Notice: This material will be included in a forthcoming (summer 2000) book with the tentative title Experiencing Geometry in Euclidean, Spherical, and Hyperbolic Spaces. This new book will be an expanded and updated version of Experiencing Geometry on Plane and Sphere. This material is in draft form and may not be duplicated or quoted without the author's written permission, except for purposes of review or trying out the material with students. As always comments are welcome and will affect the final draft. Send comments to firstname.lastname@example.org.
Let the following be postulated: that, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
Euclid, Elements, Postulate 5 [A: Euclid]
Parallel Lines on the Plane Are Special
Up to this point we have not had to assume anything about parallel lines. No version of a parallel postulate has been necessary either on the plane or on a sphere or on a hyperbolic plane. We defined the concrete notion of parallel transport and proved in Problem 8.2 that on the plane parallel transported lines do not intersect. Now in this chapter we will look at two important properties on the plane:
If two lines on the plane are parallel transports of each other along some transversal, then they are parallel transports along any transversal. (Problem 10.1)
On the plane the sum of the interior angles of a triangle is always 180°. (Problem 10.4)
Neither of these properties is true on a sphere or on a hyperbolic plane and thus both need an additional assumption on the plane for their proofs. The various assumptions that permit proofs of these two statements are collectively termed the Parallel Postulates. Only the two statements above are needed from this chapter for the rest of the book. Therefore, it is possible to omit this chapter and assume one of the above two statements and then prove the other. However, parallel postulates have a historical importance and have a central position in many geometry textbooks and in many expositions about non-Euclidean geometries. The problems in this chapter are an attempt to help people unravel and enhance their understanding of parallel postulates. Comparing the situations on the plane with a sphere and a hyperbolic plane is a powerful tool for unearthing our hidden assumptions and misconceptions about the notion of parallel on the plane.
Since we have so many (often unconscious) connotations and assumptions attached to the word "parallel," we find it best to avoid using the term parallel as much as possible in this discussion. Instead we will use terms like "parallel transport," "non-intersecting," and "equidistant."
Problem 10.1. Parallel Transport on the Plane
Show that if l1 and l2 are lines on the plane such that they are parallel transports along a transversal l, then they are parallel transports along any transversal. Prove this using any assumptions you find necessary. Make as few assumptions as you can, and make them as simple as possible. Be sure to state your assumptions clearly.
What part of your proof does not work on a sphere or on a hyperbolic plane?
This problem is by no means as trivial as it, at first, may appear. In order to prove this theorem, you will have to assume something there are many possible assumptions, so use your imagination. But at the same time, try not to assume any more than is necessary. If you're having trouble deciding what to assume, try to solve the problem in a way that seems natural to you and see what develops.
On a sphere, try the same construction and proof you used for the plane. What happens? You should find that your proof does not work on a sphere or on a hyperbolic plane. So, what is it about your proof (or a sphere and hyperbolic plane) that creates difficulties?
Again, you may be tempted to use "the sum of the angles of a triangle is 180°" as part of your proof. As in many other cases before, there is nothing wrong with doing the problems out of order you can use Problem 10.4 to prove Problem 10.1 as long as you don't also use Problem 10.1 to prove Problem 10.4. Most people find it much easier to prove Problem 10.1, first, and then use it to prove Problem 10.4.
Problem 10.1 emphasizes the differences between parallelism on the plane and parallelism on a sphere. On the plane, non-intersecting lines exist, and one can "parallel transport" everywhere. Yet, as was seen in Problems 8.2 and 8.3, on a sphere two lines are cut at congruent angles if and only if the transversal line goes through the center of the lune formed by them. That is, on a sphere two lines are locally parallel only when they can be parallel transported through the center of the lune formed by them. Be sure to draw a picture of the lune locating the center and the transversal. On a sphere (and on a hyperbolic plane) it is impossible to slide the transversal along two parallel transported lines keeping both angles constant which is something you can do on the plane. In Figure 10.1a and Figure 10.1b, the line t' is a parallel transport of line t along line l, but it is not a parallel transport of t along l¢.
Figure 10.1a. Parallel transport on a sphere along l, but not along l'.
Figure 10.1b. Parallel transport on a hyperbolic plane along l but not along l'.
Pause, explore, and write out your ideas before reading further.
Parallel Circles on a Sphere
Figure 10.2. Special equidistant circles.
The latitude circles on the earth are sometimes called "Parallels of Latitude." They are parallel in the sense that they are everywhere equidistant as are concentric circles on the plane. In general, transversals do not cut equidistant circles at congruent angles. However, there is one important case where transversals do cut the circles at congruent angles. Let l and l¢ be latitude circles which are the same distance from the equator on opposite sides of it. See Figure 10.2. Then, every point on the equator is a center of half-turn symmetry for these pair of latitudes. Thus, as in Problems 8.3 and 10.1, every transversal cuts these latitude circles in congruent angles.
One of Euclid's assumptions constitutes Euclid's Fifth (or Parallel) Postulate (EFP), which says:
If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles.
For a picture of EFP, see Figure 10.3.
Figure 10.3. Euclid's Parallel Postulate.
