Deviation Actions
This delightful little thing is from engineering! So, as an engineer myself, I am very happy to say that I think it’s named horribly. It’s actually a mathematical law, but it got its current name because of a specific engineering application. In my opinion, it should be called the Linear-Quadratic-Cubic-Quartic law, though I’d settle for just calling it the Quadratic-Cubic law instead if it really came to it. I’ll explain why.
Say I have a box. This box is 1 unit by 1 unit by 1 unit, a perfect cube. The units don’t matter, be they inches or kilometers or bunny-hops, as long as they’re all exactly the same. Now, I take this box, and I make it three times bigger. Use the scale up tool, cut it larger, splash it with my giantess potion as a test, whatever. The point is, the box ends up three times larger on a side than it started.
So, I take out my ruler, and it’s now 3 units by 3 units by 3 units. Great so far, it got 3 time larger. A scale factor of 3 made distance increase by a factor of 3, a linear relationship.
Say I want to paint the box, though. Now I’d need to know the surface area, and that’s a bit different. For simplicity’s sake, we’ll say I only want to paint one face of the box. Now, I know how to figure out how much paint I need, because the area of a single side of a box is easy to calculate. Height times width, 3 times 3, and I find that it’s 9 square units. Now, hold on. I only scaled the box up 3 times, but I need 9 times the paint? This is because area, and surface area as well, has a quadratic relationship with scale. Scale it 3 times, and area increases by 9.
Okay, okay. So, now I want to fill the box. And you, being smart and all, stop me before I go on. “I get it,” you say, “volume is found by multiplying height by width by length, so the box is 3 times 3 times 3, or 27 times larger. I suppose this is a ‘cubic’ relationship – because it’s what happens to cubes.”
“But, then, what I don’t understand is why it’s called the square-cube law.” You say, being very perceptive and all. ‘I mean, dropping the quadratic bit I get, because it makes it less catchy, but they’re missing the whole linear term!”
I would tell you that this is where the engineering comes in. You see, when you scale things up, there are only really two things an engineer cares about (a mechanical engineer, for clarity. Other engineers would care about other things). They would care how much the material weighs, and they would care about how strong the material is.
“Now, hold on.” You say. “I’m no fool. I know that mass and volume are linked – something with twice the volume will have twice the mass. So, mass must be the ‘cube’ part of the law. If the only other thing that matters is strength, and the only other word in the law is ‘square’, then strength of material must be based on area!”
And you’d be exactly correct, as it happens. So, as a material increases in scale, it becomes heavier at a cubic rate but only stronger at a quadratic rate. It becomes heavier faster than it becomes stronger. Thus, something scaled up without changing materials becomes weaker in comparison to its original form.
“Okay. So, I get why you don’t like the name. It only refers to what is going on to one aspect of a thing. The law really applies to anything to do with area or volume, and it’s scaled using a linear factor. I’m with you so far. But, what on earth is the quartic part for!”
Good question, non-existent person that I’m still using as a framing device for some reason! This is actually a thing I noticed that most people left unaddressed, and it is inspired by giantesses directly! The thing is, we all know that giantesses don’t get weaker as they get bigger. They stay the same relative strength, no matter the size! But, they can’t be becoming tougher arbitrarily – imagine if your skin suddenly became twice as tough. You’d notice. It would get stiff and weird on you. No, they have to be getting tougher at the exact rate that they’re getting bigger, so that they always feel like nothing is happening. The difference between a quadratic and a cubic function is a linear factor, so a giantess becomes tougher at a linear rate.
Combine a linearly increasing toughness with a cubically increasing volume that has to get damaged, and you end up with the fact that a giantess becomes tougher to kill at a quartic rate! Scale her up 3 times, and she’s already 81 times harder to kill. 10 times bigger, and she’s 10,000 times harder to hurt. A relatively small girl, one just 30 meters tall, would be almost 100,000 times tougher to injure, and would already be able to brush off most sub-nuclear weaponry. It turns out, the ‘giantess can’t be injured by any human weaponry’ thing is actually realistic – required, even, by the necessary secondary powers that lets them exist in the first place.
This effect is because force increases by a quartic value, and damage is related to force. The force of something is equal to, (thank you newton,) mass times acceleration. Mass increases by a cubic term, acceleration increases by a linear term, so force increases by a quartic term. A girl 10 times larger is 10,000 times harder to hurt, and also 10,000 times as strong. Good luck with that.
Linear Quadratic Cubic Quartic
1 ---- 1 ---- 1 ---- 1
2 ---- 4 ---- 8 ---- 16
3 ---- 9 ---- 27 ---- 81
4 ---- 16 ---- 64 ---- 256
5 ---- 25 ---- 125 ---- 625
6 ---- 36 ---- 216 ---- 1296
7 ---- 49 ---- 343 ---- 7401
8 ---- 64 ---- 512 ---- 4096
9 ---- 81 ---- 729 ---- 6561
10 ---- 100 ---- 1000 ---- 10,000
15 ---- 225 ---- 3375 ---- 50,625
20 ---- 400 ---- 8000 ---- 160,000
25 ---- 625 ---- 15,625 ---- 390,625
50 ---- 2500 ---- 125,000 ---- 6,250,000
100 ---- 10,000 ---- 1,000,000 ---- 100,000,000
[n] ---- [n^2] ---- [n^3] ---- [n^4]
