What does # cu. ft. even mean?

[jiriku]

Woot for calculators! I will readily acknowledge that I am a massive, massive gaming geek, but tech is really the way to go. I develop games at the PC, store monster stats as document files, and build dungeon maps as more document files, using drawing tools. Frequently, if there's a contentious ruling, I just pull up my browser (which is already loaded with an online DM screen), open another tab and surf to the Playground, and start a thread with my rules question. We can generally get 2-4 opinions on the subject before the encounter is even over.

When you're that extensively wired, finding the answer to a simple question is trivial (Dear Google, "What is the weight of one cubic foot of steel?").

[Vizzerdrix]

This topic makes my head hurt

What about a re write of the spell based on sizes instead of feet?

[rubycona]

This topic makes my head hurt

What about a re write of the spell based on sizes instead of feet?

Thing is, while "size" works great with things like, I dunno, an ogre, it's less fantastic with things like water. How much is a "Huge" or "Large" amount of water, as opposed to "small"?

Ultimately, it breaks down to this: there will be DMs/players who care, and DMs/players who don't.

The DMs/players who don't care will look at "2 cubic feet / caster level" and go, "whatever, let's say you can shrink this amount."

The DMs/players who Do care, and actually want to try using these kinds of spells in a more precise way... it is Only for these people that this thread exists. And if you want to take care of these things in a more precise way, then the volume measurement is better than the arbitrary "size" measurement.

No doubt plenty of DMs decided it was 20*20*20, and ran with it, some did fine, others thought it was broken and nerfed it. Simple, fixed, done.

But for those of us who don't mind a little number crunching (or saying Hi to our beloved friend and comrade, Google) when the situation requires precision, and who want to actually Have that precision, then the whole "how do you determine exact volume" thing comes into play.

Even as a general rule, though, knowing what a cubic foot IS (which the original poster and his Math major buddy made a common mistake with) really helps, even for hastily half-randomly made judgement calls.

So all in all, I find the cubic feet thing to be best, even without the aid of the computer/internets.

As a side note, I am utterly thrilled to actually have a thread to which I can make a solid contribution! Who'd have guessed that my college chemistry classes would help in a D&D discussion? (Ok, when put that way, maybe it's not so surprising >.>)

[jiriku]

As a side note, I am utterly thrilled to actually have a thread to which I can make a solid contribution! Who'd have guessed that my college chemistry classes would help in a D&D discussion? (Ok, when put that way, maybe it's not so surprising >.>)

If it is deadly to catgirls, then it must be good!

[Renchard]

I think this makes more sense for the layman in context of an example:

Say, wood. I typed in Google, "Density of Pine," and of the results, I picked 530 (which this site clarifies is, 530kg/cubic meter.) Well, we have our spell in cubic feet, not meters. So let's convert to cubic meters. Say, we've got 20 cu feet for our spell.

I plug that into Google (20 cubic feet to cubic meters), and I get 0.566336 cubic meters. So, I multiply 0.566366 times 530kg/cubic meter, which gives me the result in kilograms, or 300.158 kgs. Say you don't like kilograms, so type into Google, 300.158 kg to pounds, which gets 661.735 pounds.

That's a light wood. So let's try again, a heavier one. Let's go with Oak, upwards of 930kg/cu meter. Let's do our math here:

20 cu feet -> cu meters = 0.566336. Multiply that by the 930 we got from the search result, for 526.69kgs. Plug in 526.69kg to lbs and you get 1,161 lbs.

2 minutes on Google and you know you can shrink, as a 10th level caster, 661 lbs of pine, and 1,161 lbs of oak. Not bad! By 20th level, that doubles, more if you've got CL increases.

The fact that some people consider this par for the course for their gaming more than adequately explains the difference between the 3.5 and 4e camps.

[jiriku]

Heck yeah! 3.5 forever baby!

On a more realistic note, calculations of this nature mark you forever and anon as a powergamer. Regular gamers can get by quite fine by guestimating.

[TheMadLinguist]

"Okay, so I cast shrink item on the boat, but not the air inside."

"Your boat instantly shrinks to a tiny fraction of its former size. It then, having exceeded the speed of light, travels backwards in time, overlapping its past self. Nuclear fusion is initiated, resulting in the destruction of the boat and a large spray of radiation."

If you're bringing physics into it, you have to bring them all in

[rubycona]

Heck yeah! 3.5 forever baby!

On a more realistic note, calculations of this nature mark you forever and anon as a powergamer. Regular gamers can get by quite fine by guestimating.

Haha! Actually, I'm not a powergamer at all. I really suck at feat-item-class-etc combinations for brokenness. (One of my players, mind you, DOES powergame, and it drives me nuts).

