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**HallsofIvy** No, that's not true. "Non-Euclidean Geometry" has nothing to do with dimensions (you may have found examples of non-Euclidean geometries in which other dimensions were added for other, perhaps physics, reasons).

Strictly speaking, non-Euclidean geometry is any geometry in which at least one of Euclid's postulates is denied. However, it is most commonly used to mean geometries in which the fifth postulate is denied. The most commonly used rephrasing of Euclid's fifth postulate is "Playfair's axiom" that "through a given point, P, not on a given line, l, there exist one and only one line through P parallel to l".

There are two ways to deny that:

1) Assert that there are NO parallels to l through p which leads to the statement that no line has any parallels. In order to be consistent with other postulates, you must also drop postulate that two lines intersect in, at most, one point. That is called "elliptic geometry". Spherical geometry, where "lines" are great circles on a sphere, is a model for elliptic geometry.

2) Assert that the exist two or more parallels to l thorugh p. This is "hyperbolic geometry". The Poincare "half plane" and "disc" models are the simplest models for hyperbolic geometric although you can also use the "hypersphere" (a sort of sphere with "negative" radius) to mimic the sphere model for elliptic geometry.