The Flat Sphere......?

I would say infinity. As the sphere increases in size to infinity, its curvature drops to zero.

But somehow I feel like I'm missing something here. Some other thoughts occur to me: if the universe is not infinite and is positively curved, then perhaps the solution would be that the sphere is the same size as the universe. Just like a circle going around the Earth with the same radius as the Earth (the equator, for example), would be considered a "straight" line within the realm of the surface of the Earth.

Or if it is negatively curved, then perhaps there is a solution where the curvature of the sphere is precisely balanced by the curvature of space, resulting in flatness. However I don't know enough about non-Euclidean geometry to say any of this for sure, nor do I know exactly how you would define "flat" in a non-Euclidean space.

The sphere cannot be a 0 dimensional point because it violates condition number 4 and 1. A 0 dimensional point has no length, no width, and no depth, it doesn't occupy space therefore it can't occupy either time or space. Also, if it's zero dimensional then this violates the mathematical ratio of the sphere, which is π. Everywhere on the sphere, the ratio of the circumference to the diameter is π which is impossible for a 0 dimensional point, therefore this eliminates this posit as an explanation. In order for the sphere to be completely flat we still can't violate this mathematical relation because it's always spherical. The only way for the sphere to be spherical and flat is if an object can travel to the left side and end up on the right side, sort of like in the game "asteroids" where the ship travels along a 2 dimensional plane, but never falls off because it's a 2 dimensional sphere (well, the surface is) and on a sphere you can go in one direction and end up at the same point. This is the only way in which a sphere can be "flat" at any size except 0 is if the sides are somehow linked by an extra dimension, yet given the circumstances one would measure a curvature to the surface. So, relative to an outside observer the surface would appear to be flat, but for anyone on the surface, they would measure a curvature. This sphere is hypothetically existent in the universe and there is nothing in the laws of physics that denies extra dimensions or higher dimensional passage ways such as wormholes that could merit a 2 dimensional plane the characteristics of a sphere.

Very interesting question by the way, it's probably the first question on here that made me think this much. 10 trillion times better than the "2012" questions on here.

Is this correct?

I would instinctively say that this isn't possible, because, as one prior response said, it is like an asymptote of a hyperbola. The curvature will come close to but not become 0. But then I realized that I know of no way to measure curvature (That I know of, not saying there isn't a way).

I once read a book that said for a long time that real numbers were incomplete. That is, how do we factor (x^2+1). The book went on to say that real numbers were complete when the relevance of sqrt(-1) was discovered. Maybe this has something to do with it. (I am typing while I think about it)

Now I think we should find an equation that describes it. This is difficult because the information given is difficult to visualize. But how do we measure curvature (I can't visualize this...damn). Initially one might think of it as the sphere is being stretched, but that cant be because it will no longer be spherical.

Uhhh, I can't figure this out... I am fairly certain that the sphere will never be flat in a euclidean sense. I suppose in a non-euclidean sense that it already is flat, no matter the curvature.

Just spitting out ideas... this better not be a trick.

The way I brain it, a sphere losing or gaining curvature as a function of scale is irrelevant. A small circle has the same round shape as a big one. If I enlarged a sphere and zoomed out on it, it would appear exactly the same, the edge would curve round at the same rate. If you are referring to the way a surface looks to an observer on the surface, the earth does a pretty good job of appearing flat. To answer what I think, the sphere would be infinitely large so as to be unmeasurable.

As Adaviel said, it's physically impossible.

As the size increases, the sphere surface would lose curvature. No matter how large the sphere is, however, it would never be completely flat. Of course, this works in the opposite direction as well. No matter small a sphere is, it could never reach a state of infinite curvature.

If you were to graph the correlation between the size of a sphere and it's curvature, it would be a hyperbola with one asymptote representing a completely flat surface, and other representing an infinitely curved surface.

The arms of the hyperbola would get increasingly closer to the asymptotes, but never actually touch them.

Even if your sphere were only two, or even one, atoms thick, it STILL would not be mathematically "flat", because even atoms have three spatial dimensions. HYPOTHETICALLY, yes the sphere could be only two dimensional, but that is physically impossible in the real world. Or, as someone else has already said, so he should get the credit, expand the volume of the sphere until it becomes flat. Again, that is hypothetical, not reality. Or shrink the volume of the sphere until it becomes a point. Then it would be flat, mathematically, but it would also have no volume if it was a mathematical point. ????Oh, right , when you divide by zero the result is not infinity. The result is undefined. OK. The mathematical logic holds up. How can something exist, yet have no volume. Oh, it's flat - no volume. Can something exist if it is undefined????? Oh, right. This is hypothetical and mathematical, not reality. Whew!

I'm going with shrinking the sphere until it becomes a point, because then it would be flat mathematically, for my answer.

It would never be flat. However, depending on the sensitivity of your measuring equipment, and/or the greatest measurable distance, it could appear to show no measurable curvature at a large enough radius. There is no fixed size it would have to be to show this, as the other two variables would also come into play.

<Added>

Given your history of contributing to cosmology related topics, I wonder if this is a "What's the minimum size of the universe beyond our observable horizon?" question in disguise. If that is the case, someone better at mathematics than me should be able to answer this with ease.

Well, to be able to have a completely flat sphere, you'd have to convert this 3 dimensional object into a 2 dimensional object with only length and width but no depth or height. The only way to make this flat is make it 2 dimensional. Although it would depend on which way you look at it. In three dimensional world of ours, we could place a sphere in front of us in a room with equal lighting all around at eye level and look at that sphere with one eye closed. The ball would appear to be flat since since we cannot see the depth of the sphere. In a 4 dimensional world, we would however be able to see the 3 dimensional sphere all around like we would see a picture.

In conclusion, the only way I could think of turning this sphere flat is to have it travel to 2 dimensional existence. I guess that would be possible since no laws of physics or any laws really apply to your sphere. Therefore it doesn't matter what size, it just matters in placement.

It will never be completely flat. If you have some definition of flatness like say 1mm in 10m good enough for an Olympic icerink or something then you can work out the radius that will give that deformation from a plane surface.

I let the ball shrink to zero size. Then it doesn't have a surface and therefore no curvature.

Having no curvature is the definition of flat (at least, I hope it is :P )