I understand the circle abstraction that you are using. It happens to occasionally make the 4e stuff seem to work out if you use it.
You have to use the circle abstraction to get it to work for the race scenario, but the fact that it works there doesn't mean that it actually corrects for all 4e geometry problems (edit: I call them problems, but they aren't problems for everyone, and indeed, the HEad-in-the-Sand argument of just ignoring them will work fine for many people. Perhaps you would prefer to call them 'idiosyncrasies'. What I'm really saying is that just calling the squares a radial circle sweeping out from the origin is insufficient as a topological correction)--I believe that the only way to completely correct for it is to consider the world to be on a hyperbolic surface rather than a flat surface (though I haven't tested whether that will fix everything yet). For example, the circle abstraction fails to fix the following race, no matter what you do:
The PCs go straight forward some distance r (r being the radius of our circle we're sweeping out from the origin) to a goal. Let's say the NPCs have a faster move speed, so they get a handicap--they have to run a distance r/2 in a 45 degree angle away from the goal, then turn directly towards the goal and run the remainder of the distance. The NPC will always win, even though the PC is running in a straight line and the NPC isn't. No abstraction will fix the fact that there are a variety of equidistant paths from Point A to Point B.
But since arguing topology isn't really fun for most people, let's assume that the circle somehow did work. At that point, we're left with:
That's only if you consider the battlemat as an accurate representation of the imaginary universe. It's not. It isn't really meant to be. It's a placeholder, a form of notation to keep track of the rules aspects of combat. It doesn't need to correspond 1:1 with the narrative in order to be useful. The fact is, squares in 4e are actually circles, each square on the battlemat represents one 5ft circle. The "leftover" space between those circles doesn't exist in the game universe, it's an artifact of transferring an imaginary world onto an imperfect, real-world game-aid, nothing more.
and
If it were merely an aesthetics issue, why did you bring up the race scenario as an example of how 1-1-1 diagonal movement doesn't work? It seemed to me, from that example, you weren't really understanding the underlying assumptions that make the race scenario perfectly workable with the new system.
The key misconception you have (again, assuming that the circle heuristic always worked) is again a failure to be generous in understanding how other people think differently from you. For you, the circle idea is so ingrained as being okay that you couldn't possibly see how that could
be an aesthetics issue in and of itself, perhaps even
the aesthetics issue. Instead, you were forced to leap to the conclusion that I didn't understand the circle heuristic, etc. It's not that I don't know--it's that in full knowledge of everything you know (but with different tastes, my opinion is still not yours.
No, you stop jumping through philosophical hoops to insist there are implications. The battlemat is a representation of the rules, it doesn't map the gameworld or the real world, in any meaningful way. You're creating your own problem here.
But people think differently, all depending on perspective and frame of reference. You see me jumping through hoops, whereas in my frame of reference, I'm standing still and you're jumping through hoops. I totally believe you that you see yourself as standing still because I know that these things work. And anyway, we're both jumping through hoops in some third frame of reference.
