Difference between revisions of "Chapter 2: An ancient theorem and a modern question"

This is the fancy name for the basic geometry we are familiar with where parallel lines do not intersect. The rules or "postulates" of Euclidian geometry are as follows.

This is the fancy name for the basic geometry we are familiar with where parallel lines do not intersect. The rules or "postulates" of Euclidian geometry are as follows.

# A straight line segment can be drawn joining any two points

# A straight line segment can be drawn joining any two points

Euclid's fifth postulate cannot be proven as a theorem, although this was attempted by many people. Euclid himself used only the first four postulates ("absolute geometry") for the first 28 propositions of the Elements, but was forced to invoke the parallel postulate on the 29th. In 1823, Janos Bolyai and Nicolai Lobachevsky independently realized that entirely self-consistent "non-Euclidean geometries" could be created in which the parallel postulate did not hold. (Gauss had also discovered but suppressed the existence of non-Euclidean geometries.)

Euclid's fifth postulate cannot be proven as a theorem, although this was attempted by many people. Euclid himself used only the first four postulates ("absolute geometry") for the first 28 propositions of the Elements, but was forced to invoke the parallel postulate on the 29th. In 1823, Janos Bolyai and Nicolai Lobachevsky independently realized that entirely self-consistent "non-Euclidean geometries" could be created in which the parallel postulate did not hold. (Gauss had also discovered but suppressed the existence of non-Euclidean geometries.)

=== Radians and <math> \pi </math> ===

=== Radians and <math> \pi </math> ===