Filling different shaped volumes

[Hoku]

Ok, I need some math info and I think this is the right section for my question, but I may be wrong. Is it physics? General math? I don't know.

I'd like to know if you can determine the shape of a volume by the rate differences with which it fills up.

Let's say you are filling a ball with water. The ball is marked at regular intervals from the bottom to the top. If you want the rate at which the water level reaches each marked interval to remain constant, you must increase the amount and/or speed of the water entering as the water level approaches the "equator", right?

So, if you didn't already know you are filling a ball, is it possible to deduce that it IS a ball by the rate of increase if you know that you are maintaining the "interval constant"? I'm sure it is possible. I guess I'm just looking for a formula or something for how to figure out the shape. Maybe even just a specific topic that I can get a book on to figure it out.

I hope my question makes sense.

 

[rasmhop]

No it's not possible. What you know is the surface area at all points (actually its derivative, but since you know it at all points it's the same).

Consider for instance a 1x1x1 cube. And instead consider a box with bottom 2 x 0.5 and height 1. These two containers are different, but they fill up at the same rate.

In general if the rate of fill-up is $\rho(h)$ at height h you can construct a solid which at height h has area $1 \times \rho(t)$. Such a solid also fills at a rate of $\rho(h)$.

 

[Hoku]

Thanks rasmhop! You're right, if you're filling a volume that has all opposite sides parallel, then you can't determine the exact shape, but you CAN determine that all sides are parallel. So you still learn something about it's shape.

I think it get's a little easier if you're filling a different shaped volume, like a pyramid or a ball. In these cases, I think you can differentiate which is which based on the changes of the rate of the fill.

 

[rasmhop]

Thanks rasmhop! You're right, if you're filling a volume that has all opposite sides parallel, then you can't determine the exact shape, but you CAN determine that all sides are parallel. So you still learn something about it's shape.

I think it get's a little easier if you're filling a different shaped volume, like a pyramid or a ball. In these cases, I think you can differentiate which is which based on the changes of the rate of the fill.

I'm not sure what you mean by all opposite sides parallel. You could also construct a cylinder with radius $$\sqrt{1/\pi}$$ and height 1. That would fill at the same rate as the two containers I mentioned. You could also have something that transforms from being circular at the bottom to square at the top (but it would be a bit harder to describe).

In general the only information you get from the rate of change is the cross-sectional area at all heights. If two constructions have the same cross-sectional area at all heights, then they fill up at the same rate. This is the information you gets.

 

[Hoku]

In general the only information you get from the rate of change is the cross-sectional area at all heights. If two constructions have the same cross-sectional area at all heights, then they fill up at the same rate. This is the information you gets.

Yes! This is what I'm looking for. By piecing together each cross sectional area you can determine if the shape is a cube/cylinder or if it's a pyramid or a ball. Various shapes of cubes and cylinders can be mixed and matched without knowing the difference but you can't really do that for a pyramid or ball. With a pyramid or ball you'd see the alteration.

So is there a formula that you can plug rates of change into to determine something about the shape?

 

[Ben Niehoff]

No, because the only information you get is the cross-sectional areas at different heights. You get NO information about the shape of those cross-sections; only their areas.

So for example, you can't tell the difference between a square pyramid of base A x A, and a cone of base radius $A/\sqrt{\pi}$. You also can't tell whether the pyramid "twists" as it rises, or if it shears to one side. You also can't tell whether, at some point partway up, the cross-section of the shape changes discontinuously from a square to a circle of the same area!

 

[Hoku]

Interesting. I know that you are both far more knowledgeable than I in the math department. That's why I'm asking. I appreciate that you are both saying it isn't possible, although it really surprises me. I'd like to understand this better. Maybe I'm not asking the right questions.

You both suggested that you can go from a square to a circle without knowing it, but I'm pretty sure that only holds true for the first transition between the two. The more cross-sections you add, the more you can't hide the differences...

