Some further information about non-Euclidean geometry:

Our handout on the parallel postulate and logic.

A few other statements equivalent to the Parallel Postulate:

• The Pythagorean Theorem
• The sum of the interior angles of a triangle is equal to the sum of two right angles
• There exists a triangle whose interior angles sum to the sum of two right angles
• Any three noncollinear points lie on a circle
• If three angles of a quadrilateral are right angles, so is the fourth

Spherical geometry:
brief overview
longer treatment

Hyperbolic geometry:
introduction ("neutral geometry" means the first four postulates)
one of the most common visualizations
Escher pictures that are hyperbolic: one two
miscellaneous hyperbolic geometry links

Back