** 2.4.1 Plane Separation Postulate
***Printout*

*Whoever… proves his point and demonstrates the prime truth
geometrically should be believed by all the world, for there we are captured.*

**—****Albrecht Dürer*** (1471–1528)*

** **

An important
axiom that is often not considered in a high school geometry course is the
Plane Separation Axiom. The Missing Strip plane is a model that satisfies all
the axioms we have discussed to this point. It is included for the sole purpose
of demonstrating the necessity for the Plane Separation Axiom.

** Definition.**
A set

Consider the interiors of the above three figures. Note that the first and last figures have segments that are not completely contained in their interiors. The first and last sets are not convex. It is important to note that the definition of convex depends on a segment; a set that is convex in the Poincaré Half-plane may not be convex as a set in the Euclidean plane. Consider the interior of the following quadrilateral constructed in the Poincaré Half-plane. The set is convex in the Poincaré Half-plane but is not a convex set in the Euclidean plane.

** Postulate 9.**
(

Note that the line and the two half-planes are disjoint sets; that is, a line forms three disjoint sets, the line and two half-planes. To illustrate the Plane Separation Postulate, consider the Cartesian plane, Missing Strip plane, and Poincaré Half-plane.

The
line in the Cartesian plane separates the plane into two convex sets, the
region above the line and the region below the line. The line in the Missing Strip
plane does ** not** separate the plane into two convex sets. See the
illustration, the segment shown does not intersect the line since the light
blue region between 0 and 1 does not exist in the plane. The line in the
Poincaré Half-plane separates the plane into two convex sets.

A statement that is sometimes used in place of the Plane Separation Postulate is Pasch's Postulate. It can be proven that the Plane Separation Postulate and Pasch's Postulate are equivalent,

The above diagrams illustrate
one example of Pasch's Postulate in the Euclidean plane,
Missing Strip plane, and Poincaré Half-plane. Is Pasch's Postulate satisfied in
each plane?

** Reminder. **Prepared
Geometer's Sketchpad sketches and GeoGebra sketches with tools for constructions in the Missing Strip
plane and Poincaré Half-plane are available in Appendix B of the Course Title
Page - Geometer's Sketchpad
and GeoGebra Prepared Sketches and Scripts. Also, an on-line java based
program called

** Exercise 2.28. **
Is
the Plane Separation Postulate discussed or mentioned in the high school
geometry book you are using as a reference?

© Copyright 2005, 2006 - Timothy Peil |