To place objects equidistantly on an Archimedean (arithmetic) spiral, the arc length of the spiral has to increase linearly between the objects.
This is what I have so far: The length of a spiral is determined by $ l = \frac{a}{2}\left[\varphi\cdot\sqrt{1+\varphi^2}+\ln \left(\varphi+\sqrt{1+\varphi^2} \right)\right] $ I presume that solving this equation for $\varphi$ will give me what I need. But trying that with WolframAlpha leads to a timeout.
Is solving this equation for $\varphi$ really the right thing to do? If yes, how can I solve it?