this post was submitted on 10 Jun 2025
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But D&D uses Chebyshev distance, not Euclidean. No need for Pythagoras. And Pathfinder alternates between Chebyshev and Manhattan to approximate Euclidean.
There's no grid in the sky, though
Fair point. I actually don't know what, if anything, the D&D (or Pathfinder) rules say on this matter. I've always just treated it as a natural 3D extension of the 2D grid rules. If they're three squares in one direction, same square in the other, and 10 feet up, I'd treat that as 15 feet away because of Chebyshev rules.
I believe that's how it's handled in D&D too, or at least how my table has always done it. I meant more as a practical matter, you're very unlikely to have a vertical wall grid and some kind of stand of the correct height for your minis, so you can't just count squares like you would for horizontal movement. That's when the Pythagorean Theorem comes up in my experience.
In PF1e you'd still alternate between Manhattan and Chebyshev. I used to know the rules to that so well I'd run it without the book for reference.
As it happens I've just looked up the 5e rules for this for the sake of another comment, and their rules are that, like PF1e, if you go vertical, you follow the same rules (i.e., Chebyshev by default, optional alternating) as on a flat plain.
I've not looked up the PF2e rules, but I feel safe in assuming it's the same in this regard as 1e.