I'm looking for a spiral that satisfies a seemingly simple condition.

Consider an arbitrary point u on the spiral. Let the Euclidean distance between the origin and the point u be d. Let the arc length of the spiral between the origin and u be l. Then l = 2

^{d}.

The moral of the story is that if I cut across the arms of the spiral, I'll get to my destination exponentially faster than by following the rules of the road and riding along the arms.

Does this beast have a name? Has it already been under the scrutiny of countless mathematicians? Does it have a "nice" formula? Or perhaps have I uncovered something so ugly that no one wants to look at it?

The furthest I got in determining the formula for this thing was to write down this awful parametric differential equation with parameter u:

Trouble is, I never did learn how to solve these things, and I have no idea where to start. Perhaps parametric is the wrong way to go about this? I can't really figure out how to write the above in polar notation either though. I just need a nudge in the right direction, but won't complain if I get a treatise