- #1

lavinia

Science Advisor

Gold Member

- 3,145

- 563

## Summary:

- Is the existence of parallels provable in Euclidean and Hyperbolic plane geometry?

## Main Question or Discussion Point

Parallels exist in both Euclidean and Hyperbolic geometry. Yet each includes a separate postulate that declares the number of parallels to a line in a plane through a given point. But it seems that if both geometries have parallels then their existence - as opposed to how many of them - should be provable without a parallel postulate. One should be able to derive them from the other postulates, the postulates that describe points and lines and their intersections and how they separate the plane.

My question is: Is there a proof of the existence of parallels without Euclid's Fifth Postulate or its equivalents such as Playfair's Axiom? Same question for Hyperbolic geometry.

The following postulates describe simple properties of the plane that it seems one would have to use to prove the existence of parallels without a parallel postulate.

- Given two points in a plane there is a unique line that contains both of them.

- Two lines that intersect intersect in exactly one point

- A point on a line separates the line into two rays - or half lines. The two rays intersect in the point.

- A line separates the plane into two half planes. These two half planes intersect in the line.

- Let two lines ##L_1## and ##L_2## intersect in the point ##P##. Let ##R_1## and ##R_2## be the two rays on ##L_1## determined by ##P##. Then ##R_1## lies completely in one of the half planes determined by ##L_2## and ##R_2## lies completely in the other.

Note: These postulates exclude Elliptic geometry since, there, two lines intersect in two points. One might try identifying each pair of intersection points to get only one point but then a line will not separate the plane into two disjoint half planes. If one pictures this geometry as a sphere with great circles as the lines then identifying opposite poles creates the projective plane. In the projective plane the projections of the great circles do intersect in a single point but they do not separate the plane into two pieces. This is because the inverse image of the projective plane minus the projection of the circle is two hemispheres. But these are identified in the projective plane. Also on a sphere opposite poles do not determine a unique line. So it seems that an axiomatic version of this geometry would have to say something like if two points are not "opposite" then they determine a unique line but if they are "opposite" they do not. It would be interesting to derive spherical geometry axiomatically.

Geometric intuition says that two perpendicular lines to a given line ##L## must be parallel. Why is this? Suppose they are not. Then they must intersect in one of the half planes and not there other. But which half plane would that be? The lines make right angles to ##L## in both half planes so there is no difference in the way they enter into the half planes. One would think therefore if they intersected in one they would have to intersect in the other. There seems to be a principle of symmetry here that is implicit in the intuition but is not stated in the postulates. Still one would think that this symmetry is intrinsic to the idea of these two geometries and is independent of any parallel postulate.

So is there a proof or is some additional postulate dealing with symmetry needed?

My question is: Is there a proof of the existence of parallels without Euclid's Fifth Postulate or its equivalents such as Playfair's Axiom? Same question for Hyperbolic geometry.

The following postulates describe simple properties of the plane that it seems one would have to use to prove the existence of parallels without a parallel postulate.

- Given two points in a plane there is a unique line that contains both of them.

- Two lines that intersect intersect in exactly one point

- A point on a line separates the line into two rays - or half lines. The two rays intersect in the point.

- A line separates the plane into two half planes. These two half planes intersect in the line.

- Let two lines ##L_1## and ##L_2## intersect in the point ##P##. Let ##R_1## and ##R_2## be the two rays on ##L_1## determined by ##P##. Then ##R_1## lies completely in one of the half planes determined by ##L_2## and ##R_2## lies completely in the other.

Note: These postulates exclude Elliptic geometry since, there, two lines intersect in two points. One might try identifying each pair of intersection points to get only one point but then a line will not separate the plane into two disjoint half planes. If one pictures this geometry as a sphere with great circles as the lines then identifying opposite poles creates the projective plane. In the projective plane the projections of the great circles do intersect in a single point but they do not separate the plane into two pieces. This is because the inverse image of the projective plane minus the projection of the circle is two hemispheres. But these are identified in the projective plane. Also on a sphere opposite poles do not determine a unique line. So it seems that an axiomatic version of this geometry would have to say something like if two points are not "opposite" then they determine a unique line but if they are "opposite" they do not. It would be interesting to derive spherical geometry axiomatically.

Geometric intuition says that two perpendicular lines to a given line ##L## must be parallel. Why is this? Suppose they are not. Then they must intersect in one of the half planes and not there other. But which half plane would that be? The lines make right angles to ##L## in both half planes so there is no difference in the way they enter into the half planes. One would think therefore if they intersected in one they would have to intersect in the other. There seems to be a principle of symmetry here that is implicit in the intuition but is not stated in the postulates. Still one would think that this symmetry is intrinsic to the idea of these two geometries and is independent of any parallel postulate.

So is there a proof or is some additional postulate dealing with symmetry needed?

Last edited: