Originally posted by Astner
Comprehension failure.Example: Rectangle (i.e. not perfect cube)
Dimensions:
Length: 2 m
Width: .5 m (2/4)
Breadth: .25 m (2/8)
Volume: 1 * 0.5 * 0.25 = 0.25 m^3
We want the volume of the same type of rectangle with an arbitrary length L.
Apply the same calculation.
[Large (volume)] = [Small (volume)] * ([new length]/[previous length])^3 = 1/4 * L^3 / 8 [b]= L^3 / 2^5
Now lets control check:
[length]*[width]*[breadth] = L * L/4 * L/8 = L^3 / 2^5
Of course this works for any given function of x, y, z as explained beneath.[/B]
Several things:
1. Not a rectangle, but a rectangular cuboid.
2. Where are you getting your numbers from? Those seem arbitrary. Also, if those numbers are supposed to be used for Gama's sword, they are way off.
3. 1*.5*.25 does not equal .25*.25*.25. I don't even have to do that math on that to know that you're way off base, there. I have no idea what you were trying to accomplish, there. No rearranging of decimals or adding in a factor corrects for your .25 error.
4. You're volume measure is correct. If you run across a congruent sword that has a depth measure (because that would be the quickest and easiest way to measure it) of 100cm, it is 100^3 times voluminous (see, very easy to do). Simply multiply 1000cm (the small sword's volume) by 1,000,000 (the cube of the factor) to get the right answer of 1,000,000,000cm^3. If you don't believe me, here everything is, worked out, the long way: length 10000cm (100*100cm), width 1000cm (100*10cm), depth 100 (100*1cm). Volume = 1,000,000,000cm^3. That's just assuming a rectangular cuboid, though. It's not as accurate as other forms of measure, but it will be close than other things you could do. It's just faster that way.
5. Reading over your first post, I see that there is not even congruency with what you just posted, above. You originally posted this:
([Length of Gamabuta's katana]/[Length of a regular katana])^3 * [Density of a regular katana] = [Mass of Gamabunta's katana]
Then calculated this:
(2/.05)^3 * 2 = 250 000 kg (= 250 tons)
That has so many problems with it, it's ridiculous.
Here are all the problems:
a.)The Density of a regular katana is significantly greater than 2. It's closer to 7. Density of high quality Japanese Katana steel is probably above 8, or right below 8. Hilt will decrease the density, significantly, of course, but probably not much below 7.
b.) You incorrectly estimated the length of the sword as 2 meters. The length is significantly greater than that and any degree of miss-measure is very significant to the final outcome because the factor is cubed.
This scan here shows us a different story:
http://www.mangafox.com/manga/naruto/v19/c170/16.html
The top right panel shows a Tsunade, at an angle to the width of the blade (not a straight on perpendicular shot), has Tsunade almost standing all the way up. The next shot shows the sword's smallest length on (the depth of the hilt). It's even "longer" than Tsunade standing straight up (even account for a bit of her legs being cut off, due to the "upwards" angle of the "shot"😉. Tsunade is 163cm in height. I used a straight edge and measured, and came up the hilt being about 3 times as "wide" as Tsunade is tall. (I accounted for the angle and Tsuande's slightly bent knee. That's 163*3 = 489cm.
c.) Let's check out the triple integral.
Take rectangular cuboid "A" whose dimensions are (x, y, z): 4, 2, 3cm. Volume of A is 24cm^3.
Take congruent figure "B". We only know one side of B and that is z. It is 6. (simply by that, we know all of the other sides, as well: 8 and 4.) Using your math, that comes to:
24cm^3 (old volume) * (6(new length)/3(previous length)^3) =
8*24cm = 192
Does that check?
6*8*4 = 192
Yes.
But, wait...is that the same thing you were talking about earlier?
