Is the universe round or flat?

Does the universe's flatness mean that non Euclidean geometry isn't important in physics?

 Quote by kaushik_s I am a bit confused on what the current model of the universe is. some are saying it is round, while others are saying it is flat. which is true? If the universe is flat, then is it that only some part of it is flat or the whole universe is flat? how can we prove it theoratically and mathematically?
According to the depicted universe described in special relativity, particulary concerning the Lorentz transformations, the universe is flat with a .08 margin of deviation due to the differentiation between co-motion systems. In a difference of say, a rotational axis, certain points experience alternate rates of fluctuations to compensate for their differing inertial reference frames, specifically, when evaluating that circular system of rotation, when an side undergoes a contraction by:
L0√1-v2/c2
the radius will grant an invalid conclusion in the description of the area of a circle, namely A=∏r2, which is mathematically impossible in the Euclidean geometry of a classical universe, hence the necessitation of a curved space-time.

 Quote by neginf Does the universe's flatness mean that non Euclidean geometry isn't important in physics?
It is flat on average over very large scales. Locally it has curvature, we call that gravity and without it the Earth wouldn't orbit the Sun so yes, non-Euclidean geometry remains important.
 flat or round, spherical or trumpet - does the universe have a thickness ? is it centrally thicker like 2 cymbals placed together ? how do we know if it is symmetrical ? is it getting thinner in the centre as it expands peripherally? why should it be symmetrical - was the singularity that preceded the big bang spinning in order to give it such symmetry ?

 Quote by urodoc flat or round, spherical or trumpet - does the universe have a thickness ? is it centrally thicker like 2 cymbals placed together ? how do we know if it is symmetrical ? is it getting thinner in the centre as it expands peripherally? why should it be symmetrical - was the singularity that preceded the big bang spinning in order to give it such symmetry ?
Urodoc, when people speak of the shape of the universe, they are speaking of a natural curvature that spacetime has over large distances. The universe doesn't have a 'shape' in the literal since, as it has no boundary.

Keep in mind that the expansion of the universe occurs everywhere at once. There is no center or boundary. Also, the universe is symmetrical because it underwent a period called 'inflation' in it's very early history. Inflation was an enormous expansion, of unimaginable magnitudes. It essentially functioned to 'iron out' any inhomogeneities in the early universe.

Mentor
 Quote by GeorgeDishman It is flat on average over very large scales. Locally it has curvature, we call that gravity and without it the Earth wouldn't orbit the Sun so yes, non-Euclidean geometry remains important.
 Quote by Mark M when people speak of the shape of the universe, they are speaking of a natural curvature that spacetime has over large distances.
When people talk about the curvature of the universe, they mean the curvature of space, not the curvature of spacetime. For example, in a flat FRW universe, space is Euclidean and flat, while spacetime is neither flat nor Euclidean.

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