Absolute geometry is a geometry 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 can be included in absolute geometry. It is sometimes referred to as neutral geometry, as it is neutral with respect to the parallel postulate. Table of Geometry, from the 1728 Cyclopaedia. ... a and b are parallel, the transversal t produces congruent angles. ... Behavior of lines with a common perpendicular in each of the three types of geometry The term non-Euclidean geometry (also spelled: non-Euclidian geometry) describes both hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. ... A triangle immersed in a saddle-shape plane (an hyperbolic paraboloid), as well as two diverging parallel lines. ... Elliptic geometry (sometimes known as Riemannian geometry) is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. ... Euclid Euclidean geometry is a mathematical system due to the Hellenistic mathematician Euclid of Egypt. ... The frontispiece of Sir Henry Billingsleys first English version of Euclids Elements, 1570 Euclids Elements (Greek: ) is a mathematical and geometric treatise, consisting of 13 books, written by the Hellenistic mathematician Euclid in Egypt during the early 3rd century BC. It comprises a collection of definitions, postulates...

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.

See also

• Axioms of Neutral Geometry