If we look at the cross-sections of a cube or a cylinder we would find the ratio between each cross-section to be 1/1 all the way up. Right? Not so with a ball or a pyramid. The first cross section of a ball is not at a 1/1 ration with the second because the second cross-section is larger.

I would expect the ratio between cross-sections of a ball to be constant except for the swap after they pass the equator. I would expect a pyramid to also have a constant ratio between each cross-section, but I would expect that ratio to be different from that of a ball.

So, unless you have a really odd shape that is made up of all sorts of different shapes at each cross section, you should be able to find a pattern that is consistent with one of these shapes. Right?

 

[rasmhop]

You both suggested that you can go from a square to a circle without knowing it, but I'm pretty sure that only holds true for the first transition between the two. The more cross-sections you add, the more you can't hide the differences...

Remember that a square is two-dimensional surface (and in this case the cross-section) not the whole figure itself. Similarly a circle is two-dimensional while the 3-dimensional equivalent is a ball.

If every cross-section is a circle of the same radius r, then we get a cylinder, but imagine constructing the following solid:
Take a cylinder of height 1 and radius $$\sqrt{1/\pi}$$. On top of it place a box of height 1 and base 1x1. Then for heights between 0 and 1 our cross-section is a circle with area $$\pi\sqrt{1/\pi}^2 = 1$$. The cross-section for heights between 1 and 2 is a square of area 1. Thus the rate of fill-up is 1 at all heights from 0 to 2. Therefore this is indistinguishable from a box of height 2 with base 1x1. This is what is meant by going from a circle to a square in a non-continuous manner.

So, unless you have a really odd shape that is made up of all sorts of different shapes at each cross section, you should be able to find a pattern that is consistent with one of these shapes. Right?

Yes, but the problem is that you can't rule out all shapes unless you severely limits the accepted constructions. For instance if you say that the only allowed constructions are cubes and balls, then you will always be able to identify the exact solid (if it has a constant rate of fill-up it's a cube, otherwise it's a ball). However when you add a few more permissible constructions you will have several shapes it could be. Consider for instance a ball in xyz-space (3-dimensional space) where I take z to be the direction of height. Then I could scale the ball with a factor 1/2 in the x-direction and a factor 2 in the y-direction. This would preserve the rate of fill-up.

Along another line consider the unit-ball centered at 0 (i.e. the ball of radius 1). Then if we consider the cross-section circle at height h we know from the Pythagorean theorem that it has radius $$\sqrt{1-h^2}$$ so it will have area $\pi-\pi h^2$ (here h of course lies between -1 and 1 since our ball has radius 1). Thus we will not be able to distinguish this from another solid with the same property. We could for instance choose to at height h let the cross-section be a square centered around the z-axis with side-length $$\sqrt{\pi-\pi h^2}$$. Whether you allow such configurations are of course up to you.

As the problem is given in the original post the rate is far too little information to completely classify the solid, but it will often be enough to rule out certain solids.

 

[Hoku]

As the problem is given in the original post the rate is far too little information to completely classify the solid, but it will often be enough to rule out certain solids.

I've definitely gained some interesting insights from the discussion thus far. You're right, rasmhop, the problem as was given in my original post is insufficient. I was trying to get specific information for a problem that I was stating too generally. Hopefully it's not too late to salvage this.

I'm trying to solve a problem but don't have the knowledge of which equations I need to solve it. All I want to know is, 1) if the problem can be solved and 2) what equations or principles do I need to know to solve it...

The problem relates to the "shape" of our expanding universe, based on two assumptions: 1) that spacetime existed before the big bang, and 2) that the big bang happened at the edge of spacetime and not the middle.

We know that the universe is exanding at an increased rate. So, if there were some sort of law requiring the exploded matter to fill spacetime from the one edge to the other at a constant rate, then, if spacetime were a type of rounded shape, the matter in the universe would have to expand at an icreased rate as it approached the equator. This scenario would make matter in all directions appear to be going farther from all other matter, as is the case.

So, assuming such a constant existed, can we approximate the shape of this spacetime by knowing 1) how old the universe is, and 2) the rate increase of its expansion?