Parallel Principle Revisited Help

Introduction to Geometry Postulates

Conventional geometry is based on five axioms, also called postulates , that were first stated by a Greek mathematician named Euclid who lived in the 3rd century Everything we have done in this book so far—even the theoretical problems involving four dimensions—has operated according to Euclid’s five axioms. We have been dealing exclusively with Euclidean geometry . That is about to change.

Euclid’s Axioms

Let’s state explicitly the things Euclid believed were self-evident truths. Euclid’s original wording has been changed slightly, in order to make the passages sound more contemporary. Examples of each postulate are shown in Fig. 11-8.

• Any two points P and Q can be connected by a straight line segment (Fig. 11-8A)
• Any straight line segment can be extended indefinitely and continuously to form a straight line (Fig. 11-8B) 

Fig. 11-8 . Euclid’s original five axioms. See text for discussion.

• Given any point P , a circle can be defined that has that point as its center and that has a specific radius r (Fig. 11-8C)
• All right angles are congruent; that is, they have equal measure (Fig. 11-8D)
• Suppose two lines L and M lie in the same plane and both lines are crossed by a transversal line N . Suppose the measures of the adjacent interior angles x and y sum up to less than 180° (π rad). Then lines L and M intersect on the same side of line N as angles x and y are defined (Fig. 11-8E)

The Parallel Postulate

The last axiom stated above is known as Euclid’s fifth postulate . It is logically equivalent to the following statement that has become known as the parallel postulate :

• Let L be a straight line, and let P be some point not on L . Then there exists one and only one straight line M , in the plane defined by line L and point P , that passes through point P and that is parallel to line L

This axiom—and in particular its truth or untruth—has received enormous attention. If the parallel postulate is denied, the resulting system of geometry still works. People might find it strange, but it is logically sound! Geometry doesn’t need the parallel postulate. There are two ways in which the parallel postulate can be denied:

• There is no line M through point P that is parallel to line L
• There are two or more lines M ₁ , M ₂ , M ₃ , . . . through point P that are parallel to line L

When either of these postulates replaces the parallel postulate, we are dealing with a system of non-Euclidean geometry. In the 2D case, it is a non-Euclidean surface . Visually, such a surface looks warped or curved. There are, as you can imagine, infinitely many ways in which a surface can be non-Euclidean.

Geodesics

In a non-Euclidean universe, the concept of “straightness” must be modified. Instead of thinking about “straight lines” or “straight line segments,” we must think about geodesics .

Suppose there are two points P and Q on a non-Euclidean surface. The geodesic segment or geodesic arc connecting P and Q is the set of points representing the shortest possible path between P and Q that lies on the surface. If the geodesic arc is extended indefinitely in either direction on the surface beyond P and Q , the result is a geodesic .

The easiest way to imagine a geodesic arc is to think about the path that a thin ray of light would travel between two points, if confined to a certain 2D universe. The extended geodesic is the path that the ray would take if allowed to travel over the surface forever without striking any obstructions. On the surface of the earth, a geodesic arc is the path that an airline pilot takes when flying from one place to another far away, such as from Moscow, Russia to Tokyo, Japan (neglecting takeoff and landing patterns, and any diversions necessary to avoid storms or hostile air space).

When we re-state the parallel postulate as it applies to both Euclidean and non-Euclidean surfaces, we must replace the term “line” with “geodesic.” Here is the parallel postulate given above, modified to cover all contingencies.