Circle becomes Sphere in 3d? I was watching a movie Interstellar when it came out. Its been months but now I cant keep it inside and I must ask this question. So a guy was talking about wormhole in movie and ask the other guy whats a circle in 3d and other guy said, a Sphere. But I do not understand what is the rule to convert a 2d shape to 3d.
The simple rule I understand is to add 1 dimension that 2d shape do not have. That is a height. So if I add height to a circle by stacking it over and over. It makes a cylinder, not a sphere. Same goes for rectangle and square. They become box(cuboid) and cube.
To make sphere from circle, you need to stack them, not by overlapping in height but by rotating it on x or y axis and at 0 pivot. But if that's the rule, why rectangle and square doesn't become cylinder ?
 A: There is not a good mathematical or physical answer to what a circle becomes in 3D except to say it stays a circle.  The statement that a circle becomes a sphere is an analogy: As CuriousOne points out, one is extracting the important thing about a circle, that it is all points at a given distance from the center, and carrying that to 3D to get a sphere.  One could make an argument that a circle becomes a cylinder by translating it along the new dimension. One could also argue for a circular helix as the most general curve that is self-congruent.  Most people would reject these choices.  If you ask what a square becomes in 3D, it seems a cube is the right answer.  A rectangle becomes a parallelepiped, but nothing indicates what the third dimension should be.  Douglas Hofstadter has written a lot of interesting pieces about how we decide what is the important feature that should be maintained when you make analogies.
A: In 2D a circunference is 
$$(x-x_c)^2 + (y-y_c)^2 = r^2$$
A circle is:
$$(x-x_c)^2 + (y-y_c)^2 \leq r^2$$
In 3D sphere is:
$$(x-x_c)^2 + (y-y_c)^2 + (z-z_c)^2 = r^2$$
A ball is:
$$(x-x_c)^2 + (y-y_c)^2 + (z-z_c)^2 \leq r^2$$
You can even have a "1D ball/circle":
$$(x-x_c)^2 \leq r^2$$
which is just a line segment.
A 1D circunference/sphere is
$$(x-x_c)^2 = r^2$$ which are just 2 points
An $n$-dimentional sphere is
$$\sum_i^n(x_i-x_{ic})^2 = r^2$$
where $x_i$ are orthogonal coordinates in a Cartesian system.
As you can notice it is just a matter of the equivalent structure in higher dimentions. But the 3D equivalente of a circle is a ball, not a sphere, since a sphere is just a surface. The sphere is the 3D equivalente of a circunference.
