and plane graphs.
Our interest in them is disproportionate.
Any class bipartitions the universe; few so violently.
Yet some constraints only become meaningful when everything is forced to appear at once.
Graphs do not inhabit space; they are equivalence classes of adjacency.
Yet, when embedded into dimensions, some give particular dissonances.
The failure is combinatorial.
On the presence of or or similar.
On , invisible. On , boring.
Planarity survives because it fails loudly. One crossing suffices. As does one minor. The violation is simultaneous and shallow.
Two dimensions are not special; only unforgiving. Below them, structure trivializes. Above them, it hides.
Euler’s appears. Edge density stabilizes. Separators shrink. Duals emerge.
None of these belong to the graph; all of them persist after the embedding disappears.
This preference is not mathematical. It is epistemic.
We return to objects that tolerate total exposure. We distrust those whose properties require routing, depth, or deferred interaction.
Planar graph theory records one such boundary. The boundary generalizes.