The randomization is to prevent bias in the sampling, as they illustrate with the grid. They then overlay random "grid shapes" to roughly approximate a sphere.

The randomization is to prevent bias in the sampling, as they illustrate with the grid. They then overlay random "grid shapes" to roughly approximate a sphere. I can't prove it right now, but I'm betting that if you extended this approach to infinite hashes you would eventually get a perfect circular fallof. So, the randomization is not the only way to prevent bias: you can also just not have it to begin with by cutting to the chase and using circular falloff to begin with. But what I'm asking is how would one implement such a thing.