Hypothesis: On fast, undulating fairways, the player is often left with an uphill lie for his next shot.
Let's say you are playing golf in Bandon or Scotland, or one of the great new sand-based designs under construction. The fairways are firm, fast and undulating.
I don't have mathematical proof, though one is probably findable. Common sense tells me a common result for a struck golf ball is to finish on the upslope, leaving the player with a uphill lie. Balls that hit the flat or downslope areas race forward until slowing to a stop on a uphill slope. Balls that strike the uphill slope bounce straight up in the air and stay there.
I wonder if there is any compiled data on this subject. It just occurred to me one day walking around Ballyneal that I might end up hitting shots all day long of an uphill lie. I think it's the one shot you've got to have to succeed at a course like that. I'm not very good at that shot; it's easy to hit a pull hook off that lie, and difficult to transfer your weight forward as you finish the swing. Last time in Bandon, I had some luck taking an exaggerated open stance and playing to fade/slice, which resulted in some nice straight approaches.
I'll restate: On firm, undulating fairways, a disproportionate percentage of balls result in an uphill lie. Has this been considered by architects? Should we expect this, or does the architect accommodate this fact by shaping the holes to yield a more balanced set of uneven lies?