# Absolute geometry

Absolute geometry

Absolute geometry is a geometry based on an axiom system for Euclidean geometry that does not assume the parallel postulate or any of its alternatives. The term was introduced by János Bolyai in 1832.[1] It is sometimes referred to as neutral geometry,[2] as it is neutral with respect to the parallel postulate.

## Relation to other geometries

The theorems of absolute geometry hold in hyperbolic geometry, which is a non-Euclidean geometry, as well as in Euclidean geometry.[3]

Absolute geometry is an extension of ordered geometry, and thus, all theorems in ordered geometry hold in absolute geometry. The converse is not true. Absolute geometry assumes the first four of Euclid's Axioms (or their equivalents), to be contrasted with affine geometry, which does not assume Euclid's third and fourth axioms. Ordered geometry is a common foundation of both absolute and affine geometry.[4]

Absolute geometry is inconsistent with elliptic geometry: in that theory, there are no parallel lines at all, so Euclid's parallel postulate can be immediately disproved; on the other hand, it is a theorem of absolute geometry that parallel lines do exist.[5]

It might be imagined that absolute geometry is a rather weak system, but that is not the case. Indeed, in Euclid's Elements, the first 28 Propositions avoid using the parallel postulate, and therefore are valid in absolute geometry. One can also prove in absolute geometry the exterior angle theorem (an exterior angle of a triangle is larger than either of the remote angles), as well as the Saccheri-Legendre theorem, which states that a triangle has at most 180°.[6]

## Incompleteness

Absolute geometry is an incomplete axiomatic system, in the sense that one can add extra independent axioms without making the axiom system inconsistent. One can extend absolute geometry by adding different axioms about parallel lines and get incompatible but consistent axiom systems, giving rise to Euclidean or hyperbolic geometry. Thus every theorem of absolute geometry is a theorem of hyperbolic geometry and Euclidean geometry. However the converse is not true.

## Notes

1. ^ Various Geometries
2. ^ Greenberg cites W. Prenowitz and M. Jordan (Greenberg, p. xvi) for having used the term neutral geometry to refer to that part of Euclidean geometry that does not depend on Euclid's parallel postulate. He says that the word absolute in absolute geometry misleadingly implies that all other geometries depend on it.
3. ^ Indeed, absolute geometry is in fact the intersection of hyperbolic geometry and Euclidean geometry when these are regarded as sets of propositions.
4. ^ Coxeter, pgs. 175-176
5. ^ This can be proved using a familiar construction: given a line l and a point P not on l, drop the perpendicular m from P to l, then erect a perpendicular n to m through P. By the alternate interior angle theorem, l is parallel to n. (The alternate interior angle theorem states that if lines a and b are cut by a transversal t such that there is a pair of congruent alternate interior angles, then a and b are parallel.) The foregoing construction, together with the alternate interior angle theorem, do not depend on the parallel postulate and are therefore valid in absolute geometry (Greenberg, p. 163).
6. ^ One again sees the incompatibility of absolute geometry with elliptic geometry, because in the latter theory all triangles have more than 180°.

## References

• Greenberg, Marvin Jay Euclidean and Non-Euclidean Geometries: Development and History, 4th ed., New York: W. H. Freeman, 2007. ISBN 0-7167-9948-0
• Coxeter, H. S. M Introduction to Geometry, 2nd ed., New York: John Wiley & Sons, 1969.
• Pambuccain, Victor Axiomatizations of hyperbolic and absolute geometries, in: Non-Euclidean geometries (A. Prékopa and E. Molnár, eds.). János Bolyai memorial volume. Papers from the international conference on hyperbolic geometry, Budapest, Hungary, July 6-12, 2002. New York, NY: Springer, 119-153 , 2006.

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