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This reminds me of a Putnam problem from a few years back, was something along the lines of:

Construct a set of discs in R^2 s.t. no infinite straight line can be drawn without intersecting at least one disc, and the sum of the areas of all the discs is finite.

Take a set of discs with radii 1/n. Their total area is finite but the sum of radii is infinite, so we can just use them to cover the X and Y axes. Yeah, this is pretty similar to Gabriel's horn.

(approximate) logarithmic spiral of discs whose radii are the harmonic sequence?

(Awarded 1 point of 10 for being unable to specify the precise piecewise-linear spiral function)

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