I guess you could describe the area as how much paint it would require to “fill” a shape while perimeter would be the effort it takes to “draw” said shape. Say you had a square that was 10×10 meters. Now say you cut out a 1×1 meter square from one side and pasted it to another. Your area remains 100 m, you have not technically added anything that wasn’t already there, but your perimeter has increased from 40 m to 44 m. You can do this indefinitely with smaller squares but the area will always remain the same.

When measuring a coastline, the little nooks and crannies can drastically increase the perimeter of your coast, sometimes increasing it multiple fold. Obviously in real life, you can only measure to a certain degree, but “coastline paradox” probably stuck better than “fractal curve paradox”.

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