I am a very thorough DM, though, in the sense of the world. In one game, we spent about an hour arguing about how long standard lantern oil would burn when spread on the cliff face. Man, we looked up everything... types of lantern oil, volume of a barrel, trying to figure out the surface area of the cliff based on size and roughness, the amount of spread and splatter... we almost got into the viscosity of the oil when I said, screw it! It's lasting this long. Conversation over. No more arguing with the DM.

So it's not just that I'm a numbers-intensive DM (though, I'm the worst of the lot). My players happen to be a group that drags me into it XD

[BobVosh]

Thing is, while "size" works great with things like, I dunno, an ogre, it's less fantastic with things like water. How much is a "Huge" or "Large" amount of water, as opposed to "small"?

Find the weight of the large, huge, and small water elementals. Find out how many gallons of water it takes to equal out these weights. Wa-la!

Probably the easiest way to do this is off weight rather than size anyway. So you can reduce 20 lbs. The object was 40 lbs. It is now half as large. Simplistic.

[Tetsubo 57]

For me this is a fairly clear cut issue. Did the spell say 20 cubic feet OR a 20' cube? If it's the former, it's 20 cf. If it's the latter it's a cube 20' on a side. And if someone is trying to argue the latter when it is saying the former, they are being a ****.

[Ravingdork]

Just thought of this, this morning:

Most everyone can agree that 1 cubic foot gives us a cube with 1 foot on a side.

12 cubic feet is NOT a 12x12x12 foot cube.

So answer me this: How come 12 cubic inches ISN'T a 12x12x12 inch cube?

Shouldn't 12 cubic inches give us the exact same cube as 1 cubic foot? Mathematically, doing it in inches gives us a much smaller cube. Why is that?

EDIT: Feel free to discuss the question, but I think I figured it out on my own.

[Philistine]

Just thought of this, this morning:

Most everyone can agree that 1 cubic foot gives us a cube with 1 foot on a side.

12 cubic feet is NOT a 12x12x12 foot cube.

So answer me this: How come 12 cubic inches ISN'T a 12x12x12 inch cube?

Shouldn't 12 cubic inches give us the exact same cube as 1 cubic foot? Mathematically, doing it in inches gives us a much smaller cube. Why is that?

Because you've got the order of operations wrong.

EDIT: Perhaps it would be more helpful to first look at the case of area, rather than volume. A square foot is an area that is 1'X1', or 12" by 12". That gives us 1 sq ft as equal to 144 sq in. To get volume we multiply again: 1x1x1 = 1, and 12x12x12=1728, and so 1 cu ft = 1728 cu in.

EDIT 2: To try to clarify further, the 1 cu ft = a cube with sides of 1ft length is due to a unique property of the number '1' - namely, that is the multiplicative identity. When you use any number other than one, that no longer holds.

[tyckspoon]

Just thought of this, this morning:

Most everyone can agree that 1 cubic foot gives us a cube with 1 foot on a side.

12 cubic feet is NOT a 12x12x12 foot cube.

So answer me this: How come 12 cubic inches ISN'T a 12x12x12 inch cube?

Shouldn't 12 cubic inches give us the exact same cube as 1 cubic foot? Mathematically, doing it in inches gives us a much smaller cube. Why is that?

For exactly the same reason that 12 cubic feet isn't the same size as a 12-foot cube. If you state it as 12-inches-cubed, then the units are equivalent, but if you're starting with a single cubic inch and then trying to stack together twelve of those to fill the same volume as a cubic foot.. well, you're doomed to failure.

[Philistine]

The long form of my previous post...

The original question in the thread was, "Why isn't '20 cu ft' the same thing as 'a cube with sides of length 20 ft'?" The subsequent question about the number of cubic inches in a cubic foot is essentially a rehash of the same issue, and the two can be addressed as one problem.

To begin, consider the general case of the expression (x * y) ^ n. Standard rules for the order of operations dictate that this can be expanded as (x ^ n) * (y ^ n); expanding it as x * (y ^ n) is not correct.

If we have a cube with sides of length (x * y), then the volume of that cube will be calculated as (x * y) ^ 3, or (x ^ 3) * (y ^ 3). Now take our cube with sides of length 20 ft, whose volume is calculated as 20 ft * 20 ft * 20 ft, or (20 ft) ^ 3. Next, we expand the expression 20 ft to 20 * 1 ft, and substitute it back into our volume calculation, which now becomes (20 * 1 ft) ^ 3. Referring back to the general case, the proper expansion of this expression is (20 ^ 3) * (1 ft ^ 3); and since (1 ft) ^ 3 = 1 cu ft, this gives us our volume as 8000 cu ft. (Similarly, a cube with sides of 12 in has a volume of 1728 cu in, demonstrating that the relation 1 ft = 12 in is only valid for linear measurements; it does not hold for measurements of either area or volume.)

