# Axiom of parallels

The axiom of parallels is a much discussed axiom of Euclidean geometry . In a frequently used phrase going back to John Playfair , it says:

"In a plane there is for every straight line and every point outside of exactly one straight line that is too parallel and goes through the point ." ${\ displaystyle \ alpha}$ ${\ displaystyle g}$${\ displaystyle P}$${\ displaystyle g}$${\ displaystyle g}$${\ displaystyle P}$

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

This uniquely determined straight line is called the parallel to through the point${\ displaystyle g}$${\ displaystyle P}$ .

Point of intersection S of h and k if α + β <180 °.

In the elements of Euclid , this sentence is found as the fifth postulate (parallel postulate ) in the following formulation: “The following should be required: ... that if a straight line [ ] cuts with two straight lines [ and ] causes that inside on the same side resulting angles [ and ] together become smaller than two rights, then the two straight lines [ and ] when extending into infinity meet on the side [of ] on which the angles [ and ] lie, which are together smaller than two rights. " ${\ displaystyle g}$${\ displaystyle h}$${\ displaystyle k}$${\ displaystyle \ alpha}$${\ displaystyle \ beta}$${\ displaystyle h}$${\ displaystyle k}$${\ displaystyle g}$${\ displaystyle \ alpha}$${\ displaystyle \ beta}$

This says in a modern formulation that every line and every point no more than a parallel to through can give. That there is at least one such parallel can be proven from the other postulates and axioms of Euclid , so that the formulation given at the beginning is justified. ${\ displaystyle g}$${\ displaystyle P}$ ${\ displaystyle g}$${\ displaystyle P}$

The naming of the parallel postulate fluctuates in the literature. Often it is called the Fifth Postulate of Euclid ( Elements, Book 1), but sometimes it has also been called the 11th axiom or the 13th axiom.

## history

This postulate stands out clearly from the other postulates and axioms because of its length and complexity. It was already felt in ancient times as a flaw (unattractive feature) in the theory of Euclid. There have been repeated attempts to derive it from the others and thus to show that it is dispensable for the definition of Euclidean geometry. Historically, this task is known as the Parallels Problem and has remained unsolved for over 2000 years. There have been unsuccessful attempts by, for example

 Archimedes (3rd century BC) Poseidonius (2nd / 1st century BC) Ptolemy (2nd century) Proclus (5th century) Agapius (5th / 6th century AD, pupil of Proklos), quoted by Al-Nayrizi Simplicity al-Abbas ibn Said al-Jawhari 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 Saccheri-Viereck Johann Heinrich Lambert (18th century) Adrien-Marie Legendre (18th / 19th century)

Carl Friedrich Gauß was the first to recognize that the parallel problem is fundamentally unsolvable ; but he did not publish his findings. However, he corresponded with various mathematicians who pursued similar ideas ( Friedrich Ludwig Wachter , Franz Taurinus , Wolfgang Bolyai ).

## Equivalent formulations

A number of statements have also been found which are equivalent to the Euclidean parallel postulate, assuming the other axioms of plane Euclidean geometry . The underlying axioms are the planar 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 system of axioms 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 the same."
• "Through a point inside an angle there is always a straight line that intersects the two legs."
• "There is a circle through three points that are not on a straight line." ( Farkas Wolfgang Bolyai )
• "Three points that lie on 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 other axioms of Euclidean geometry apply, but not the axiom of parallels, Lobachevskian or hyperbolic geometry . This proved that the axiom of parallels cannot be derived from the other axioms of Euclidean geometry.

Independently of this, János Bolyai achieved similar results almost simultaneously.

This led to the development of non-Euclidean geometries , in which the postulate was either deleted entirely or replaced by others. In part, non-Euclidean geometries violate other axioms of Euclidean geometry in addition to the axiom of parallels.

### Elliptical parallel axiom

The difference between affine and projective arrangement.

So it is not possible in an elliptical plane that Hilbert's arrangement axioms (group II) and the congruence axioms for lines (III.1, III.2 and III.3) are fulfilled at the same time . Here, in the sense of congruence, one can "meaningfully" only introduce an arrangement ("separation relationship" through four instead of three points in a Hilbertian intermediate relationship ) as for projective planes , because elliptical planes in the sense of metric absolute geometry are also projective planes, their "elliptical “(Actually projective) axiom of parallels simply reads:“ There are no non-intersecting lines, two different straight lines of the plane always intersect at exactly one point ”see Elliptical Geometry # Identification .

The figure at the top right illustrates the difference between an arrangement on an affine straight line at the top of the image and a projective straight line, represented by the circle at the bottom of the image. A Hilbert interrelation can be defined on an affine straight line if the coordinate range can be arranged. Every affinity that maps the (disordered) pair set onto itself also forms the “route” , that is the set of intermediate points from onto itself. In fact, there are exactly four such affinities: Two of them (identity and vertical axis reflection on ) hold the straight line as a whole, the other two, the vertical axis reflection and the point reflection at the (affine) line center of, exchange the points and the half-straight lines . ${\ displaystyle g}$${\ displaystyle h}$${\ displaystyle g}$${\ displaystyle \ {G_ {1}, G_ {2} \}}$${\ displaystyle g_ {1}}$${\ displaystyle (G_ {1}, G_ {2})}$${\ displaystyle G_ {1} G_ {2}}$${\ displaystyle (G_ {1}, G_ {2})}$${\ displaystyle G_ {1}, G_ {2}}$ ${\ displaystyle g_ {2}, g_ {3}}$

