Kazimierz Kuratowski

How do mathematicians define the threshold where a graph goes from flat to tangled? This leads us to one of my absolute favorite theorems in the entire book, Kuratovsky's Theorem from 1930.

Oh, Kuratowski's theorem is a masterclass in identifying the root cause of a structural failure.

It proves that there is a universal barrier to planarity, and it boils down to the presence of just two very specific, highly toxic substructures, the graph K5 and the graph K33.

Okay,

You have three houses, and you must draw a line from each house to a water plant, a gas plant, and an electrical plant without any lines crossing.

Which is famously impossible on a flat sheet of paper.

Kuratowski proved that a graph is planar if and only if it does not contain a subdivision, essentially a topological minor, of either K5 or K.3.

My way of visualizing this is to view these two subgraphs as structural knots.