Painter’s Paradox Reimagined

Consider the Coastline Paradox where the smaller the unit of measurement or ruler you pick out to measure the total length of the coastline of any landmass, the larger it gets. This occurs because of the fractal-like property of the coastline wherein the area bound by the curve or coastline is finite, but the perimeter or the length of the coastline approaches infinity. This is how Norway gets the second-longest coastline in the world due to its long and deep fjords placing it ahead of much bigger countries like Russia in terms of the length of coastline. In practice, the resolution of the GIS data determines the measurement of such geographical features.

Koch Snowflake

In the same manner, Gabriel’s horn, a geometry figure having infinite surface area but finite volume introduces Painter’s paradox where the horn could be filled with a finite quantity of paint and yet the paint would not be sufficient to coat its surface. To resolve the paradox there would be certain things to consider such as the limitation of a physical paint, its thickness, viscosity, flowrate etc or it can be avoided by considering a hypothetical paint for the sake of the problem.

Now let’s talk about painting the whole surface area of a country in a quest to determine the true surface area with all of its rich topographical features — mountains, hills, valleys, water bodies, and buildings. Unlike the length of coastlines, different sources providing data regarding the areas of the countries are rather consistent. The region is assumed perfectly flat and thus the area is calculated. Elevation changes are mainly disregarded in the calculation of the area as the effect of elevation difference is relatively small in the grand scheme of calculating the total area of the region. Say, for a 300 x 300 sq. km region a fairly high elevation difference of 3 km (9842 feet) i.e. 1% of the 2D stretch of the country appears to be pretty much negligible and as smooth as a baby’s bum. For the majority of the countries being relatively smaller, there is no substantial effect due to the roundness and the curvature of the earth. For mountainous countries like Switzerland or Nepal, a measurement resolution of 100 sq. m would probably only make 5% to 10 % difference in the land area.

Relief Map of the Indian subcontinent
Exaggerated Relief Map of the Indian subcontinent

However, in my little to no knowledge of geomorphometry, the actual surface area to be painted for a country, given all the mountains, troughs, buildings, and nooks and crannies — poses the same kind of paradox as the Coastline paradox does. Of course, the volume of the paint would be a finitely huge number but the painted surface area would approach an extremely unthinkable number if not infinity.

Consider this: you have a hill that is 1 km in diameter. When you measure it with a 1 sq. km resolution, it will look like 1 sq. km in area, but when you go to measure it with the 1 sq. m resolution, it will look like more than a million, because the sides of the hill have a slope to them. I believe, the economic surveys of smaller regions and the local geospatial data certainly take into account these elevation differences.

Mathematically, the shape of the terrain in a mountainous region would render a larger surface area due to its fractally rough and sloped surface and if we try to go microscopically to the atomic level taking into account all the flora, rocks, and soil particles not to mention the chaotic water bodies.

“As our circle of knowledge expands, so does the circumference of darkness surrounding it.”

— Albert Einstein

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