Nonlinear Tessellations

Background

Certainly, the best-known of Euclid's postulates is the parallel postulate. The familiar form is

Through a point not on a given line, exactly one line is parallel to the given line.

The parallel postulate has such a different charater from the other four that for twenty centuries people sought to derive the parallel postulate from the other four.

Eventually, these attempts were shown to be in vain because there are models of geometry in which the first four postulates and a negation of the parallel postulate are true.

This would be impossible if the parallel postulate were a consequence of the other four.

The parallel postulate can be negated in two ways:

Through a point not on a given line, no line is parallel to the given line.

Through a point not on a given line, more than one line is parallel to the given line.

The first is spherical geometry, the second hyperbolic geometry.

Return to background.