Definitions

# Absolute geometry

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

## Relation to Other Geometries

The theorems of absolute geometry hold in some non-Euclidean geometries, such as hyperbolic geometry, as well as in Euclidean geometry.

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.

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° .

## Incompleteness

Absolute geometry is an example of an incomplete postulational system. Consider the proposition "The sum of the measures of the angles in every triangle is equal to the measures of two right angles". This proposition is not provable in absolute geometry. Were the proposition provable, it would be true in hyperbolic geometry, where the sum of the measures of the angles in a triangle is less than the sum of the measures of two right angles. The proposition's negation—that there exists a triangle the sum of whose angle measures does not equal the measures of two right angles—is also unprovable. Were the negation provable, it would be provable in Euclidean geometry, where the sum of the measures of the angles of a triangle equals the sum of the measures of two right angles. Therefore, the proposition is undecidable in absolute geometry.