Non-Euclidean Geometry Quiz

Non-Euclidean Geometry: A Summary

• Non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry: hyperbolic geometry and elliptic geometry.

• The essential difference between the metric geometries is the nature of parallel lines.

• Euclid's fifth postulate, the parallel postulate, is equivalent to Playfair's postulate.

• In hyperbolic geometry, there are infinitely many lines through a point A not intersecting l, while in elliptic geometry, any line through A intersects l.

• Non-Euclidean geometries began to be developed in the 19th century, with the work of mathematicians such as Lobachevsky, Bolyai, and Riemann.

• Euclidean geometry includes some of the oldest known mathematics, and geometries that deviated from this were not widely accepted as legitimate until the 19th century.

• Several modern authors still use the generic term non-Euclidean geometry to mean hyperbolic geometry.

• The Cayley–Klein metrics provided working models of hyperbolic and elliptic metric geometries, as well as Euclidean geometry.

• There are many kinds of geometry that are quite different from Euclidean geometry but are also not necessarily included in the conventional meaning of "non-Euclidean geometry", such as more general instances of Riemannian geometry.

• Euclidean geometry can be axiomatically described in several ways, with Hilbert's system consisting of 20 axioms most closely following the approach of Euclid.

• Models of non-Euclidean geometry are mathematical models of geometries which are non-Euclidean, such as a sphere for elliptic geometry or a pseudosphere for hyperbolic geometry.

• In the elliptic model, for any given line l and a point A, which is not on l, any line through A intersects l.Non-Euclidean Geometry: Models, Properties, and Importance

• The discovery of non-Euclidean geometries challenged the authority of Euclidean geometry as the mathematical model of space.

• In two dimensions, there are two non-Euclidean geometries: elliptic and hyperbolic geometry.

• The pseudosphere model answers the question of the existence of a model for hyperbolic geometry.

• In the hyperbolic model, for any given line l and a point A not on l, there are infinitely many lines through A that do not intersect l.

• Three-dimensional non-Euclidean geometries include Euclidean, elliptic, and hyperbolic geometries, mixed geometries, twisted versions of mixed geometries, and anisotropic geometries.

• Non-Euclidean geometries have properties that distinguish them from Euclidean geometry, such as the behavior of lines with respect to a common perpendicular.

• The discovery of non-Euclidean geometries had a ripple effect beyond mathematics and science, affecting philosophy, theology, and Victorian England's intellectual life.

• Planar algebras use non-Euclidean geometries to explain non-Euclidean angles.

• Hyperbolic geometry found an application in kinematics with the physical cosmology introduced by Hermann Minkowski in 1908.

• Non-Euclidean planar algebras support kinematic geometries in the plane.

• Non-Euclidean geometry often appears in works of science fiction and fantasy.

• The study of non-Euclidean geometry represents a scientific revolution in the history of science.