n-spheres.

[BigMoosie]

Does it make sense to say that two points is a zero-sphere?

 

I read that an n-spehere is the structure made from all the points being exactly the same radius from a single point in (n+1) dimensions (thats of the top of my head). But also shouldnt the resulting structure be of n-dimensions itself? I'm not sure whether two points would be called 0 dimensional would it?

 

Also, could one visualise a 3-sphere (picturing it with 3 spatial dimensions and the fourth as time) as a point that grows to a sphere and then shrinks back down?

[woelen]

Does it make sense to say that two points is a zero-sphere?

Yes, it does. A countable set of points is said to have dimension 0.

 

I read that an n-spehere is the structure made from all the points being exactly the same radius from a single point in (n+1) dimensions (thats of the top of my head). But also shouldnt the resulting structure be of n-dimensions itself? I'm not sure whether two points would be called 0 dimensional would it?

An n-sphere indeed is a (curved) n-dimensional space of finite n-volume. For 1-dimensional spaces (a circle), the 1-volume is the length of the circle, for 2-dimensional spaces (an ordinary sphere, the surface of a ball), the 2-volume is the surface of the sphere, etc. The n-volume of an n-sphere is proportional to r^n, with r being the radius of the n-sphere, e.g. the 1-volume of a circle is 2*pi*r, the surface of a sphere equals 4*pi*r^2. For a 0-sphere, the 0-volume does not depend on the radius, it can be written as k*r^0, being a constant k.

 

Also, could one visualise a 3-sphere (picturing it with 3 spatial dimensions and the fourth as time) as a point that grows to a sphere and then shrinks back down?

A three-sphere has nothing to do with time. It simply is a geometrical object. A nice way to visualize a 3-sphere is the following: Suppose you are on the sphere floating around in space (this is imaginable, because you are 3D as well I hope ) and start moving forward. If you keep on moving forward in a "straight line" in the same direction, then you'll eventually reach your original position again.

If you cannot imagine this, then think of the analogon of a 2-sphere with you being on the surface (of e.g. a planet). If you walk to a certain direction and you keep on moving, then you'll end up at your initial position again.

 

I place the words "straight line" between quotes, because of the fact, that such thing not really exists on a sphere. A sphere is not an euclidian space, although locally it approaches an n-dimensional euclidian space. In curved spaces, the concept of straight line must be replaced by the more general concept of "geodesic". Google is your friend on this subject.

[BigMoosie]

That is interesting about ending up where you start from, I see what you mean. But I am trying to visualise the sphere still.

 

If a hyper-plane was cutting the 3-sphere slowly from one end to the other. Me being in the hyper-plane would see a regular sphere appear, grow and then reach a maximum and shrink back down into a point?

[woelen]

That is interesting about ending up where you start from' date=' I see what you mean. But I am trying to visualise the sphere still.

 

If a hyper-plane was cutting the 3-sphere slowly from one end to the other. Me being in the hyper-plane would see a regular sphere appear, grow and then reach a maximum and shrink back down into a point?[/quote']

Yes, you would see a regular sphere (hollow ball) appearing out of 'nothing', growing, shrinking back and disappearing again.