Non-Euclidean geometry this is not the case. As an example in Euclidean geometry the sum of the interior angles of a triangle is 180°, in In non-Euclidean geometryĪ shortest path between two points is along such a geodesic, or "non-Euclidean line".Īll theorems in Euclidean geometry that use the fifth postulate, will be altered when you rephrase the In non-Euclidean geometry, the concept corresponding to a line is a curve called a geodesic. If you redefine what you mean by a line, you may have that two parallel lines either converge towards each other or diverge from one another. The parallel postulate is seemingly obvious only if you assume that parallel lines look like railroad tracks.
There is nothing in the definition indicating that the distance between two parallel lines is the Definition number 23 states that two lines are The theorems in the Elements of Euclid are built on a number of postulates and a number of definitions. Distances, angles, and lines Euclidean, elliptical, and hyperbolic triangle. The given line can be drawn through the point. If the parallel postulate is replaced by: It can be shown that if there is at least two lines, there are in fact infinitely "parallel to the given line through the point". The converse statement to the parallel postulate, as stated above, is that there are either no lines, or at least two lines Is consistent if it does not contain any contradiction.
Furthermore, itĬan be shown that non-Euclidean theories can be just as consistent as Euclidean theory. Such geometries are called non-Euclidean. To build a theory of geometry where the fifth postulate is not true. The fifth postulate is independent of the other postulates. Many attempts have been made to prove the fifth postulate using the other four postulates. The Elements of Euclid is built upon five postulates.įor over 2000 years the fifth postulate has been considered to be less intuitive than the other postulates, and not sufficiently As a statement that cannot be proven, a postulate should be self-evident. It cannot be proven using previous result. Exactly one line through the green point is parallel to the thick white line.Ī postulate (or axiom) is a statement that acts as a starting point for a theory.
However, a seemingly valid statement is not good enough. The postulate is not true in 3D but in 2D it seems to be a valid statement. Parallel to the given line can be drawn through the point.
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