Ordered geometry

Builds on Line Betweenness Pasch's axiom

Definition

Let $\Pi$ be a set of points with a family of subsets called lines. Let $[ABC]$ be a ternary relation on $\Pi$ , read " $B$ lies between $A$ and $C$ ". Then $\Pi$ is an ordered plane if:

Two distinct points lie on exactly one line, each line has at least two points, and there are three points not on one line.
If $[ABC]$ , then $A$ , $B$ , $C$ are three distinct points of one line, and $[CBA]$ .
For distinct points $A$ and $C$ , there is a point $B$ with $[ACB]$ .
Of three distinct points of a line, at most one lies between the other two.
Pasch's axiom holds.

Ordered geometry is the study of the consequences of these axioms, and of their analogues in higher dimensions.

Remarks

Example: the plane $F^2$ over an ordered field $F$ , with the usual lines and with $[ABC]$ meaning $B = A + t(C - A)$ for some $t \in F$ with $0 < t < 1$ , is an ordered plane. Non-example: take $\R^2$ with the usual lines, and let $[ABC]$ mean that $A$ , $B$ , $C$ are three distinct points of one line. This fails clause 4, since each of three collinear points then lies between the other two. One proves from the axioms that, of three distinct collinear points, exactly one lies between the other two. The Sylvester–Gallai theorem holds in every ordered plane.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Betweenness concept
Line concept
Triangle concept
Pasch's axiom statement
Ordered geometry definition

Tag	`N56DHWH283`
URI	`m2://N56DHWH283`
Kind	definition
Writing	W0 stub
MSC	51G05
Defines	Ordered geometry
Uses	Line, Betweenness, Pasch's axiom

Ordered geometry

Definition

Remarks

Reading path

Further reading

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