Absolute geometry is ageometry based on anaxiom system forEuclidean geometry without theparallel postulate or any of its alternatives. Traditionally, this has meant using only the first four ofEuclid's postulates.[1] The term was introduced byJános Bolyai in 1832.[2] It is sometimes referred to asneutral geometry,[3] as it is neutral with respect to the parallel postulate. The first four of Euclid's postulates are now known to be an insufficient basis for Euclidean geometry, so other systems (such asHilbert's axioms without the parallel axiom) are used instead.[4]
InEuclid'sElements, 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 theexterior angle theorem (an exterior angle of a triangle is larger than either of the remote angles), as well as theSaccheri–Legendre theorem, which states that the sum of the measures of the angles in a triangle has at most 180°.[5]
Proposition 31 is the construction of aparallel line to a given line through a point not on the given line.[6] As the proof only requires the use of Proposition 27 (the Alternate Interior Angle Theorem), it is a valid construction in absolute geometry. More precisely, given any linel and any pointP not onl, there isat least one line throughP which is parallel tol. This can be proved using a familiar construction: given a linel and a pointP not onl, drop the perpendicularm fromP tol, then erect a perpendicularn tom throughP. By the alternate interior angle theorem,l is parallel ton. (The alternate interior angle theorem states that if linesa andb are cut by a transversalt such that there is a pair of congruent alternate interior angles, thena andb are parallel.) The foregoing construction, and the alternate interior angle theorem, do not depend on the parallel postulate and are therefore valid in absolute geometry.[7]
In absolute geometry, it is also provable that two lines perpendicular to the same line cannot intersect (i.e., must be parallel).[8]
The theorems of absolute geometry hold inhyperbolic geometry, which is anon-Euclidean geometry, as well as inEuclidean geometry.[9] Absolute geometry is inconsistent withelliptic geometry orspherical geometry: the notion of ordering or betweenness of points on lines, used to axiomatize absolute geometry, is inconsistent with these other geometries.[10]
Absolute geometry is an extension ofordered 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 withaffine geometry, which does not assume Euclid's third and fourth axioms.(3: "To describe acircle with any centre and distanceradius.",4: "That allright angles are equal to one another." ) Ordered geometry is a common foundation of both absolute and affine geometry.[11]
Thegeometry of special relativity has been developed starting with nine axioms and eleven propositions of absolute geometry.[12][13] The authorsEdwin B. Wilson andGilbert N. Lewis then proceed beyond absolute geometry when they introducehyperbolic rotation as the transformation relating twoframes of reference.
A plane that satisfies Hilbert'sIncidence,Betweenness andCongruence axioms is called aHilbert plane.[14] Hilbert planes are models of absolute geometry.[15]
Absolute geometry is anincompleteaxiomatic system, in the sense that one can add extra independent axioms without making the axiom system inconsistent. One can extend absolute geometry by adding various axioms about parallel lines and get mutually incompatible but internally 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.
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