D&D 5E Square grid, what's the difference between a square and a circle?

From the blue square, a character with a 20 ft range weapon can hit all the white squares and a character with a 20 ft move can move to all of the white squares. Should a fireball dropped on the blue square affect all the white squares? Note: I know that fireball is a point-centered, which doesn't match the diagram, so imagine this a thunderwave or similar.



Or, to put it more simply, is the area of effect of a fireball a square?

I know there are other ways to deal with this (like hex grids or no grids) but I'm curious to get the community's view on this for square grids.

You are correct; a circle of radius r centered at the origin in this geometry is a square of side length 2r+1. In 4e, this is exactly how it worked. For convenience, it uses chessboard/Chebyshev distance (the L_∞ metric) and foregoes a correction for off-axis movement.

The long and short of it is that you cannot have all of the following:
A: discretized, integer-indexed space ("a grid")
B: completely covered space (all points are contained in, or at the edge of, a tile)
C: movement without imperfect/complex conversions (e.g. no "2 diagonal squares = 3 horiz/vert squares")
D: movement that preserves uniform velocity (e.g. moving diagonally, north/south, and east/west are all equally 'fast')

3e chose to take points A, B, and (essentially) D, and thus dropped C; you have to perform a calculation to account for the 'inefficiency' of moving diagonally. However, this calculation is imperfect; it makes every diagonal square count as 1.5 units, when it should be sqrt(2) = ~1.414, thus it technically violates point 4 as well. This effect is essentially imperceptible on the scale of typical battles, but if you assume that this is true, then there is a difference in efficiency between travelling diagonally and travelling partially by straight line and partially diagonally.

Each of the other requirements follows from the act of trying to discretize space. Real, physical space is not discrete (and no, quantum physics doesn't show it's discrete at the quantum level either, but that's an unrelated subject). We live with the Euclidean (L₂) metric, where there is one unique shortest distance between two points, which requires solving a right-triangle to calculate and in all but a subset of cases (that is, Pythagorean triples) is an irrational number. Irrational numbers cannot be handled by a discrete, integer grid placed on space, so you have to "break" something.

Even hex grids aren't immune. They meet points A, B, and C perfectly, but they fail at point D. Depending on how you orient the hex grids, moving north/south will either be slower or faster than moving east/west--it depends on whether your hexagons have vertices on their north/south bisection, or their east/west bisection. Travelling NE or SE will never be less efficient than travelling due north/south or due east, but may be more efficient. Again, though, these considerations are small for the areas typically used by tabletop battle maps, which is why people like the hex grid, though it makes "square" structures a little awkward. A game like Civilization or Age of Wonders, where you may cover hundreds of hexes exploring or attacking, it adds up to enough to be worth caring about.

Because squares are simple to work with, and because people think in perpendicular lines better than they think in lines at 60-degree angles (by a small margin), 4e and games like it eschew being particularly "accurate" to real-space geometry. They value simplicity of representation over physical-world accuracy, assuming that it is an acceptable break from reality to make the game easier to run and adjudicate.

If you really want to blow your mind, you should check out the Manhattan distance (the L₁ metric). In that, a circle of radius r centered at the origin is a "square," rotated 45 degrees (so that its "vertices" are on the axes), with 2r tiles on each "side." (I used quotes on "square," "vertices," and "side" because it's not a true square: it has jagged edges).

It's the difference between 4 and Pi

Square AoEs are a sensible house rule for massively speeding play if you use grids but the rule in the book is circles, including eyeballing whether at least half of a grid square is covered.

You know when I want combat faster I want things like square fireballs not the removal of interesting options.

Pickles III said:

You know when I want combat faster I want things like square fireballs not the removal of interesting options.

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Well-said, and completely agreed. I'd rather have a convenient, but "real-world" incorrect, shorthand than a perfect physical simulation that takes significantly longer to evaluate.

You can easily convert to radias combat, but it has to be everything not just AoE spells. Circle spell on square grid means you have partial squares, so you would have to account for those in the rules. You could use a pixel circle, but that is complex and doesn't add much for the trouble

Areas, positioning, and movement are all highly abstract anyway. An orc is not really standing still in one square and waiting for its turn to come up before moving. A Fireball is not really a homogeneously filled, perfectly rounded sphere. Square AoE effects are fine with me because it's all just a rough approximation anyway. Compared to how ridiculously abstract turn-based combat itself is, this issue doesn't really matter.

[MENTION=84122]leinadvirgo[/MENTION] Rules do account for spheres on the grid.

On the Grid variant, fireball is a sphere, centered on a grid intersection, affecting the squares that are at least 50% covered by the sphere.

Thunderwave is a cube, and I'd say the cube has to be aligned with the grid.

Scorpio616 said:

[MENTION=84122]leinadvirgo[/MENTION] Rules do account for spheres on the grid.

On the Grid variant, fireball is a sphere, centered on a grid intersection, affecting the squares that are at least 50% covered by the sphere.

Thunderwave is a cube, and I'd say the cube has to be aligned with the grid.

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I had thought that that is what Leinadvirgo meant: you can do a "pixellated circle" with whatever square-inclusion rules you prefer (the above image is difficult to say for sure, but it looks like the "at least 50% covered" rule), but that is fairly significant work for...what, you don't hit 12 out of the 64 squares you "could" have hit with a "square" fireball? Is that 20%-25% difference really going to matter in the grand scheme of things?

EzekielRaiden said:

I had thought that that is what Leinadvirgo meant: you can do a "pixellated circle" with whatever square-inclusion rules you prefer (the above image is difficult to say for sure, but it looks like the "at least 50% covered" rule), but that is fairly significant work for...what, you don't hit 12 out of the 64 squares you "could" have hit with a "square" fireball? Is that 20%-25% difference really going to matter in the grand scheme of things?

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I'd say so, 20' radius spells are potent enough as it is, giving them an area boost just because folks are playing on a grid is like giving an iron club to an oni.

Turning it into a firebox also stifles creativity and intelligence play as that robs the characters from airbursting spherical areas to reduce area size at ground level

Scorpio616 said:

I'd say so, 20' radius spells are potent enough as it is, giving them an area boost just because folks are playing on a grid is like giving an iron club to an oni.

Click to expand...

Do you really find yourself in situations where those extra 12 squares are going to add substantially more targets, without adding more allies? I find that...frankly difficult to believe, except in rare/unusual circumstances. The smaller cases (e.g. things with only a 5' or 10' radius) have a much more dramatic difference percentagewise and are much more likely to end up filled/spanned...but even for those, it feels like just a lot of work for so little "gain."

Plus...if we're talking a design consideration here...just design the spells to use the simpler-to-run method in the first place, and there's no "potency boost" in the first place.

More or less, a preoccupation with geometric precision--"cones," "circles," etc. that are always a compromise to begin with--strikes me as being exactly the same kind of thing as a preoccupation with "weapon speed" or "iterative attacks." Yes, in the games that were designed with those in mind, just dropping them for a simpler alternative will probably make something go pear-shaped. But if we're talking about our "preferences" for how a game does something, my preference is that the designers just cut out the largely-superfluous geometry, timing, and bonus-variation, and make a system that works properly without needing those things.