Volume???

[ConfusedKid]

Hi,

 

My name is Aimee and I have a question about volume.... basically how do you work it out? If you have two differnet objects then how would you work out the difference (in %) of volume between the two???

 

Thanks

 

 

Aimee

[Dave]

This is quite an interesting question. Since you posted in the mathematics forum, I'm assuming you want a maths-style answer, so I'll attempt to give you one

 

There's a number of ways that one might define an area/volume. Probably the most intuitive is an area of mathematics called measure theory, originally created by Lebesgue at the turn of the 20th century.

 

One object that we have a clear definition of area/volume for is a rectangle (or oblong in three dimensions). It's clear that if we have a rectangle [math]R = \{ (x,y) \ | \ a \leq x \leq b, c \leq y \leq d \}[/math] with [math]b > a, d > c[/math] then its area is simply given by (b-c)*(c-d).

 

Now, the problem of measuring the area of any set can be approached as follows. We cover for the set by a union of disjoint rectangles. Then we look at smaller and smaller covers and try to find the greatest lower bound of the area.

 

In more mathematical notation, we define [math]\mu^* : \mathbb{P}(\mathbb{R}^2) \to \mathbb{R}[/math] by:

 

[math]\mu^*(A) = \inf_{\cup_k R_k \supset A} \left\{ \sum_k (b_k - a_k)(d_k - c_k) \right\}[/math]

 

where [math]R_k = \{ (x,y) \ | \ a_k \leq x \leq b_k, c_k \leq y \leq d_k \}[/math]. This can be easily extended to calculate volumes of sets in [math]\mathbb{R}^3[/math].

 

I'm terribly sorry if you don't get this - I'll try to provide a more intuitive post if so!

[the tree]

Generally, there is an equation for the volume of any simple geometric shape and more complicated shapes can be derived as sum of the simpler shapes that they're made up of. I think that is what Dave was getting at, that any shape's area can be calculated by reducing it to lots of rectangles. Although, like the Trapezium Rule in integration (exactly like, I guess) isn't it a little crude?

[Dave]

Generally, there is an equation for the volume of any simple geometric shape and more complicated shapes can be derived as sum of the simpler shapes that they're made up of. I think that is what Dave was getting at, that any shape's area can be calculated by reducing it to lots of rectangles. Although, like the Trapezium Rule in integration (exactly like, I guess) isn't it a little crude?

 

Essentially this is the idea. You cover your shape by rectangles, and make an approximation. However, the inf term I stated above is the greatest lower bound. So the answer is exact; it's sort of like taking a limit.

[Sayonara]

Or, if you don't like maths, you can simply take a leaf out of Archimedes' book and measure how much water the two objects displace.

 

http://www.juliantrubin.com/bigten/archimedesprinciple.html

 

Eureka!

[Dave]

True However it won't work so well if you're trying to measure the volume of, let's say, a Sierpinski pyramid.

[Sayonara]

Context selection Dave!

 

Expect a username change request from "ConfusedKid" to "BewilderedKid" any hour now"

[Dave]

Context selection Dave!

 

Expect a username change request from "ConfusedKid" to "BewilderedKid" any hour now"

 

My money's on "CompletelyConfusedKid"

[the tree]

Essentially this is the idea. You cover your shape by rectangles, and make an approximation. However, the inf term I stated above is the greatest lower bound. So the answer is exact; it's sort of like taking a limit.

O.k. then, so it's an idea about how it could be done but not a very serious idea then. I presume you'd never use that if you actually did want to know the volume of something, it's just some very interesting theory.

 

Oh, and since a Sierpinski pyramid involves essentially taking chunks out of a regular pyramid every iteration, wouldn't a true one have a volume of zero? (if it makes sense to have volumes of fractals, I'm not sure it does)

[Dave]

More generally we're interested in something like the Hausdorff dimension for a fractal, but I shouldn't derail the topic. It does make sense to have some notion of volume,

 

It's not meant to be a practical idea (I assumed, as the OP posted in Maths a physical answer would have been a bit pointless). But it does form part of an extremely important theory of measures, which we can use to define integrals and a whole bunch of other stuff.

[the tree]

Okay, but by Lebesgue mesurement would I be correct in thinking that the area of a Sierpinski triangle is zero, because you couldn't find a solid rectangular area that wasn't full of triangles cut out of it, now matter how small you wanted the rectangle to be? (I have no idea how to say that formally, I apologise).

Wikipedia says that a Sierpinski carpet has an area of zero, so I'm assuming that the triangle also does.

 

I guess it might be an idea to wait for the OP to ask something more specific.

[Dave]

Yes, you're correct. I can do a simple demonstration with something like the Cantor set if people are interested.