Ordered geometry

Ordered geometry is a special kind of geometry that helps us understand the relationships between points, lines, and planes. Unlike some other types of geometry, it does not focus on measuring distances or angles. Instead, it uses the idea of "betweenness," which tells us when a point lies between two other points on a line.

One important feature of ordered geometry is that it leaves out the basic idea of measurement, much like projective geometry. This makes it a useful tool for building other kinds of geometry. It serves as a common framework for several important geometries, including affine, Euclidean, absolute, and hyperbolic geometry. However, it does not work for projective geometry, which is another type that also lacks measurement.

Because ordered geometry focuses on the order of points rather than exact measurements, it helps mathematicians study the deeper structures and relationships in space. This makes it an important area of study in both pure mathematics and its applications.

History

Moritz Pasch first described a type of geometry without using measurements in 1882. Later, other mathematicians like Peano, Hilbert, and Veblen improved his ideas. Even long ago, Euclid had similar thoughts in his famous book The Elements.

Primitive concepts

The only basic ideas in ordered geometry are points, like A, B, C, and so on, and a special way to describe how points are connected. We use a ternary relation written as [ABC], which means that point B is between point A and point C. This idea of "betweenness" helps us understand the order of points without needing to measure distances.

Main article: Primitive notions
Links: points, ternary relation

Definitions

In ordered geometry, we describe shapes and positions using special terms. A segment AB is all the points P that lie between points A and B. An interval AB includes the segment AB and its end points A and B. A ray A/B_ is the set of points P that start at A and go away from B. A line AB is made up of the interval AB and the two rays starting at A and B.

We also have angles, which have a point in the middle called the vertex and two rays coming out from it. A triangle is formed by three points that aren’t on the same line, called vertices, and the three segments connecting them. When four points aren’t all on the same flat surface, they form a space, which includes all points that line up with pairs of points from any of the four flat surfaces of the shape made by these four points, called a tetrahedron.

Axioms of ordered geometry

Ordered geometry is a type of geometry that includes the idea of "betweenness" — knowing which point lies between two others — but does not measure distances like length or angle. It helps us understand how points, lines, and planes relate to each other.

Some important rules, or axioms, in ordered geometry include: there must be at least two points; if you have two points, there is always another point between them; and points can be arranged in lines and planes in specific ways. These ideas form the basis for many kinds of geometry, such as Euclidean geometry, which is the geometry most people learn in school.

Main articles: Axiom of Pasch, Axiom of dimensionality, Hilbert's axioms of order

Results

Ordered geometry can be used to prove the Sylvester–Gallai theorem, which deals with points and lines arranged in special ways.

Important thinkers like Gauss, Bolyai, and Lobachevsky studied the idea of parallel lines, which can also be described using ordered geometry. They showed that for any point and a line not passing through that point, there are exactly two special directions from the point that never meet the line. This helps define what we mean by a line being parallel to another. However, ordered geometry cannot prove that parallelism works in the same way for all lines, meaning it does not create a consistent relationship between all parallel lines.