# What direction is a shrinking cube going + logical problem

1. Jul 16, 2013

### Caveat

- Is the "center" of an object an entirely logical construct?

I draw a circle, find it's center and mark a point there. This point, no matter how small I mark/draw it takes up space (both physically and visually in my mind). Anything that takes up space has a center point, and this process goes on infinitely

If I apply maths to this question, it tells me the center point doesn't even exist. For example, if the diameter of a circle is 1, then it's center point would be at 0.5, but adding 0.5 and 0.5 (radius) equals 1, so where does this center point lie along it's diameter?

- Does the distance between the faces of the cube and it's center point decrease as it shrinks?

Firstly, why isn't the center point of the cube part of the cube?

Assuming the entire cube is shrinking would suggest that even it's center point would shrink with it (if that makes sense), that way the cube would shrink infinitely and the distance between its faces/vertices and center would remains in equal ratio

Another way to think of this is to imagine a man at the center of a perfectly square room but for every step he takes towards one of it's walls he shrinks to half his size. Notice that what determines when he shrinks is his steps, which is a subjective form of measurement since they decrease WITH him, so he never reaches the walls of the room?

Last edited: Jul 16, 2013
2. Jul 16, 2013

### Staff: Mentor

The center of an object is a mathematical concept. What do you mean with "logically construct"?

Your mark of this point takes up space, the point itself does not.
Exactly in the middle, of course.
Sure
It is inside the volume of the cube.
A point has no (non-zero) size, it cannot shrink.
There is no meaningful ratio you could calculate.

3. Jul 16, 2013

### Caveat

Sorry that was a typo I meant "logical construct"

How can the point not take up space then? this is where I'm confused, you're saying the point doesn't exist? please explain this in layman's terms

I'm sorry, but that doesn't make sense to me? Put it this way, if we go back to the shrinking cube, at what distance does one of the faces meet the center of the cube? can't be 0.5, because that's when the face meets the opposite face. Assuming the faces even meet is presuming space is quantified/finite

Why is it exempt from shrinking then?

Physically, that makes no sense. That's the reason I asked if it were a logical construct, because how can something have no size?

It would just remain 1:1 I suppose

Last edited: Jul 16, 2013
4. Jul 16, 2013

### Staff: Mentor

What do you mean with "logical construct"?

Where is the problem? It is a mathematical concept, not a physical object.
Its existence is similar to the existence of the number 3. The concept of the number 3 does not need any space either.

At a side length of 0, exactly when the opposite faces meet.
No.

Shrinking reduces all lengths. A point has no length (or 0, if you like, and 0*anything=0 where anything is your scale factor).

A point is not a physical object.

A ratio of what? 1:1 means the same size.

5. Jul 16, 2013

### Caveat

Something like Zeno "half of half" paradox, where the center of an object is never reached because we'd have to find the "center of the center of the center etc", so it turns into a logical construct that requires us to suspend certain modes of reasoning

I did some googling about zero dimensional stuff and I finally understand what you're referring to, but I'm still wondering if a physical description of the center of an object makes any sense

Exactly, it doesn't really shrink at all and that's the paradox. We could only tell that it were shrinking in a relative sense to us

^for example, the guy walking towards the wall and shrinking in relation to the wall but not to himself (his parts)

Last edited: Jul 16, 2013
6. Jul 16, 2013

### DrewD

This is a bit of a tangent, but perhaps it will help convince you that a point takes up no space. To be simple, just think about the normal number line. Take the point zero. This is certainly a valid place on the number line. So your question translates into, "what length does zero have?" Assume (to the contrary) that zero has some finite length. Then we can draw (not necessarily physically, but in our mind) a line from one end to the other and that will have length $X$. But, then this line will extend to $X/2$ and $-X/2$ because I have assumed that the point zero should be centered at zero and half the length should be on either side. But this is all clearly impossible because, no matter how small we choose $X$ (literally ANY choice of X), it will be a number greater than zero and therefore cannot be part of the "point" zero.

While it seems that a point should have some size, this is actually does NOT make sense for a point to have a size.
The paradox is found if points DO have size. If every point had a size, every line would be infinitely long since every line (even finite ones) have infinitely man points. If each point were $y mm$ long, then the line would be infinitely long.

Also, this is a math question not a physics question.

7. Jul 16, 2013

### Caveat

Don't get me wrong I fully understand how a point takes up no space as more of a location, my problem is that I took a literal interpretation of what a point was, but I get it now

What my problem was akin to is that in reality I suppose the true "center" of an object makes no sense

8. Jul 16, 2013

### DrewD

Geometrically, it does make sense. There may or may not be a physical particle at that point, but center of an object is as well defined as the object is. If you can define the distance between the edges of the object, the center is halfway between (not so simple in more than one dimension).

There is a difference between something not making sense logically and not being a physical object that can be held. If that's what you mean, then you are right, but I would be more careful with the phrase "makes no sense". A lot of important concepts in physics are not made of physical objects that you can hold.

Also, there is a solution to Zeno's Paradox since each length is traverse in half the time (assuming constant velocity) and the full distance is then traversed in finite time. The paradox only arises if each length is traversed in the same amount of time... but that means that you are slowing down, so it isn't really a paradox, you just slowed down.

Again, this is math not physics.