He might vary his path slightly to see different things, but he'd simply be make the circle bigger. He can never know what lay outside the circle with his given modus operandi. From my perspective, true knowing, is being what you know. Which implies a great deal on what is truly knowable. If you look at what we're used to here, we have belief and knowing implicitly understood in statistical terms.
Belief seems to be predicated on the existence of doubt.
True knowing has no constraints of doubt. To know is to be one with that knowledge. This from a mystic standpoint, is true faith. Faith is knowing. I propose the best way to do this is to know what one is after specifically, then solve the problem of achieving it.