Absolute geometry

From Wikipedia, the free encyclopedia

Absolute geometry is a geometry based on an axiom system that does not assume the parallel postulate or any of its alternatives.

Its theorems are therefore true in non-Euclidean geometries, such as hyperbolic geometry and elliptic geometry, as well as in Euclidean geometry. In Euclid's Elements, the first 28 Propositions avoid using the parallel postulate, and therefore are valid in absolute geometry.

Absolute geometry is an example of an incomplete postulational system. Consider the statement "The sum of the angles in every triangle is equal to two right angles". This is not provable in absolute geometry, because if it was, it would be true in hyperbolic geometry, and the sum of the angles in a hyperbolic triangle is less than two right angles. However, the negation of the statement, that there exists a triangle whose angles don't add up to two right angles, is not provable either, because if it was, it would be provable in Euclidean geometry, and the sum of the angles in Euclidean geometry is always two right angles. Therefore this proposition is undecidable in absolute geometry.