Absolute geometry

Absolute geometry is a special kind of geometry. It studies shapes and spaces using only a few basic rules. It does not use a rule called the parallel postulate. This rule is part of traditional Euclidean geometry. It talks about lines that never meet.

The idea of absolute geometry was first introduced by a mathematician named János Bolyai in 1832. He wanted to see how geometry would look without the parallel postulate. Because of this, absolute geometry is sometimes called neutral geometry. This means it doesn’t decide if the parallel postulate is true or not.

Today, we know that the first four rules of Euclid's postulates alone aren’t enough to describe all of Euclidean geometry completely. Mathematicians often use other sets of rules, like Hilbert's axioms without the parallel axiom, to build absolute geometry. This helps us understand the foundations of geometry better.

Properties

In Euclid's Elements, the first 28 Propositions and Proposition 31 do not need the parallel postulate. This means we can still prove important ideas like the exterior angle theorem. This theorem tells us that an exterior angle of a triangle is larger than either of the remote angles. We can also prove the Saccheri–Legendre theorem. This theorem says that the angles in any triangle add up to at most 180°.

Proposition 31 shows how to draw a parallel line to a given line through a point not on that line. This can be done without using the parallel postulate. We only need basic ideas about angles and perpendicular lines. In absolute geometry, we also know that two lines perpendicular to the same line will never meet. This means they must be parallel.

Relation to other geometries

The theorems of absolute geometry work in both hyperbolic geometry and Euclidean geometry. They do not work in elliptic geometry or spherical geometry because these have different rules about points and lines.

Absolute geometry builds on ordered geometry, so all the rules of ordered geometry also apply to absolute geometry. It uses the first four of Euclid's basic ideas, unlike affine geometry, which skips two of these ideas. The ideas of absolute geometry helped start the development of the geometry of special relativity.

Hilbert planes

A Hilbert plane is a special flat space that follows basic rules, called axioms. These rules explain how points and lines relate to each other. Hilbert planes help us understand absolute geometry. Absolute geometry is a type of geometry that does not need one special rule that Euclidean geometry uses.

Incompleteness

Absolute geometry is an incomplete axiomatic system. This means you can add new rules without causing problems.

By adding different rules about parallel lines, you can create two different types of geometry. One is Euclidean geometry, and the other is hyperbolic geometry.

Every idea in absolute geometry also works in both Euclidean and hyperbolic geometry. However, not every idea from those two geometries works in absolute geometry.