Non-Euclidean Geometry: From Parallel Postulate to Models, Lecture notes of Geometry

Download Non-Euclidean Geometry: From Parallel Postulate to Models and more Lecture notes Geometry in PDF only on Docsity!

NON-EUCLIDEAN

GEOMETRY

FROM PARALLEL POSTULATE TO MODELS

GREEK GEOMETRY

Greek Geometry was the first example of a deductive system with axioms, theorems, and proofs. Greek Geometry was thought of as an idealized model of the real world. Euclid (c. 330-275 BC) was the great expositor of Greek mathematics who brought together the work of generations in a book for the ages.

Considering Euclid’s

Postulates

One reason that Euclidean geometry was at the center of philosophy, math and science, was its logical structure and its rigor. Thus the details of the logical structure were considered quite important and were subject to close examination. The first four postulates, or axioms, were very simply stated, but the Fifth Postulate was quite different from the others.

Postulates I-IV

I. A straight line segment can be drawn

joining any two points.

II. Any straight line segment can be

extended indefinitely in a straight line.

III. Given any straight line segment, a

circle can be drawn having the segment

as radius and one endpoint as center.

IV. All right angles are congruent.

A picture of Postulate V

If the sum of angles 1 and 2 is less than a straight angle, then the lines intersect at a point C on the same side of the line AB as the two angles. 2 1 A B 2 1 C A B

Why Postulate V is the

Parallel Postulate

The Postulate does not mention the word parallel, but for a line m through A and any line n through a point B not on m, this rules out the possibility that line n is parallel to m except when two interior angles add up to a straight angle. So there is only one possible line through B parallel to m. It can be proved that this line is in fact parallel.

Attempts to Prove V

The attempts to prove postulate V dig deep into subtleties of basic geometry. Here are some theorems, that if proved, imply V: The distance between two parallel lines is finite. (Proclus, 410-485) A quadrilateral with two equal sides perpendicular to the base is a rectangle. Saccheri, 1667-1733) A quadrilateral with 3 right angles is a rectangle. (Lambert, 1728-1777)

Assuming V False to

Prove Euclid Right

Such mathematicians as Saccheri attempted proofs of Postulate V by contradiction. They assumed that V is false, then proved many theorems based on this assumption -- with the goal of finding a contradiction. Saccheri never really found a contradiction but he concluded that Euclid was vindicated anyway because the theorems were so odd.

Controversy

The idea of such a new geometry shook mathematics and science to its foundations. There was much doubt and debate about this new geometry. It seemed very strange, and there was no proof that the next week a contradiction would not be discovered. However, the doubts were resolved by the discovery of models for the new geometry.

Models

A mathematical model of an abstract system such as non-Euclidean geometry is a set of objects and relations that satisfy as theorems the axioms of the system. Then the abstract system is as consistent as the objects from which the model made. So if a model of non-Euclidean geometry is made from Euclidean objects, then non-Euclidean geometry is as consistent as Euclidean geometry.

Strange Theorems

Thus we can prove theorems in non-Euclidean geometry by proofs about the models. Some parallel line pairs have just one common perpendicular and grow far apart. Other parallels get close together in one direction. Angle sums of triangles are less than 180 degrees. There are no rectangles at all.

Curvature

Riemann was the first geometer who really sorted out a key concept in geometry. He made a general study of curvature of spaces in all dimensions. In 2-dimensions: Euclidean geometry is flat (curvature = 0) and any triangle angle sum = 180 degrees. The non-Euclidean geometry of Lobachevsky is negatively curved, and any triangle angle sum < 180 degrees. The geometry of the sphere is positively curved, and any triangle angle sum > 180 degrees.

Why is non-Euclidean

Geometry Important?

The discovery of non-Euclidean geometry opened up geometry dramatically. These new mathematical ideas were the basis for such concepts as the general relativity of a century ago and the string theory of today. The idea of curvature is a key mathematical idea. Plane hyperbolic geometry is the simplest example of a negatively curved space. Spherical geometry has even more practical applications.