11 Facts About Elliptic geometry

Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold.

Elliptic geometry has a variety of properties that differ from those of classical Euclidean plane geometry.

The reason for doing this is that it allows elliptic geometry to satisfy the axiom that there is a unique line passing through any two points.

On scales much smaller than this one, the space is approximately flat, Elliptic geometry is approximately Euclidean, and figures can be scaled up and down while remaining approximately similar.

Therefore any result in Euclidean geometry that follows from these three postulates will hold in elliptic geometry, such as proposition 1 from book I of the Elements, which states that given any line segment, an equilateral triangle can be constructed with the segment as its base.

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Elliptic geometry is like Euclidean geometry in that space is continuous, homogeneous, isotropic, and without boundaries.

Elliptic geometry space has special structures called Clifford parallels and Clifford surfaces.

Notable property of the projective elliptic geometry is that for even dimensions, such as the plane, the geometry is non-orientable.

Elliptic geometry is obtained from this by identifying the antipodal points and, and taking the distance from to this pair to be the minimum of the distances from to each of these two points.

Spherical elliptic geometry can be modeled as, for example, a spherical subspace of a Euclidean space, it follows that if Euclidean geometry is self-consistent, so is spherical elliptic geometry.

Tarski proved that elementary Euclidean Elliptic geometry is complete: there is an algorithm which, for every proposition, can show it to be either true or false.