I was looking at this neat page on logarithmic spiral tilings when a question popped up:

http://www.uwgb.edu/dutchs/symmetry/log-spir.htm

It seems that in all of the tilings shown, the area of each tile is exponentially increasing as a function of the distance to the origin. Are there any radial- or spiral-type tilings (or "tiling-like" configurations) in which the area of each tile is a *polynomial* function of the distance to the origin, say $r^{1/2}$ or $r^2$? I don't require similarity of shapes or anything, just a simple way to fill the plane with shapes possessing (or approximately possessing) this property. I guess it would be nice if the shapes stay reasonable rounded and convex (i.e. not really long and skinny).