Seeing Then Believing

A good diagram is a precious thing. Words are amazing workhorses but reading never stops being that acquired skill a semi-arcane technology. There’s translation effort there. Pictures leap into your head as if they were born there and we have effortless recall. Data visualization is a powerful way of aggregating a information and I have great respect for those who can do it well.

The canonical example for most school kids is the inscribed squares that show the proof of the Pythagorean theorem. Geometry and topology are full of these moments to the point that other branches of mathematics deride them for practicing “proof by picture”. But most people will probably find them impenetrable so let’s look at some of Seth Kadish’s work.

I loved wandering the streets in London. My home town is small and most of the few streets fall into a grid. The jogs in the road are where it dodges around some bit of forest or hill. Very few alleys, very few surprises. London was nothing but narrow winding roads and surprises. There were streets running parallel to each other separate by a single row of shops but you wouldn’t know if you had not glanced down that alley two intersections back. Four right turns may not leave you facing your original direction. I’ll never complete The Knowledge but if I lived there, the urge to map the place would be irresistible.

Consider our expectations of an ancient country’s boundaries. At its smallest level, it would arise naturally from divisions between neighboring tribes and villages. Since these presume a single central point, they would appear as a rough circle with jagged edges for natural features that make easier delimitation marks. Aggregating several of these together will deform the circle we’d expect to still be able to discern some small number of fixed central points. In fact, the cantons of Switzerland form more-or-less these shapes. Thus we expect that for self-determined countries, the graphs in the visualization above would form roughly the same shape as the continent for which they’re drawn. The long outliers are indicators that decisions about a border was not made at the local level. Assuming a continuous habitation of the entire landmass, this implies that people were not given a choice about which country they would join and so were never given the chance to agree to that countries version of a social contract.

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