Explain to me how a sphere can exist in higher dimensions?

[Quantum1332]

could someone explain to me how a sphere can exist in higher dimensions? Would it just be a 3d sphere suspended within a 4d hyperspace? Lastly, caould the reason be as to why we can't see these other dimensions be because light goes underneath these other dimensions?

 

[Cexy]

Well, a n-dimensional sphere is just a mathematical object defined in (n+1)-dimensional Euclidean space by the equation

$$x_1^2 + \cdots + x_{n+1}^2 = 1$$

This immediately tells us that a 1-dimensional sphere is a circle with $$x^2+y^2=1$$, and a 2-dimensional sphere is what we usually think of when we say the word 'sphere'. Note that when I say a 2-dimensional sphere' I mean something that lives in 3d space - one of the properties of spheres is that they are curved, and so you can't 'fit' them into a flat space with the same number of dimensions.

In more dimensions it's hard, if not impossible, to imagine what a sphere might look like. To imagine a 3d sphere you'd have to be able to think in 4 dimensions, and I don't know anyone who can do that - although some people claim to be able to!

This is why we do everything with mathematics when we get above a certain number of dimensions. We live in 3 spatial dimensions, and so it's incredibly hard for us to get any kind of intuition about how things 'should' happen in 4 dimensions.

The most often quoted reason for why we don't see any higher dimensions is that they're curled up very small - we only see the three spatial dimensions that are large and mostly flat. Imagine a hosepipe - the surface certainly has two dimensions, the 'large' dimensions along the length of the hosepipe, and the 'small' dimensions which is curled around the pipe. If the small dimension was small enough, we would only be able to see the large, extended dimension.

 

[daveb]

The reason some folks like to say that our universe in embedded in a higher dimension hypersphere is as follows (this was the explanation a professor used once). Imagine a universe of 2 dimensional folks living in a 2D universe (say the surface of a balloon). Stretch the balloon so it is no longer flat, as viewed from 3D. The 2D folks would never be able to tell that their universe isn't flat. Then our non-flat 3D universe can be viewed similarly as appearing non-flat in 4D. We can't see the non-flatness.

Similarly, suppose the 2D folks saw gravimetric lensing in their universe. Locally, the 2D space doesn't appear flat (even though the universe as a whole appears as such), and they can work all kinds of calculations in 3D to show that this bending can be explained by geodesics if you consider them in 3D space, giving rise to the belief that things work better (mathematically) if you apply local non-flatness to a higher dimension. Then our universe can be similarly viewed.