Greywander
2022-05-27, 08:00 PM
Well yes but actually no but actually yes. This is a discovery I made while thinking about size after posting the "Gargantuan isn't really that big" (https://forums.giantitp.com/showthread.php?645929-Gargantuan-isn-t-really-that-big) thread.
As you probably know, the size rules for 5e have some quirks to them. We're given a few general rules, and a couple exceptions, but the underlying math isn't really explained. Fortunately, I've figured it out and explained it before.
I'll spare you the math, but the square-cube law (https://tvtropes.org/pmwiki/pmwiki.php/Main/SquareCubeLaw) states that if we double a creature's size (that is, double all three spatial dimensions), it becomes 8 times heavier but only about 4 times stronger. Now, D&D's size rules don't care so much about weight, but they do have a relationship between size and strength. Specifically, increasing a creature's size category both increases their carry weight and increases the area they occupy. Does this match with the square-cube law? Let's find out.
A large creature occupies a 10x10 foot square. A gargantuan creature occupies a 20x20 foot square. There's a bit of ambiguity here, particularly in regards to height, but I think most of us can agree that a gargantuan creature is at least "roughly" twice the size of a large creature. According to the rules, a gargantuan creature can also carry 4x as much as a large creature with the same STR score. So it seems that it does follow the square-cube law, at least as far as size and strength are concerned. There's some nuance here, particularly since a creature's own weight doesn't encumber themselves, whereas the entire point of the square-cube law is to show that things eventually get so big they can't support their own weight. But it's close enough, all that's really happened is that some of the math has been simplified a bit.
So we see that going up two size categories will both double your size and quadruple your carry weight. It's easy to infer that going up a single step will increase your size by √2 and double your carry capacity. We can see that this (roughly) holds true for huge as well. A huge creature should and does have twice the carry capacity of a large creature, and should occupy a space that is a 10 * √2 foot square, or a square slightly larger than a 14x14 foot square. Now, since 5e uses 5 foot squares as its basic unit, it makes sense to simply round up to a 15x15 foot square, which matches what we see in the rules. This type of rounding ends up being the cause of some of the oddities we see in the size rules.
This also kind of holds true for tiny as well. Due to some special exceptions made for the small size, it's not completely true, but we can still see the essence of the square-cube law here. Increasing one step from tiny to small does indeed double our carry weight as we would expect. And increasing two steps from tiny to medium does indeed double our size as we would expect. The only reason why the square-cube law isn't 100% accurate here is due to some of the special exceptions made specifically for the small size. But I'll get back to that in a bit.
Because the small size isn't the only oddity here. As we've established, increasing size by one step should multiply our size by √2, and it should require two steps to double our size. And yet we can double our size in a single step by going from medium to large. It would seem that there's actually a size between medium and large that is missing from the game, one that should be about 7x7 in size (exactly half the size of huge). Now, why this size is missing is actually pretty straightforward. D&D uses 5 foot squares as its basic unit, and a 7x7 creature is too big to fit in a single 5 foot square. So we round up to a 10x10 foot square. But now there's not really a meaningful distinction between the missing 7x7 size and the large 10x10 size, except as it relates to the grappling and carry weight rules. Such distinctions weren't considered worth tracking, so the missing size (which I have referred to as "big" before) was simply merged into large. (As an aside, I find this a shame, because this "big" size would have been perfect for races such as centaurs and goliaths.)
You should now be able to guess what happened with the small size. Well, except for the carry weight. I'm pretty sure small and medium creatures have the same carrying capacity because they didn't want different PCs to use different calculations. Small and medium are "PC sizes", which is why you've never seen a large or tiny playable race, so the differences are downplayed to make them more similar. But that's not actually why they both occupy a 5 foot square. I mean, it might have played a role, but I think the true reason comes back to rounding up. See, going up one step from tiny, or down one step from medium, should give a size that is roughly 3.5x3.5 feet. But obviously that's too big to fit more than one small creature into a 5 foot square, so again, it rounds up.
But what if I told you that wasn't actually true?
Okay, fine, if we throw two halflings into a 5x5 foot pit, there's no way they can position themselves so that their 3.5 foot squares aren't overlapping. But let's say we had an infinite featureless plain with an infinite number of halflings. We want to pack those halflings as close together as we can without their squares overlapping. This won't line up perfectly with the 5 foot grid, but what would be the average number of halflings per 5 foot square? Well, it turns out that you can fit exactly two halflings per 5 foot square.
