absolute geometry

This commonality is the subject of absolute geometry (also called neutral geometry).

It might be imagined that absolute geometry is a rather weak system, but that is not the case.

A good example is the relative consistency of neutral geometry or absolute geometry with respect to the theory of the real number system.

Absolute geometry is an incomplete axiomatic system, in the sense that one can add extra independent axioms without making the axiom system inconsistent.

This follows since parallel lines exist in absolute geometry, but this statement says that there are no parallel lines.

The existence of triangle with angle sum of 180 degrees in absolute geometry implies Euclid's parallel postulate.

In absolute geometry, the Saccheri-Legendre theorem asserts that the sum of the angles in a triangle is at most 180 .

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

Indeed, in Euclid's Elements, the first 28 Propositions avoid using the parallel postulate, and therefore are valid in absolute geometry.

Alexiou later created Solar Houses, minimalistic constructions of cane and reeds which share a fragile feeling of absolute geometry.

Thus every theorem of absolute geometry is a theorem of hyperbolic geometry and Euclidean geometry.

Negating the Axiom of Euclid yields hyperbolic geometry, while eliminating it outright yields absolute geometry.

Lines and points are undefined terms in absolute geometry, but assigned meanings in the theory of real numbers in a way that is consistent with both axiom systems.

One can extend absolute geometry by adding different axioms about parallel lines and get incompatible but consistent axiom systems, giving rise to Euclidean or hyperbolic geometry.

Absolute geometry - an extension of ordered geometry that is sometimes referred to as neutral geometry because its axiom system is neutral to the parallel postulate.

It is a fundamental geometry forming a common framework for affine geometry, Euclidean geometry, absolute geometry and hyperbolic geometry.

The theorems of absolute geometry hold in hyperbolic geometry, which is a non-Euclidean geometry, as well as in Euclidean geometry.

Versions of a tropical geometry, of an absolute geometry over a field with one element and an algebraic analogue of Arakelov geometry were realized in this setup.

Absolute geometry is an extension of ordered geometry, and thus, all theorems in ordered geometry hold in absolute geometry.

Absolute geometry assumes the first four of Euclid's Axioms (or their equivalents), to be contrasted with affine geometry, which does not assume Euclid's third and fourth axioms.

In any of these systems, removal of the one axiom which is equivalent to the parallel postulate, in whatever form it takes, and leaving all the other axioms intact, produces absolute geometry.

However, in the presence of the remaining axioms which give Euclidean geometry, each of these can be used to prove the other, so they are equivalent in the context of absolute geometry.

As the first 28 propositions of Euclid (in The Elements) do not require the use of the parallel postulate or anything equivalent to it, they are all true statements in absolute geometry.

Shiro Kuramata, one of Japan's most influential and radical designers, mastered the first two in the postwar years, when he was among the first to break with the modernists' death grip on absolute geometries.

Pambuccain, Victor Axiomatizations of hyperbolic and absolute geometries, in: Non-Euclidean geometries (A. Prékopa and E. Molnár, eds.)