Absolute geometry
Absolute geometry is a geometry based on an axiom system for Euclidean geometry without the parallel postulate or any of its alternatives. Traditionally, this has meant using only the first four of Euclid's postulates. The term was introduced by János Bolyai in 1832. It is sometimes referred to as neutral geometry, as it is neutral with respect to the parallel postulate. The first four of Euclid's postulates are now considered insufficient as a basis of Euclidean geometry, so other systems are used instead.
Properties
In Euclid's Elements, the first 28 Propositions and Proposition 31 avoid using the parallel postulate, and therefore are valid in absolute geometry. One can also prove in absolute geometry the exterior angle theorem, as well as the Saccheri–Legendre theorem, which states that the sum of the measures of the angles in a triangle has at most 180°.Proposition 31 is the construction of a parallel line to a given line through a point not on the given line. As the proof only requires the use of Proposition 27, it is a valid construction in absolute geometry. More precisely, given any line l and any point P not on l, there is at least one line through P which is parallel to l. 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 foregoing construction, and the alternate interior angle theorem, do not depend on the parallel postulate and are therefore valid in absolute geometry.
In absolute geometry, it is also provable that two lines perpendicular to the same line cannot intersect.
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. Absolute geometry is inconsistent with elliptic geometry or spherical geometry: the notion of ordering or betweenness of points on lines, used to axiomatize absolute geometry, is inconsistent with these other geometries.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, 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.
The geometry of special relativity has been developed starting with nine axioms and eleven propositions of absolute geometry. The authors Edwin B. Wilson and Gilbert N. Lewis then proceed beyond absolute geometry when they introduce hyperbolic rotation as the transformation relating two frames of reference.