3.5 feet is actually an approximation. Let's start with a tiny space, which is exactly 2.5x2.5 feet, and then increase in size by one step. This means our size increases by √2, so each side of our square is now 2.5 * √2 feet long. So what's the area of our square? Well, it's going to be (2.5 * √2) * (2.5 * √2) square feet. We can rearrange this to (2.5 * 2.5) * (√2 * √2). When you multiply a square root by itself, you just get the number that it's a square root of. So √2 * √2 is just 2. This means our area comes out to (2.5 * 2.5) * (√2 * √2) = (6.25) * (2) = 12.5 square feet. And what's the area of a medium square? It's 5 * 5 = 25 square feet. So a medium space has twice the area of a small space, meaning we can fit exactly two small creatures in a medium space.
But what do we do with this forbidden knowledge? Nothing good, that's for sure. And yes, that's a SsethTzeentach reference.
It's true that there's no way to actually arrange two small creatures so that they can both fit inside a single medium space, but then again size categories are already abstractions. There's not really a reason we couldn't just allow two small creatures to occupy a medium space together, in much the same way that up to four tiny creatures can. This then adds a lot of interesting ways to play a small creature that can help balance out some of the disadvantages of being small, particularly if you decide to nerf their carrying capacity to bring them back in line with the square-cube law.
However, while thinking about this very topic, I made another interesting discovery. Because it turns out we can perfectly represent the space occupied by small, big, and huge creatures, exactly, not with approximations. Representing it numerically without rounding or using a square root is impossible, but representing it geometrically is actually really easy. We use diamonds instead of squares.
So let's divide size categories into "even" and "odd" sizes. "Even" sizes are ones with an exact numerical representation with no rounding or square roots; so tiny, medium, large, and gargantuan. Even sizes occupy a square of the specified size, e.g. a medium size is 5x5 feet. "Odd" sizes would need to use a square root in order to have an exact numerical representation; so small, big, and huge. Odd sizes occupy a diamond that fits into a square one size larger.
For example, a small creature occupies a diamond that fits exactly into a medium square. The corners of the diamond touch the midpoints of the sides of the square. The diamond doesn't fill the whole space, in fact, it only occupies half the space, with each of the corners of the square being left unoccupied. An easy way to visualize this is to imagine a four halflings arranged in a 2x2 group (that's 2x2 squares, not feet). Each one occupies a diamond at the center of a 5x5 foot square, but between the four of them is an empty space that's just big enough to fit a fifth halfing between them. Instead of sitting on the center of a square, this halfling is sitting on the corner between four squares. This allows our halflings to pack a little closer together without letting two of them occupy a single 5x5 square, and it represents their true size exactly, which is the really neat part for me.
The reason this works is because the diagonal of a square is equal to √2 times the length of that square's side. As we previously established, the length of the side of a small space is 2.5 * √2 feet, and by fitting a diamond inside a 5x5 foot space, each side of that diamond is the diagonal of a 2.5x2.5 foot square.
Something to note about this is that a huge creature now occupies a diamond that fits inside a 20x20 foot square. But you'll notice that the 5x5 foot squares at each corner of the 20x20 foot square are actually totally unoccupied! This means that there's no reason you couldn't move into those spaces when surrounding a huge creature.
Now, I suspect the ideal way to handle this, particularly if using a VTT, is to actually use circular tokens that can snap to a grid in a couple different ways. The circles of the tokens would then likely be inscribed inside of the square or diamond, e.g. a medium token would be a circle with a diameter of 5 feet, while a small token would have a diameter of roughly 3.5 feet. You could arrange small tokens on the grid such that you could fit five of them inside a 10x10 foot area, making them function like diamonds, but visually there would be no difference between diamond and square tokens, and a huge creature could fit into either a 20x20 foot square as a diamond or a 15x15 foot square. With a pure diamond shape, huge creatures would find it difficult to fit through 15 foot wide hallways, since the diamond fits inside a 20x20 foot square, but the circular token would allow it to fit.
Anyway, I just thought this was all really interesting. Not sure how useful it actually is, but I will definitely consider using some or all of this in homebrew and/or an original system.
PS. Do the random bolded words help with readability? It does feel a bit weird sometimes, like I'm verbally stressing the bolded words as I say them, and sometimes it sounds awkward. It kind of reminds me of old comic books that would often bold words for seemingly no reason. But I think it does help with readability, which is why I continue to do it.
