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Euclid gave the definition of parallel lines in Book I, Definition 23 [2] just before the five postulates. [3] Euclidean geometry is the study of geometry that satisfies all of Euclid's axioms, including the parallel postulate. The postulate was long considered to be obvious or inevitable, but proofs were elusive.
Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry , there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines are usually assumed to intersect at a single point (rather than two).
The various attempted proofs of the parallel postulate produced a long list of theorems that are equivalent to the parallel postulate. Equivalence here means that in the presence of the other axioms of the geometry each of these theorems can be assumed to be true and the parallel postulate can be proved from this altered set of axioms.
Independence of the parallel postulate ; Infinite monkey theorem (probability) Integral root theorem (algebra, polynomials) Initial value theorem (integral transform) Inscribed angle theorem ; Integral representation theorem for classical Wiener space (measure theory) Intermediate value theorem ; Intercept theorem (Euclidean geometry)
The fundamental property that singles out all projective geometries is the elliptic incidence property that any two distinct lines L and M in the projective plane intersect at exactly one point P. The special case in analytic geometry of parallel lines is subsumed in the smoother form of a line at infinity on which P lies.
For examples, elliptic geometry (no parallels) and hyperbolic geometry (many parallels). Both elliptic and hyperbolic geometry are consistent systems, showing that the parallel postulate is independent of the other axioms. [2] Proving independence is often very difficult. Forcing is one commonly used technique. [3]
The very old problem of proving Euclid's Fifth Postulate, the "Parallel Postulate", from his first four postulates had never been forgotten. Beginning not long after Euclid, many attempted demonstrations were given, but all were later found to be faulty, through allowing into the reasoning some principle which itself had not been proved from ...
Elliptic geometry a type of non-Euclidean geometry (it violates Euclid's parallel postulate) and is based on spherical geometry. It is constructed in elliptic space. Enumerative combinatorics an area of combinatorics that deals with the number of ways that certain patterns can be formed. Enumerative geometry