The situation is different on an affine circle and a projective straight line. Two points divide the affine circular line into two circular arcs. Affinities of the plane, which map and the circular line on themselves, also map the two arcs on themselves, unless they are on the same diameter, then and can also be interchanged, namely by the point reflection at the center of the circle and by the (vertical) Axis reflection on the diameter . ${\ displaystyle \ {H_ {1}, H_ {2} \}}$${\ displaystyle H_ {1}, H_ {2}}$${\ displaystyle h_ {1}}$${\ displaystyle h_ {2}}$${\ displaystyle H_ {1} H_ {2}}$

You can see the circle as a model of projective line over an arranged body regarded by it from a point of this circular line from the center to the point opposite circle tangent projected . The point is thus assigned to the far point of . For a projective plane , for any two points that are assigned to the points on the circular line in the image , there are projectivities of the plane that map the point set to themselves, but interchange point sets that correspond to or  . In short: on an arranged projective straight line one cannot distinguish “inside” and “outside” in a projectively invariant manner! ${\ displaystyle K}$${\ displaystyle P = H _ {\ infty}}$${\ displaystyle t}$ ${\ displaystyle P}$${\ displaystyle t}$${\ displaystyle K}$${\ displaystyle T_ {1}, T_ {2} \ in t}$${\ displaystyle H_ {1}, H_ {2}}$${\ displaystyle \ {T_ {1}, T_ {2} \}}$${\ displaystyle t_ {1}, t_ {2} \ subset t}$${\ displaystyle h_ {1}}$${\ displaystyle h_ {2}}$

Note that even with the straight line in the picture above, if one understands it as a real, projective straight line, the complementary set of the closed affine segment , which then also contains the far point of , is a connected subset of with regard to the order topology ! ${\ displaystyle g}$${\ displaystyle c = g \ setminus s}$${\ displaystyle s = g_ {1} \ cup \ {G_ {1}, G_ {2} \}}$${\ displaystyle g}$${\ displaystyle g}$

### Hyperbolic parallel axiom according to Hilbert

Hilbert's hyperbolic axiom of parallels in the Klein disk model of (real) hyperbolic geometry. The formulation of Hilbert's axiom with half-lines presupposes an arrangement of the hyperbolic plane in the sense of Hilbert's axioms. Note that only the points within the circular line (gray) are points of the hyperbolic plane.

In 1903 David Hilbert gave the following formulation for an axiom of parallels in hyperbolic geometry , compare the figure on the right:

If there is any straight line and a point not located on it, then there are always two half-lines that do not make up one and the same straight line and do not intersect the straight line , while every half-line located in the angular space formed by, and starting from , intersects the straight line .${\ displaystyle b}$${\ displaystyle A}$${\ displaystyle A}$${\ displaystyle a_ {1}, a_ {2}}$${\ displaystyle b}$${\ displaystyle (a_ {1}, a_ {2})}$${\ displaystyle \ angle (a_ {1}, a_ {2})}$${\ displaystyle A}$${\ displaystyle b}$

The angular space is marked in the figure on the right by an arc (light blue). All half-straight lines with a starting point that are not in this angular space do not intersect the straight line . ${\ displaystyle \ angle (a_ {1}, a_ {2})}$${\ displaystyle A}$${\ displaystyle b}$

In the above axiom system of Hilbert can be the Euclidean parallel axiom (IV Hilbert) by Hilbert hyperbolic parallel axiom replace . This gives (for the level to which Hilbert restricts himself here, i.e. from the group of incidence axioms only I.1 to I.3 are required) a consistent system of axioms for which there is exactly one model (except for isomorphism) : The real, hyperbolic plane, which can be modeled for example by the (real) Klein disc model within the real Euclidean plane. He sketches the proof himself in its basics . Complete proof was given by Johannes Hjelmslev in 1907 .

## literature

• David Hilbert : Fundamentals of Geometry . 14th edition. Teubner, Stuttgart / Leipzig 1899, ISBN 3-519-00237-X ( edition from 1903  - Internet Archive - The axiom system of real-Euclidean geometry and real-hyperbolic geometry (Appendix III) formulated in this publication represents the 20th century the most important basis for the discussion of the axiom of parallels and non-Euclidean geometries).
• Paul Stäckel , Friedrich Engel : The theory of the parallel lines from Euclid to Gauss . Teubner, Leipzig 1895 (On the "premodern" history of the term).
• Heinz Lüneburg : The Euclidean plane and its relatives . Birkhäuser, Basel / Boston / Berlin 1999, ISBN 3-7643-5685-5 ( google-books [accessed on July 26, 2013]).
• Sibylla Prieß-Crampe : Arranged structures . Groups, bodies, projective levels (=  results of mathematics and its border areas . Volume 98 ). Springer, Berlin / Heidelberg / New York 1983, ISBN 3-540-11646-X (detailed discussion of the possible arrangements for projective planes, from the algebraic (arranged coordinate range), synthetic (separation relationship on straight lines) and (order) topological standpoint).