Diagonal wonkiness scenarios

Ulorian said:

Does anyone see a hole in this logic?

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Yes (and note I'm poking holes in my own argument here, I hope this is taken as fairmindedness and not multiple personality disorder ). If you use that explanation, but then map out walls and obstacles on the battlemat using 90 degree angles and "true" gameworld measurements, then the battlemat doesn't accurately represent the gameworld. Someone standing in the middle of a square room in the gameworld might not be able to reach the corner of the room with their movement, whereas someone standing in the middle of a square room mapped on the battlemat would. In order to get the battlemat representation to perfectly match the real world situation regarding perpendicular walls, etc. you'd have to draw rooms that were a few squares wider at the corners than in the middle (like a 4 pointed star).

So yeah, the "circles" concept does generate inconsistencies with the battlemat representation of the imaginary world, but after having played WFRP for the last couple of years using battlemats and a 1-1-1 diagonal movement system (and squares rather than feet or yards as a measure), I can honestly say that these cases where the metagame recordkeeping doesn't mesh 1:1 with the measurements in the game world are so unusual and so insignificant to the overall narrative, that I've never noticed anything that would make me consider, even for a moment, using some other system. It's at least as accurate as using the 1-2-1-2 system with players who can't keep track of which diagonal they are on.

Okay, I asked someone who knows topology, and he says:

A hyperbolic surface is insufficient. You actually need a Minkowski space to break the triangle inequality (and whatever way you slice it, 4e does break the triangle inequality). Minkowski spaces exist in relativity--they're the 4D space implied by general relativity, so now 4E is using 4D geometry. The only issue is that in order for 4e to work with a Minkowski space, you have to be time traveling.

Since this is starting to break my brain, hopefully we can all agree that the hack required to actually make 4e geometry work perfectly is too complicated for anyone to use, especially compared to 1-2-1-2-1. However, for those willing to ignore the edge cases who have issues with 1-2-1-2-1 for counting or speed reasons, I can definitely see where 1-1-1-1 can help.

Rystil Arden said:

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.

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The goblins are laughing at the race-architect, because everyone knows that 45 degrees isn't a handicap. Make your angle > 90 degrees, and run the race again.

-Hyp.

Rystil Arden said:

Okay, I asked someone who knows topology, and he says:

A hyperbolic surface is insufficient. You actually need a Minkowski space to break the triangle inequality (and whatever way you slice it, 4e does break the triangle inequality). Minkowski spaces exist in relativity--they're the 4D space implied by general relativity, so now 4E is using 4D geometry. The only issue is that in order for 4e to work with a Minkowski space, you have to be time traveling.

Since this is starting to break my brain, hopefully we can all agree that the hack required to actually make 4e geometry work perfectly is too complicated for anyone to use, especially compared to 1-2-1-2-1. However, for those willing to ignore the edge cases who have issues with 1-2-1-2-1 for counting or speed reasons, I can definitely see where 1-1-1-1 can help.

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How do you cope with the idea that moving on a diagonal at all requires your character to shrink down to zero width (in the direction normal the velocity vector)? How else can you get to a diagonally adjacent square without entering any others? Correct me if I'm wrong, but since you actually enter the squares along the diagonal, you cannot stay in that collapsed 2 dimensional state, right?

Instead you must undulate between 2 and 3 dimensions. Right?

PS

Rystil Arden said:

(and whatever way you slice it, 4e does break the triangle inequality)

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This is what I was asking in my original post: can someone definitively prove that there is a triangle inequality in Ourph's model? Ourph seems to think so according to his last post, but can someone come up with a concrete example?

Storminator said:

How do you cope with the idea that moving on a diagonal at all requires your character to shrink down to zero width (in the direction normal the velocity vector)? How else can you get to a diagonally adjacent square without entering any others?

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This is a problem with the "occupation" rules, not the movement rules. IMO, moving diagonally between two diagonally occupied squares should carry the same penalties as moving through an occupied square. You choose which occupied diagonal square you're taking the penalties from and just go from there. This also solves the problem of enemies not being able to form a defensive line on a diagonal.

This has probably been mentioned somewhere, but:

If you don't want your fireballs to be firesquares, switching back to 1-2-1 will not solve any problems for you. You will also need to change the entire targeting mechanism for radius effects from 'target a square' back to 'target an intersection'. Otherwise you will still have 'firesquares', they will just be offset 45 degrees and look like 'fire diamonds'.

Ulorian said:

This is what I was asking in my original post: can someone definitively prove that there is a triangle inequality in Ourph's model? Ourph seems to think so according to his last post, but can someone come up with a concrete example?

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I don't think the triangle inequality applies here because the problem only arises when you fail to treat the battlemat as its own self-contained universe, where every measurement is relative to every other measurement. The problem comes in when you translate non-battlemat measurements from the gameworld onto the battlemat without first converting them. If a room is square on the battlemat, it would actually be round in the gameworld (i.e. all points on the walls would be equidistant from the center). However, most gameworld rooms won't be round, they will be square and drawing a square gameworld room on the battlemat, appropriately converted for battlemat geometry, would be cumbersome. So you're mixing geometries by drawing a square gameworld room as square on the battlemat, and that produces errors. The triangle inequality presupposes that all measurements of the triangle are based on the game geometry. However, in the case of the square room, the geometry of the hypotenuse is normalized for the battlemat and the geometry of the other sides of the triangle is normalized to that of the gameworld, which breaks the rules and makes all mathematical models void.

Ourph said:

I don't think the triangle inequality applies here because the problem only arises when you fail to treat the battlemat as its own self-contained universe, where every measurement is relative to every other measurement. The problem comes in when you translate non-battlemat measurements from the gameworld onto the battlemat without first converting them.

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This is what I meant by you thinking that there is a triangle inequality: the inequality appears when you don't draw the gameworld in battlemat terms. Since it would be unreasonable to expect people to translate square rooms, straight bridges, etc. into the battlemat geometry, the result is a triangle inequality.

Ourph said:

If a room is square on the battlemat, it would actually be round in the gameworld (i.e. all points on the walls would be equidistant from the center).

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... and this is what I meant by a concrete example. Thanks for filling in the blanks.

Even if you actually had a round room and properly translated it to a square battlemap, you'd still have triangle inequality issues.

Pardon the crudely drawn diagrams. In both diagrams, the square on the right represents the square D&D 4E battlemap (it represents multiple in-game 'squares' on a grid which isn't drawn). To the left is the supposed circle that represents said square. In both figures, 4E claims that the blue and red distance in the square are equal length, and the 'Circle Heuristic' claims that if I move everything onto the true circle, it will fix all issues and make the blue and red equal length.

In Figure 1, we see that the circle indeed fixes the problem. The fact that moving to the upper-left corner of the square takes the same distance as moving straight ahead is entirely accounted for by the circle. That was my first example.

Now try Figure 2 (my second example). The guy moving straight ahead moves the radius of the circle or half the length of the big square--sure, fine, no problem. But the guy moving diagonally 45 degrees up-left and then to the same point is not traveling the same distance on the circle either. Indeed, two sides of the triangle (in any non-Minkowski space) can never add up to be equal to the third side. This issue will crop up no matter what shape you use (though props for thinking of the circle, since it does fix the first-order issues, just not the triangle inequality).

To summarise: Even if your room is actually honest-to-goodness a circle that you portray as a square, the 4E diagonals still don't work mathematically past a first order movement. Or in other words, what 4E is doing is not just the same as transforming circles into squares. You still may be okay with this, and that's fine.