The argument in the original post, that '20 cu ft' (which can also be written as 20 (ft ^ 3)) should be the same as 'a 20 ft X 20 ft X 20 ft cube', is only achievable as the result of using the incorrect expansion for the general case as stated above, as it requires you to cube only one of the factors in the expression.

That said, there is one exceptional case: when the length of a side is 1 ft (or indeed one of whatever unit you happen to be using), you can get away with doing things the wrong way around. Basically, because 1 * 1 = 1, you still get the correct volume even if you forget to cube the first term - if and only if the length of a side is exactly 1. But it is important to remember that this is a special case, not the general case.

BOOM! Mathematician'd!

[rubycona]

Accurate, high math stuff

While this may be correct, it bears the substantial problem of being very hard to understand if you're not familiar with the math already. I think I should try again to express this in simple terms, which should hopefully be understood without too much trouble.

Just thought of this, this morning:

Most everyone can agree that 1 cubic foot gives us a cube with 1 foot on a side.

12 cubic feet is NOT a 12x12x12 foot cube.

So answer me this: How come 12 cubic inches ISN'T a 12x12x12 inch cube?

Shouldn't 12 cubic inches give us the exact same cube as 1 cubic foot? Mathematically, doing it in inches gives us a much smaller cube. Why is that?

EDIT: Feel free to discuss the question, but I think I figured it out on my own.

12 cubic inches does not equal 1 cubic foot, because of the same thing I'd talked about already... 3D volume is an exponential (cubed, specifically) measurement, and straight distance (inches, feet, and so on) is linear.

Basically... inches/feet only refer to just one line, so if you increase the line, it just goes longer. Double the inches, and it's twice as long.

With 2D stuff, it goes longer AND wider. Double inches on both, and you double length AND width. Because you're increasing 2 measurements, not one, at the same time, it increases by squaring. (That's why a 2*2 square is 4 times as big, not twice as big as a 1*1 square)

With 3D stuff, it goes longer AND wider AND higher. Double inches (or feet) on all 3 measurements, and you increase it by cubing (specifically 2³, the 2 is because you've got twice as many inches, and the 3 because it's 3 dimensions, and 2³ is 8. That's why a 2*2*2 cube is 8 times as big as a 1*1*1 cube)


And finally, why 12 cubic inches don't equal 1 cubic foot. It's basically the same question you were asking in the first place (20*20*20 vs 20*1*1), but different terms.

Try to think of it in blocks. You've got a little block that's 1" by 1" by 1". You've got lots of them. Get 12 of them in a row, and it's 1 foot long, 1 inch high, and 1 inch wide, right? It takes all 12 cubic inches to increase 1 line of measurement to 1 foot.

[] -> [][][][][][][][][][][][]

How you do make a 1'*1' square with the little blocks? Well, you'll have to have them 12 blocks in another direction, too. Do that once, and you've basically got an L shape, and you've got to fill in all the empty space with more blocks.

[][][][][][][][][][][][]
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How many will it take? Specifically, 12², or 144 total (12 blocks long by 12 blocks wide = 12*12 = 144)

So just to make a flat 1 foot by 1 foot by 1 inch square out of 1 inch blocks takes 144 blocks, so 144 cubic inches to fill just the bottom of the cube you're going for. How many will it take to make it into a full 1 foot cube, or 1 cubic foot?

Just do the same thing again. You'll have to stack those blocks 12 high, and each layer will take another 144 blocks. So that means, it'll take a total of 12*144, which is 1728. Another way of saying it is 12 times 12 times 12 = 1728. Another way of saying it is 12³, which is also 1728. (Reason for the 12 is you've got 12 units. Reason for the 3 is you're increasing it in 3 dimensions, or directions (up/down, left/right, forward/back))

And that's why 12 cubic inches do not equal 1 cubic foot, because volume itself increases exponentially (3 directions), but when you add measurements together (like adding 12 cubic inches together) it adds linearly (1 direction).

So finally put, it's not 12 cubic inches to equal 1 cubic foot, it's 12³ cubic inches (1728) to equal 1 cubic foot.

To relate it to your original question, it's not 20 cubic feet to equal the 20*20*20 cube, it's 20³ (8,000) cubic feet to equal the 20*20*20 cube.

I hope I explained it in a way that a layman can follow. Sorry if it sounds patronizing to anyone, I'm trying to put it as simply as I possibly can. Hopefully, this post will clarify things once and for all.