Dimensionality of Non-Euclidean geometry

[yuiop]

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?

 

[adriank]

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]

You should read about manifolds.

...and Riemannian geometry!

 

[lavinia]

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?

If you have a 3d coordinate system that is described by lines with known lengths and angles - e.g. a geodesic coordinate system - then you do not need to embed that manifold into 4d or higher in order to see its geometry. This is the geometry you would observe by measurement on the manifold itself rather than the geometry you get by looking down on it from the outside.

There is a difference between the intrinsic geometry and the embedded geometry. Intrinsic geometry does not tell you how the manifold bends and curves in space whereas extrinsic does. However the two geometries are intimately linked. For instance, the Gauss curvature of a surface - which is intrinsic - can be determined by the way the unit normal in 3 space changes direction as you move it around on the surface.

 

[blue_raver22]

The dimensionality of non-Euclidean geometry is a complex and fascinating topic. It is true that in many introductory examples, non-Euclidean geometry is described as a 2D surface embedded in a 3D space. However, this does not necessarily mean that non-Euclidean geometry is limited to only 3 spatial dimensions.

The concept of dimensionality in geometry refers to the number of coordinates needed to uniquely describe a point. In Euclidean geometry, this is typically done using 3 coordinates (x, y, z) in a 3D space. However, in non-Euclidean geometry, the number of coordinates needed may vary depending on the type of geometry being studied.

In your example, you mention the possibility of mapping 3D Euclidean coordinates to 3D non-Euclidean coordinates in a one-to-one relationship without invoking any higher spatial dimensions. This is certainly possible in some cases, but it is not a universal truth. In fact, there are many non-Euclidean geometries that cannot be mapped onto a 3D space without introducing additional dimensions.

For example, in hyperbolic geometry, the curvature of space is negative and cannot be accurately represented in a 3D space. In order to fully describe and understand this type of geometry, it is necessary to introduce additional dimensions.

In conclusion, while it is possible to map some non-Euclidean geometries onto 3D spaces without invoking higher dimensions, this is not always the case. The dimensionality of non-Euclidean geometry is a complex and multi-faceted concept that requires a deeper understanding of the specific geometry being studied.