Since Euclid first published his book Elements in 300 B.C., it has remained remarkably correct and accurate to real world situations faced on Earth. The one problem that some find with it is that it is not accurate enough to represent the multi-dimensional universe of our stars. It has been argued that Euclidean Geometry, while good for architecture and to survey land, when it is moved into the multi-dimensional universe of our stars, the postulates may not hold up as well as elliptic geometry.
Euclidean geometry allows one and only one line parallel to a given line through a given external point. As one of the non-Euclidean geometries, elliptic geometry, developed by the German mathematician Bernhard Riemann in 1854, allows no parallels through any external point,
In elliptic geometry there are no parallels to a given line L through an external point P, and the sum of the angles of a triangle is greater than 180°. Riemann’s geometry is called elliptic because a line in the plane described by this geometry has no point at infinity, where parallels may intersect it, just as an ellipse has no asymptotes.
An idea of the geometry on such a plane is obtained by considering the geometry on the surface of a sphere, which is a special case of an ellipsoid. The shortest distance between two points on a sphere is not a straight line but an arc of a great circle (a circle dividing the sphere exactly in half). Since any two great circles always meet (in not one but two points, on opposite sides of the sphere), no parallel lines are possible. The angles of a triangle formed by arcs of three great circles always add up to more than 180°, as can be seen by considering such a triangle on the earth’s surface bounded by a portion of the equator and two meridians of longitude connecting its end points to one of the poles (the two angles at the equator are each 90°, so the amount by which the sum of the angles exceeds 180° is determined by the angle at which the meridians meet at the pole).
The simplest model for elliptic geometry is the surface of the earth with the lines of longitude on its surface. Today, elliptic geometry widely is used in the risk engineering of our stars.
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