Parallel postulate

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

"Parallel" means that the straight lines lie in a plane but do not have a common point.

This uniquely determined straight line is called the parallel to through the point .

In the Elements of Euclid, this sentence is found as the fifth postulate (parallel postulate) in the following formulation: "Required shall be: … that if a straight line [ ], when intersecting two straight lines [ and ], causes angles [and β ] formed on the inside of the same side to become together smaller than two rights, then the two straight lines [ and ] when extended to infinity meet on the side [of ] on which lie the angles [and β ] which together are less than two rights.“

In modern formulation this means that for every straight line and every point there cannot be more than one parallel to through }. That there is at least one such parallel can, however, be proved from the other postulates and axioms of Euclid, so that the formulation given at the beginning is justified.

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

History

This postulate clearly stands out from the other postulates and axioms due to its length and complexity. It was perceived as a flaw (unsightly feature) in Euclid's theory even in ancient times. Again and again there were attempts to derive it from the others and thus to show that it was dispensable for the definition of Euclidean geometry. Historically, this task is known as the parallel problem and remained unsolved for over 2000 years. Unsuccessful attempts were made, for example, by

Archimedes (3rd century BC) Poseidonius (2nd/1st century BC) Ptolemy (2nd century) Proclus (5th century) Agapius (5th/6th century AD, disciple of Proclus), quoted by Al-Nayrizi Simplikios al-Abbas ibn Said al-Dschauhari Thabit ibn Qurra (9th century) Alhazen Omar Chayyam Nasir Al-din al-Tusi (13th century) Giovanni Alfonso Borelli (17th century) John Wallis (17th century) Giovanni Girolamo Saccheri (18th century), see also Saccheri quadrangle Johann Heinrich Lambert (18th century) Adrien-Marie Legendre (18th/19th century)

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

Equivalent formulations

A number of statements were also found that are equivalent to the Euclidean parallel postulate under the condition of the remaining axioms of plane Euclidean geometry. The underlying axioms are the plane incidence axioms (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 axiom system of Euclidean geometry:

• "The sum of the angles in the triangle is two rights (180°)." (cf. Giovanni Girolamo Saccheri)
• "There are rectangles."
• "For every triangle there is a similar triangle of any size". (John Wallis).
• "Step angles at parallels are equal".
• "Through a point inside an angle there is always a straight line that intersects the two legs."
• "Through three points not lying on a straight line there is a circle". (Farkas Wolfgang Bolyai)
• "Three points that lie on one and the same side of a straight line and have congruent distances to this straight line always lie on a common straight line."

Non-Euclidean Geometry

In 1826, Nikolai Lobachevsky was the first to present a new type of geometry in which all the other axioms of Euclidean geometry apply, but not the parallel axiom, Lobachevskian or hyperbolic geometry. This proved that the parallel axiom could not be derived from the other axioms of Euclidean geometry.

János Bolyai independently arrived at similar results almost simultaneously.

This led to the development of non-Euclidean geometries, in which the postulate was either completely deleted or replaced by others. In some cases, non-Euclidean geometries also violate other axioms of Euclidean geometry besides the parallel axiom.

Elliptic parallel axiom

Thus, in an elliptical plane it is not possible for Hilbert's arrangement axioms (group II) and the congruence axioms for distances (III.1, III.2 and III.3) to be fulfilled at the same time. Here, in the sense of congruence, one can "sensibly" only introduce an arrangement ("separation relation" through four instead of three points in the case of a Hilbertian intermediate relation) as for projective planes, because elliptical planes in the sense of metric absolute geometry are also projective planes, their "elliptical" (actually projective) parallel axiom is simply: "There are no non-intersections, two different straight lines of the plane always intersect at exactly one point" see Elliptical Geometry#Marking.

The figure at the top right illustrates the difference between an arrangement on an affine straight line at the top of the picture and a projective straight line represented by the circle bottom of the picture. On an affine straight line a Hilbert intermediate relation is definable if the coordinate range can be arranged. Every affinity that maps the (unordered) pair set onto itself also forms the "line" which is the set of intermediate points of onto itself. In fact, exactly four such affinities exist: Two of them (identity and perpendicular axis reflection on ) hold the straight line as a whole, the other two, the perpendicular axis mirroring and the point mirroring at the (affine) line centre of interchange the points and the half-straights .

The situation is different on an affine circle and a projective straight line. Two points divide the affine circle line into two arcs. Affinities of the plane that map and the circular line onto themselves also map the two arcs each onto themselves, unless lie on the same diameter, then and can also be exchanged by the point reflection at the centre of the circle and by the (perpendicular) axis reflection at the diameter .

The circle line can be understood as a model of a projective straight line over an arranged body projecting it centrally from a point this circle line onto the circle opposite the point. The point is thus assigned to the far point of For a projective plane over there exist for any two points which in the picture are associated with the points on the circular line, projectivities of the plane which map the point set onto themselves, but point sets , corresponding to and interchange with each other. In short: On an arranged projective straight line one cannot distinguish "inside" and "outside" projectively invariant!

Note that also in the case of the straight line top of the picture, if it is understood as a real, projective straight line, the complementary set the closed affine line which then also contains the far point of is a connected subset of respect to the order topology!

Hyperbolic parallel axiom according to Hilbert

In 1903 David Hilbert gave the following formulation for a parallel axiom of hyperbolic geometry, compare also the figure on the right:

If any straight line and point not on it, then there are always through two half-straights which do not make up one and the same straight line and do not intersect the straight line while each half straight line located in the angle space formed by starting from intersects the straight line

The angle space is marked by an arc (light blue) in the figure on the right. All half-lines with starting point which do not lie in this angle space do not intersect the straight line .

In Hilbert's axiom system mentioned above, one can replace the Euclidean parallel axiom (IV by Hilbert) with Hilbert's hyperbolic parallel axiom. Thus one obtains (for the plane to which Hilbert restricts himself here, i.e. of the group of incidence axioms only I.1 to I.3 are needed) a contradiction-free axiom system for which there is exactly one model (except for isomorphism): The real, hyperbolic plane, which can be modelled, for example, by the (real) Klein circular disk model within the real Euclidean plane. He outlines the proof himself in his Fundamentals. A complete proof was given by Johannes Hjelmslev in 1907.

On the difference between affine and projective arrangement.


Hilbert's hyperbolic parallel axiom in Klein's circular disk model of (real) hyperbolic geometry. The formulation of Hilbert's axiom with half-straights presupposes an arrangement of the hyperbolic plane in the sense of Hilbert's axioms. Note that only the points inside the circle line (grey) are points of the hyperbolic plane.