As you probably know, the size rules for 5e have some quirks to them. We're given a few general rules, and a couple exceptions, but the underlying math isn't really explained. Fortunately, I've figured it out and explained it before.
I'll spare you the math, but the square-cube law (https://tvtropes.org/pmwiki/pmwiki.php/Main/SquareCubeLaw) states that if we double a creature's size (that is, double all three spatial dimensions), it becomes 8 times heavier but only about 4 times stronger. Now, D&D's size rules don't care so much about weight, but they do have a relationship between size and strength. Specifically, increasing a creature's size category both increases their carry weight and increases the area they occupy. Does this match with the square-cube law? Let's find out.
A large creature occupies a 10x10 foot square. A gargantuan creature occupies a 20x20 foot square. There's a bit of ambiguity here, particularly in regards to height, but I think most of us can agree that a gargantuan creature is at least "roughly" twice the size of a large creature. According to the rules, a gargantuan creature can also carry 4x as much as a large creature with the same STR score. So it seems that it does follow the square-cube law, at least as far as size and strength are concerned. There's some nuance here, particularly since a creature's own weight doesn't encumber themselves, whereas the entire point of the square-cube law is to show that things eventually get so big they can't support their own weight. But it's close enough, all that's really happened is that some of the math has been simplified a bit.
So we see that going up two size categories will both double your size and quadruple your carry weight. It's easy to infer that going up a single step will increase your size by √2 and double your carry capacity. We can see that this (roughly) holds true for huge as well. A huge creature should and does have twice the carry capacity of a large creature, and should occupy a space that is a 10 * √2 foot square, or a square slightly larger than a 14x14 foot square. Now, since 5e uses 5 foot squares as its basic unit, it makes sense to simply round up to a 15x15 foot square, which matches what we see in the rules. This type of rounding ends up being the cause of some of the oddities we see in the size rules.
This also kind of holds true for tiny as well. Due to some special exceptions made for the small size, it's not completely true, but we can still see the essence of the square-cube law here. Increasing one step from tiny to small does indeed double our carry weight as we would expect. And increasing two steps from tiny to medium does indeed double our size as we would expect. The only reason why the square-cube law isn't 100% accurate here is due to some of the special exceptions made specifically for the small size. But I'll get back to that in a bit.
Because the small size isn't the only oddity here. As we've established, increasing size by one step should multiply our size by √2, and it should require two steps to double our size. And yet we can double our size in a single step by going from medium to large. It would seem that there's actually a size between medium and large that is missing from the game, one that should be about 7x7 in size (exactly half the size of huge). Now, why this size is missing is actually pretty straightforward. D&D uses 5 foot squares as its basic unit, and a 7x7 creature is too big to fit in a single 5 foot square. So we round up to a 10x10 foot square. But now there's not really a meaningful distinction between the missing 7x7 size and the large 10x10 size, except as it relates to the grappling and carry weight rules. Such distinctions weren't considered worth tracking, so the missing size (which I have referred to as "big" before) was simply merged into large. (As an aside, I find this a shame, because this "big" size would have been perfect for races such as centaurs and goliaths.)
You should now be able to guess what happened with the small size. Well, except for the carry weight. I'm pretty sure small and medium creatures have the same carrying capacity because they didn't want different PCs to use different calculations. Small and medium are "PC sizes", which is why you've never seen a large or tiny playable race, so the differences are downplayed to make them more similar. But that's not actually why they both occupy a 5 foot square. I mean, it might have played a role, but I think the true reason comes back to rounding up. See, going up one step from tiny, or down one step from medium, should give a size that is roughly 3.5x3.5 feet. But obviously that's too big to fit more than one small creature into a 5 foot square, so again, it rounds up.
But what if I told you that wasn't actually true?
Okay, fine, if we throw two halflings into a 5x5 foot pit, there's no way they can position themselves so that their 3.5 foot squares aren't overlapping. But let's say we had an infinite featureless plain with an infinite number of halflings. We want to pack those halflings as close together as we can without their squares overlapping. This won't line up perfectly with the 5 foot grid, but what would be the average number of halflings per 5 foot square? Well, it turns out that you can fit exactly two halflings per 5 foot square.
