So I was looking at the steepness measure and calculation stats today and something really bothered me. I found that Devils Tower is 40th on the list. How can something that is nearly straight up on all sides be so low on the steepness list? The answer (and I could be wrong on this) appear to be because the steepness measure is taken from a point, and a defining characteristic of Devils Tower is that it has a summit area. This effectively means that when calculating steepness we are increasing our horizontal value associated with that angle of steepness. The end result is a diminished steepness measure for anything that looks less like a spire, and more like a mesa. However, few would argue that something that something like Devils Tower is like a spire in that there is no way to get up it easily (that is, there is no real line of weakness for accent).
A further disturbing point is that it seems much harder to come up with a quantitative way of fixing such an issue (assuming one considers this an issue). My own method for calculating steepness that I had hoped to develop using PDEs and the Finite Difference Method, does not solve the issue either. Unless, I somehow cut out the summit area from the discretization. But to define a summit area is a tricky task and would likely require and individual review of every summit that might fit the definition.
I've really run into a wall here. Most of you probably don't care, but this is really bothersome to me. Comments?