary to complete K5 could be moved outside of the pentagonal perimeter of this new graph, the perimeter line in red intersects another line that has necessarily been moved outside of the trapezoidal perimeter of the K4 subgraph already.
Thus we cannot complete the graph in the plane because it would require us to move that line back inside the pentagonal perimeter, which would necessarily intersect with a different line.
Therefore this representation of K4 cannot be adjoined with another point to form a planar representation of K5, providing the first half of our informal visual proof that K5 is nonplanar.
By Kuratowski's theorem, we also note that both of the only two nonplanar “forbidden graphs” first emerge as subgraphs of the complete graph on n vertices when n = 5 and 6 (K5 and K3,3 respectively).
As metaphor, this work explores the inevitable emergence of Adversary, as a concept, in complex dynamical systems of sufficient size, including our own society.
Adversary is represented by the fifth point because of its necessary implication of the Pentagram (with its historically infernal connotations) as a subgraph of K5, which is itself the first nonplanar “forbidden” graph we encounter when exploring the complete graphs on n points.
Nonplanarity is identified with Adversary as “that which cannot be made to lay flat or conform to patterns established prior”.
The fifth point’s apparent refusal to cohere with the established rule structure speaks to the emergence of the deviant, dissident, heretic, infidel, or maverick who cannot help but challenge the existing order plainly by the combinatorial nature of their relationship to the environment.
The mathematical discipline of Ramsey Theory teaches us that “complete disorder is impossible in sufficiently large systems”; however, it also informs us that our ideas of order come down to an aesthetic preference for one property over another, planarity being but one example out of many.
