Parallel postulate

The parallel postulate is a much discussed axiom of Euclidean geometry. In a frequently used going back to John Playfair formulation, it states:

This straight line is called the parallel to through the point. Two parallels in this plane have no common point.

In the elements of Euclid there is this sentence as the fifth postulate ( the parallel postulate ) in the following formulation: " demanded to be: ... that if a straight line [g ] causes falling on two straight lines [ h, k] that inside formed on the same side angle [ α and β ] together less than two right angles, are then the two straight lines [ h, k], if extended to infinity meet on the side of [ g], where the angle [ α and β ] system, are together less than two right angles. "

This implies in modern formulation, that it can not give more than one parallel to through every line and every point. That there is at least one such parallels, but can be proved, so that the initially specified formulation is justified from the other postulates and axioms of Euclid.

The naming of the parallel postulate varies in the literature. It is often called the fifth postulate of Euclid (Elements, Book 1 ), sometimes it was but also called Axiom 11 or Axiom 13.

History

This postulate clearly stands out because of its length and complexity from the other postulates and axioms. It was felt in ancient times as a flaw ( ugly trait) in the theory of Euclid. Again and again there have been attempts to derive it from the others and to show that it is not necessary for the definition of Euclidean geometry. Historically, this task is known as the parallel problem and remained over 2000 years unsolved. Unsuccessful attempts were the characteristics of

• Archimedes ( 3rd century BC)
• Posidonius (2nd / 1st century BC)
• Ptolemy (2nd century)
• Proclus (5th century)
• Agapius (5th / 6th century AD, a pupil of Proclus ), quoted by Al- Nayrizi
• Simplicius
• Al- Abbās ibn Said al - Jawhari
• Thabit ibn Qurra ( ninth century)
• Giovanni Alfonso Borelli (17th century)
• John Wallis (17th century)
• Giovanni Girolamo Saccheri (18th century ), see Saccheri quadrilateral
• Johann Heinrich Lambert (18th century)
• Adrien -Marie Legendre ( 18-19. Century)

Carl Friedrich Gauss was the first to parallel that the problem is basically unsolvable; he did not publish his findings, however. He corresponded with various mathematicians, pursued similar ideas ( Friedrich Ludwig Wachter, Franz taurinus, Wolfgang Bolyai ).

Equivalent formulations

There also a number of statements were found that are the Euclidean parallel postulate on the condition of the remaining axioms of plane Euclidean geometry equivalent. The underlying axioms are the axioms of incidence plane ( I.1 to I.3 ), the axioms of arrangement (group II), the axioms of congruence ( group III) and the axioms of continuity ( V.1 and V.2) in Hilbert's system of axioms of Euclidean geometry:

• " The sum of the angles in a triangle is two right angles (180 °). " (Cf. Giovanni Girolamo Saccheri )
• " There are rectangles. "
• " There is a similar triangle of any size. Each triangle" ( John Wallis ).
• " Step angle parallels are equal. "
• " Due to a point in the interior of an angle, there is always a straight line that intersects the two legs. "
• "By three are not on a straight points there is a circle. " ( Farkas Wolfgang Bolyai )
• "Three points that lie on the same side of a straight line and this line have congruent distances, always lie on a common straight line. "

Non-Euclidean geometry

Nikolai Lobachevsky, however, not presented as the first in 1826, a novel geometry before, in all other axioms of Euclidean geometry apply the axiom of parallels. It is called lobatschewskische geometry or hyperbolic geometry. Janos Bolyai reached regardless almost simultaneously to similar results.

This led to the development of non-Euclidean geometries, in which the postulate was deleted completely or replaced by others. For non-Euclidean geometries that are not part of violating the axiom of parallels also other axioms of Euclidean geometry.

Elliptical parallel axiom

So it is not possible in an elliptical plane at the same time that Hilbert's axioms of order ( group II) and the congruence of lines ( III.1, III.2 and III.3) are met. Here you can within the meaning of congruence "meaningful" to introduce as for projective planes only one arrangement ( " separation relationship " with four instead of three points in a hilbert between inter-relationship ), because elliptical planes in terms of the metric absolute geometry are projective planes, their " elliptical " (actually projective ) parallel axiom is simple: " There are no non- cutter, two different lines of the plane always intersect in exactly one point ", see also elliptic geometry # labeling.

The right figure above illustrates the difference between an array on an affine line in the picture above and a projective line, represented by the circle in the image below. On an affine line a Hilbert inter-relationship is defined when the coordinate space can be arranged. Each affinity which maps the (unordered ) pair set onto itself, forms the " route ", which is the set of intermediate points of from to itself. In fact, there are exactly four such affinities: two of them (identity and vertical axis reflection on ) keep the line as a whole determines the other two, the vertical axis mirroring and point reflection on the ( affine ) mid-span of swap the points and half-lines.

On an affine circle and a projective line, the situation is different. Two points share the affine circle into two arcs. Affinities of the plane and map the county line on its own, from the two sheets per over himself, unless lie on the same diameter, then can and will also be interchanged and through the point reflection at the center of the circle and the (vertical ) mirror image on diameter.

Can be the circle as a model of projective line interpret over a body which is arranged by being projected from a point of the circular line from the center to the point opposite circle tangent. The point is thus associated with the far point of. For a projective plane over there for any two points that are in the picture assigned to the points on the circle, projectivity of the plane, which map the set of points on, but point sets that correspond respectively, are transposed. In short: On an arranged projective line can be "inside" and "outside" not projectively invariant differ!

Note that even with the line in the picture above, if you perceives as real, projective line, the complementary set of closed affine distance, which then also includes the far point of, with respect to the order topology a connected subset of is!

A hyperbolic parallel axiom by Hilbert

David Hilbert in 1903 has given the following formulation for a parallel postulate of hyperbolic geometry, see also the illustration on the right:

The angular space is marked in the figure on the right by a circular arc ( light blue). All half-line with a starting point that does not lie in this corner of space, do not cut the line.

In the above axiom system of Hilbert can replace the Euclidean parallel axiom (IV Hilbert ) by Hilbert's hyperbolic parallel axiom. This yields ( for the plane on the Hilbert limited here, that is, from the group of the incidence axioms are only I.1 to I.3 required) a non-contradictory axiom system for which there is exactly one model ( up to isomorphism ): the real, hyperbolic plane can be modeled, for example, the (real) Klein disc model within the real Euclidean plane. The proof he sketched himself in his basics. A complete proof was given in 1907 by Johannes Hjelmslev.