View source for The Mathematics of Non-Euclidean Geometry

== Introduction ==

Non-Euclidean geometry is a branch of [[Geometry|geometry]] that deviates from the traditional concepts of Euclidean geometry, which is based on the work of the ancient Greek mathematician [[Euclid|Euclid]]. Non-Euclidean geometry encompasses two types: hyperbolic and elliptic, both of which reject Euclid's fifth postulate, also known as the parallel postulate.

[[Image:Detail-18460.jpg|thumb|center|A visual representation of a non-Euclidean space, showing curved lines and surfaces.|class=only_on_mobile]]
[[Image:Detail-18461.jpg|thumb|center|A visual representation of a non-Euclidean space, showing curved lines and surfaces.|class=only_on_desktop]]

== History ==

The concept of non-Euclidean geometry emerged in the 19th century, primarily through the work of mathematicians [[Nikolai Ivanovich Lobachevsky|Nikolai Lobachevsky]], [[János Bolyai|János Bolyai]], and [[Carl Friedrich Gauss|Carl Friedrich Gauss]]. These mathematicians independently discovered hyperbolic geometry, a type of non-Euclidean geometry, which was a significant departure from the Euclidean tradition.

== Hyperbolic Geometry ==

Hyperbolic geometry, also known as Lobachevskian geometry, is characterized by the property that given a line and a point not on the line, there are at least two lines through the point that do not intersect the given line. This contradicts Euclid's fifth postulate, which states that there is exactly one line through the point that does not intersect the given line.

[[Image:Detail-18462.jpg|thumb|center|An illustration of hyperbolic geometry, showing multiple lines passing through a single point without intersecting a given line.|class=only_on_mobile]]
[[Image:Detail-18463.jpg|thumb|center|An illustration of hyperbolic geometry, showing multiple lines passing through a single point without intersecting a given line.|class=only_on_desktop]]

== Elliptic Geometry ==

Elliptic geometry, on the other hand, denies the existence of parallel lines altogether. In this geometry, all lines intersect. This concept is often visualized on a sphere, where lines are represented by great circles, such as the lines of longitude on a globe. 

[[Image:Detail-18464.jpg|thumb|center|An image of a sphere, representing elliptic geometry, with lines of longitude intersecting at the poles.|class=only_on_mobile]]
[[Image:Detail-18465.jpg|thumb|center|An image of a sphere, representing elliptic geometry, with lines of longitude intersecting at the poles.|class=only_on_desktop]]

== Applications ==

Non-Euclidean geometry has found applications in various fields such as physics, particularly in the theory of [[General Relativity|general relativity]] proposed by [[Albert Einstein|Albert Einstein]]. In this theory, the fabric of space and time is represented as a four-dimensional non-Euclidean manifold.

== Conclusion ==

The development and exploration of non-Euclidean geometry have significantly expanded our understanding of the mathematical universe. It has challenged the traditional Euclidean perspective and has found applications in various scientific fields, demonstrating the power and versatility of mathematical thought.

[[Image:Detail-18466.jpg|thumb|center|An image of a four-dimensional non-Euclidean manifold, representing the fabric of space and time in Einstein's theory of general relativity.|class=only_on_mobile]]
[[Image:Detail-18467.jpg|thumb|center|An image of a four-dimensional non-Euclidean manifold, representing the fabric of space and time in Einstein's theory of general relativity.|class=only_on_desktop]]

== See Also ==

* [[Euclidean Geometry]]
* [[Geometry]]
* [[General Relativity]]

[[Category:Mathematics]]
[[Category:Geometry]]
[[Category:Non-Euclidean Geometry]]
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