Two types of spirals here.
One way to approximate a logarithmic spiral is with evenly space rays, having the almost curve turn at a right angle to each ray. (Here's a GeoGebra sketch of that idea: http://www.geogebratube.org/material/show/id/14470) This sketch is a generalization of that idea, allowing any angle between rays instead of just divisors of 360 degrees.
The other way is a generalization of the ancient construction of Theodorus, who made a spiral by adjacent right triangles, the first with legs 1x1. This has the wonderful property of the second triangle being root 2 by 1, and then root 3, root 4, etc. See Jennifer Silverman's sketch comparing it to an large artwork: http://www.geogebratube.org/material/show/id/43788. This sketch generalizes it by allowing you to make the original triangle any proportions.
Are the Theodorish Spirals approximations of log spirals? What do you think and why?