nLab betweenness

Idea

Betweenness is a ternary relation on a set of points on a geometric line. The notion axiomatizes the essential difference between a segment and a line by means of an ordering.

Axiomatic definition

In the context of incidence geometry, we introduce the notion of order of “strict betweenness” as follows (yielding an ordered incidence geometry).

Each line $l$ in ordered incidence geometry is equipped with a ternary relation $a b c$, where $a,b,c$ are three distinct points on $l$ (that is, incident with $l$). We then say that $b$ is between $a$ and $c$. If $a\neq c$ denote by $(a,c)$, the open interval determined by $a$ and $c$, as the set of all points on line $l$ between $a$ and $c$. The segment $\overline{a c}$ is the set of all points between $a$ and $c$ including also the end points $a$ and $c$.

Axioms (see Ben-Tal, Ben-Israel)

A1 If $a b c$ then $c b a$.

A2 If $a\neq b$ then there are $c, d$ such that $a c b$ and $a b d$.

A3 If $a,b,c$ are collinear and distinct then precisely one of them is between the other two.

A4 (Pasch’s axiom) If $a,b,c$ are three noncollinear points, and $l$ a line in their affine hull (the plane determined by $a,b,c$) then if $l$ intersects $(a,b)$ then it intersects either $(a,c)$ or $(b,c)$.

From linear order relations

One can alternatively postulate that on every line one is given two mutually opposite linear orders $\leq_1,\leq_2$ and define the non-strict betweenness $[a b c]$ if $a \leq b\leq c$ in (at least) one of the distinguished orders and a strict betweenness $a b c$ if $a \lt b \lt c$ in (precisely) one of the distinguished orders (thus excluding the possibility that $b$ equals either $a$ or $c$ in the strict case). Then one requires just the Pasch’s axiom (e.g. PavkovićVeljan).

Literature

The above approach goes back to

• M. Pasch around 1882

Further discussion:

• Aharon Ben-Tal, Adi Ben-Israel, Ordered incidence geometry and the geometric foundations of convexity theory, Journal of Geometry 30 (1987)

• M. Pasch, Max Dehn, Vorlesungen ueber die neuere Geometrie,. Springer-Verlag, Berlin, 1926 (reprinted 1976), x + 275 pp

• B. Pavković, D. Veljan, Elementarna matematika I, Školska knjiga 2004, Zagreb (in Croatian)

category: geometry