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> why does the plane look so uniformly covered?

It's a purely perceptual effect. As you track away from the center of the spiral, your eyes are following lines, which are one-dimensional, but you're tracking the lines through a plane, which is two-dimensional. The area increases rapidly as you move away from the center of the figure, but the one-dimensional lines don't have area. This gives the lines an apparent weight that they don't actually have.



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