You probably did not assume EFP in your proof of Problem 10.1. You are in good company many mathematicians, including Euclid, have tried to avoid using it as much as possible. However, we will explore EFP because, historically, it is important, and because it has some very interesting properties as you will see in Problem 10.3. On a sphere, all straight lines intersect twice which means that EFP is trivially true on a sphere. But in Problem 10.3, you will show that EFP is also true in a stronger sense on spheres. You will also be able to prove that EFP is false on a hyperbolic plane.
Thus, EFP does not have to be assumed on a sphere it can be proved! However, in most geometry text books, EFP is substituted by another postulate which, it is claimed, is equivalent to EFP. This postulate is Playfair's Parallel Postulate (PPP), and it can be expressed in the following way:
For every line l and every point P not on l, there is a unique line l¢ which passes through P and is parallel to l.
Figure 10.4. Playfair's Parallel Postulate.
Note that, on a sphere, since any two great circles intersect, there are no lines l¢ which are parallel to l in the "not intersecting" sense. Therefore, Playfair's Postulate is not true on spheres. On the other hand, if we change "parallel" to "parallel transport" then every great circle through P is a parallel transport of l along some transversal. What happens on a hyperbolic plane? In Problem 10.2, you will explore the relationships among EFP, Playfair's Postulate, and the assumptions you used in Problem 10.1.
Problem 10.2. Parallel Postulates on the Plane
On the plane, are EFP, Playfair's Postulate, and your postulate from Problem 10.1 equivalent? Why? Or, why not?
To show that EFP and Playfair's Postulate are equivalent on the plane, you need to show that you can prove EFP if you assume Playfair's Postulate and vice versa. Do the same for your postulate from Problem 10.1. If the three postulates are equivalent, then you can prove the equivalence by showing that
EFP Þ PPP Þ Your Postulate Þ EFP
or in any other order. It will probably help you to draw lots of pictures of what is going on. Note that PPP is not true on a sphere but EFP is true, so therefore your proof that EFP implies PPP on the plane must use some property of the plane that does not hold on a sphere. Look for it.
Problem 10.3. The P P on Sphere & Hyperbolic Plane
a. Show that EFP is true on a sphere in a strong sense; that is, if lines l and l¢ are cut by a transversal t such that the sum of the interior angles a + b on one side is less than two right angles, then, not only do l and l¢ intersect, but they also intersect "closest" to t on the side of a and b. You will have to determine an appropriate meaning for "closest."
To help visualize the postulates, draw these "parallels" on an actual sphere. There are really two parts to this proof first, you must come up with a definition of "closest" and, then, prove that EFP is true for this definition. The two parts may come about simultaneously as you come up with a proof. This problem is closely related to Euclid's Exterior Angle Theorem, but can also be proved without using EEAT. One case that you should look at specifically is pictured in Figure 10.5. It is not necessarily obvious how to define the "closest" intersection.
Figure 10.5. Is EFP true on a sphere?
b. On a hyperbolic plane let l be a geodesic and let P be a point not on l, then show that there is an angle q with the property that any line l¢ passing through P is parallel to (not intersecting) l if the line l¢ does not form an angle less than q with the line from P which is perpendicular to l. (See Figure 10.6.)
Figure 10.6. Multiple parallels on a hyperbolic plane.
c. Using the notion of parallel transport, change Playfair's Postulate so that the changed postulate is true on both spheres and hyperbolic planes. Make as few alterations as possible and keep some form of uniqueness.
d. Either prove your postulate from Problem 10.1 on a sphere and on a hyperbolic plane or change it, with as few alterations as possible, so that it is true on these surfaces. You may need to make different changes for the two surfaces.
Suggestions for b. and c.
Above we noted that Playfair's Postulate is not true on a sphere or a hyperbolic space, and in Problem 10.1 you should have decided whether or not your postulate is true on spheres or on hyperbolic spaces. The next step is to come up with a modified versions of the postulate, that are true on a sphere or on a hyperbolic plane. Try to limit the modifications you make so that the new postulate preserves the spirit of the old one. You can draw ideas from any of the previous problems to obtain suitable modifications. Then, prove that your modified versions of the postulate are true.
Parallelism in Spherical and Hyperbolic Geometry
Playfair's Postulate assume both the existence and uniqueness of parallel lines. In Problem 8.2, it was proven that if one line is a parallel transport of another, then the lines do not intersect on the plane or on a hyperbolic plane; that is, they are parallel. Thus, it is not necessary to assume the existence of parallel lines. On a sphere any two lines intersect. However, in Problem 8.4 we saw that there are non-intersecting lines that are not parallel transports of each other on a hyperbolic plane and on any cone with cone angle larger than 360°.
Figure 10.6 is an attempt to represent the relationships among parallel transport, non-intersecting lines, EFP, and Playfair's Postulate. Can you fit your postulate into the diagram?
Figure 10.7. Parallelism.
Problem 10.4. Sum of the Angles of a Planar Triangle
a. What is the sum of the angles of a triangle on the plane?
b. Show that the postulates in Problem 10.2 are equivalent to "The sum of the angles of a triangle on the plane is a constant."
c. What happens to your proof on spheres and hyperbolic planes?
There are many approaches to this problem. It can be done using only results from this chapter or results from other chapters may also be used. Be sure to draw pictures and to be careful about what previous results you are using.
Remember: If you assumed facts about the sum of the angles of a triangle on the plane in a previous problem, then you can not use the results of that problem here.
What happens on spheres and hyperbolic planes?