# Dimensionality of Non-Euclidean geometry

## Main Question or Discussion Point

In introductions to non Euclidean geometry, examples are often given in the form of measuring angles on a 2D surface embedded in a 3D space, such as the surface of a sphere or on a saddle surface. This gave me the initial impression that 3D non Euclidean geometry would have to be embedded in 4 or more spatial dimensions. However, it then occurred to me that if I can map 3D Euclidean coordinates to 3D non Euclidean coordinates in a one to one relationship in an unambiguous manner, then I can describe any set of Euclidean 3D coordinates in terms of non Euclidean coordinates, without having to invoke any higher spatial dimensions. Is the one to one mapping relationship true and is the conclusion true?

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You should read about manifolds. A 3-dimensional manifold locally looks like 3-dimensional Euclidean space, but in general you don't have a single 3D coordinate system (chart) for the entire manifold, but in general they're constructed by piecing together several charts.

For example: the 3-sphere S3 can be described by two charts: one defined everywhere on the sphere except the north pole, and another defined everywhere except the south pole.

quasar987
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...and Riemannian geometry!

lavinia