Let's check:
You said this:
Originally posted by Astner
([Length of Gamabuta's katana]/[Length of a regular katana])^3 * [Density of a regular katana] = [Mass of Gamabunta's katana](2/.05)^3 * 2 = 250 000 kg (= 250 tons)
No, that's not what you said earlier. You have the density of regular katana in there. Well, later, you did clarify that it was very simply the z dimension length, no doubt. So, I assume that that was just a mistake on your part and you actually meant the volume of the original object. In which case, you'd be correct (I"m a nice guy, giving you the benefit of the doubt.) However, there's still the problem with you concluding "mass" in there, totally throwing off your calculation, big time. I apologize if this comes off as rude, but, your math is totally off, there. I could understand if you did a stupid mistake, like I have, by reducing like terms in a force calculation, only ending up with weight as your final measure (trust me, I did that..forgot that my answer would be in 32.17lbs*ft/sec/sec). But this is not anything like that. It's so far out there that I have no clue what you were doing there.
However, I do have some dimensions of a bamboo "hilted" sword:
http://www.nihonzashi.com/shinken_bamboo_stick_katana.aspx
The nakago width is 2.8448 cm. Let's go with that as our measure for Gama's sword. However.....how the hell are you going to calculate the volume of your bamboo stick Katana? Oh, that's right, you'll have to do it my way. (unless you have one sitting around that you'd like to try a fluid (water) displacement measure with.) 🙂
Density? Easy. No problem. The metal is easy to figure out as we already have stuff on that. The bamboo is another problem because it's a treated nakago. Bamboo is about .97g/cm^3, untreated. Just guessing, but I'd put the treated blade at 3g/cm^3. Now we just got to figure out the volume of the hilt and then the entire blade. facepalm BTW, just looking on the internet...it looks like the average nakago "depth" is 17mm. That's the dimension you were trying to use, btw.
We can then use my way of gradually making things more accurate until we come up with a number that is close...without going too in depth. Or, dunk a blade (not preferable...unless you have one?)
Originally posted by Astner
Where x, y and z may be an arbitrary function (in other words, it also covers any other conceivable shape).For instance, apply spherical coordinates and:
x = r*cos(a)*sin(b)
y = r*sin(a)*sin(b)
z = r*cos(b)
And you'll get a sphere.
No, where x, y, and z are the Cartesian Coordinate system (to be read "x axis, y axis, and z axis.) I apologize for assuming too much as I thought everyone would know what I was talking about, especially since I clarified that just a few short sentences later.
You're way would have been this way: f(x), f(y), and f(z) (or simply f(x,y,z).
Originally posted by Astner
Once again, completely unnecessary. The formula above is derived from a triple integral that covers an arbitrary non-uniform distribution of mass over a three dimensional body.
No-no, that works. As I stated several sentences above, the measure works just fine for volume....but not mass like you tried to do.
Originally posted by Astner
Fact is, you're decreasing the accuracy of the volume by treating it as rectangles, like you do.
That's not what I was doing, btw. I was saying to use that as a starting point and then work your way down to something more accurate until you're satisfied with the volumetric results (in other words, you would get to some arbitrary point to where the volume decrease caused by adding another "edge" to the "curve", for the nakago, is too small to matter. That could be 10cm^3...it could be 1cm^3. (This is what computer animators do: the curves on their animations are actually polygons...with lots and lots of edges.)
Originally posted by Astner
Once again, it's accurate. At best your rectangles will come 90% to that of my result.
Yeah, cause a rectangular cuboid is exactly what I said you should use for the final volumetric estimate. 😐
I'll still draw pictures for you and do the measure my way....next weekend. I am out of time.
Just had an epiphany: for the hilt, it's as easy as find the radius for the curve on each side of the hilt. Then, calculate the volume of the newly formed "rectangular cuboid" left after calculating the volume of the "cylinder" and add the two volumes together. You'd only have to measure from the center of the hilt to the top edge, from the part the the hilt starts to curve. Then subject the double of that measure (cause that would be the diameter) from the top to bottom measure of the nakago/hilt to end up with the y coordinate for a the new rectangular cuboid. Again, just simply add the volumes of the two together to come up the measure. In fact, there should be a relational property between the top to bottom measure, and the radius measure: it should vary directly.