In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is rejected.

The parallel postulate in Euclidean geometry states, for two dimensions, that given a line l and a point P not on l, there is exactly one line through P that does not intersect l, i.e., that is parallel to l.

In hyperbolic geometry there are at least two distinct lines through P which do not intersect l, so the parallel postulate is false.

Models have been constructed within Euclidean geometry that obey the axioms of hyperbolic geometry, thus proving that the parallel postulate is independent of the other postulates of Euclid..

For more information about the topic Hyperbolic geometry, read the full article at Wikipedia.org, or see the following related articles:

Euclidean geometry — Euclidean geometry is a mathematical well-known system attributed to the Greek mathematician Euclid of Alexandria. Euclid's text Elements was the ...  > read more

Shape of the Universe — The shape of the Universe is a subject of investigation within physical cosmology. Cosmologists and astronomers describe the geometry of the Universe ...  > read more

Triangle — A triangle is one of the basic shapes of geometry: a polygon with three vertices and three sides which are straight line segments. Any three ...  > read more

Parallelogram — A parallelogram is a four-sided plane figure that has two sets of opposite parallel sides. Every parallelogram is a polygon, and more specifically a ...  > read more

Related Stories

Euclid Returns To Maths Lessons (December 21, 2005) — Knowing how a mathematical theory developed improves a pupil's understanding of it. This is the conclusion of Dutch researcher Iris van Gulik, who investigated how the history of mathematics can help ...  > read more

Indigenous Amazonians Display Core Understanding Of Geometry (January 23, 2006) — Researchers in France and at Harvard University have found that isolated indigenous peoples deep in the Amazon readily grasp basic concepts of geometry such as points, lines, parallelism and right ...  > read more

What Shape Is The Universe? Columbia Astronomers Have Clue (February 17, 1998) — It's probably an expanding, multidimensional equivalent of either a sheet of paper, a sphere or a saddle, according to astronomers at Columbia University, who report in the Feb. 20 issue of The ...  > read more

Penn State Researchers Look Beyond The Birth Of The Universe (May 16, 2006) — Physicists at Penn State University have used mathematical calculations to reveal a time before the Big Bang, when matter and space-time were born, according to Einstein's general theory of ...  > read more

Pioneering 3D View Of Near-Earth Magnetic 'Dance' (July 1, 2007) — Scientists have obtained the first-ever 3D picture of interconnected magnetic 'dances' in near-Earth space, known as magnetic reconnection events. The data from ESA's Cluster satellites will help to ...  > read more

Search ScienceDaily

Hurricane And Earthquake Resistant Nails