3.5 feet is actually an approximation. Let's start with a tiny space, which is exactly 2.5x2.5 feet, and then increase in size by one step. This means our size increases by √2, so each side of our square is now 2.5 * √2 feet long. So what's the area of our square? Well, it's going to be (2.5 * √2) * (2.5 * √2) square feet. We can rearrange this to (2.5 * 2.5) * (√2 * √2). When you multiply a square root by itself, you just get the number that it's a square root of. So √2 * √2 is just 2. This means our area comes out to (2.5 * 2.5) * (√2 * √2) = (6.25) * (2) = 12.5 square feet. And what's the area of a medium square? It's 5 * 5 = 25 square feet. So a medium space has twice the area of a small space, meaning we can fit exactly two small creatures in a medium space.
But what do we do with this forbidden knowledge? Nothing good, that's for sure. And yes, that's a SsethTzeentach reference.
It's true that there's no way to actually arrange two small creatures so that they can both fit inside a single medium space, but then again size categories are already abstractions. There's not really a reason we couldn't just allow two small creatures to occupy a medium space together, in much the same way that up to four tiny creatures can. This then adds a lot of interesting ways to play a small creature that can help balance out some of the disadvantages of being small, particularly if you decide to nerf their carrying capacity to bring them back in line with the square-cube law.
However, while thinking about this very topic, I made another interesting discovery. Because it turns out we can perfectly represent the space occupied by small, big, and huge creatures, exactly, not with approximations. Representing it numerically without rounding or using a square root is impossible, but representing it geometrically is actually really easy. We use diamonds instead of squares.
So let's divide size categories into "even" and "odd" sizes. "Even" sizes are ones with an exact numerical representation with no rounding or square roots; so tiny, medium, large, and gargantuan. Even sizes occupy a square of the specified size, e.g. a medium size is 5x5 feet. "Odd" sizes would need to use a square root in order to have an exact numerical representation; so small, big, and huge. Odd sizes occupy a diamond that fits into a square one size larger.
For example, a small creature occupies a diamond that fits exactly into a medium square. The corners of the diamond touch the midpoints of the sides of the square. The diamond doesn't fill the whole space, in fact, it only occupies half the space, with each of the corners of the square being left unoccupied. An easy way to visualize this is to imagine a four halflings arranged in a 2x2 group (that's 2x2 squares, not feet). Each one occupies a diamond at the center of a 5x5 foot square, but between the four of them is an empty space that's just big enough to fit a fifth halfing between them. Instead of sitting on the center of a square, this halfling is sitting on the corner between four squares. This allows our halflings to pack a little closer together without letting two of them occupy a single 5x5 square, and it represents their true size exactly, which is the really neat part for me.
The reason this works is because the diagonal of a square is equal to √2 times the length of that square's side. As we previously established, the length of the side of a small space is 2.5 * √2 feet, and by fitting a diamond inside a 5x5 foot space, each side of that diamond is the diagonal of a 2.5x2.5 foot square.
Something to note about this is that a huge creature now occupies a diamond that fits inside a 20x20 foot square. But you'll notice that the 5x5 foot squares at each corner of the 20x20 foot square are actually totally unoccupied! This means that there's no reason you couldn't move into those spaces when surrounding a huge creature.
Now, I suspect the ideal way to handle this, particularly if using a VTT, is to actually use circular tokens that can snap to a grid in a couple different ways. The circles of the tokens would then likely be inscribed inside of the square or diamond, e.g. a medium token would be a circle with a diameter of 5 feet, while a small token would have a diameter of roughly 3.5 feet. You could arrange small tokens on the grid such that you could fit five of them inside a 10x10 foot area, making them function like diamonds, but visually there would be no difference between diamond and square tokens, and a huge creature could fit into either a 20x20 foot square as a diamond or a 15x15 foot square. With a pure diamond shape, huge creatures would find it difficult to fit through 15 foot wide hallways, since the diamond fits inside a 20x20 foot square, but the circular token would allow it to fit.
Anyway, I just thought this was all really interesting. Not sure how useful it actually is, but I will definitely consider using some or all of this in homebrew and/or an original system.
PS. Do the random bolded words help with readability? It does feel a bit weird sometimes, like I'm verbally stressing the bolded words as I say them, and sometimes it sounds awkward. It kind of reminds me of old comic books that would often bold words for seemingly no reason. But I think it does help with readability, which is why I continue to do it.