Editing Euclidean geometry (section)

===20th century and relativity===
[[File:1919 eclipse negative.jpg|thumb|right|A disproof of Euclidean geometry as a description of physical space. In a 1919 test of the general theory of relativity, stars (marked with short horizontal lines) were photographed during a solar [[eclipse]]. The rays of starlight were bent by the Sun's gravity on their way to Earth. This is interpreted as evidence in favor of Einstein's prediction that gravity would cause deviations from Euclidean geometry.]]

[[Albert Einstein|Einstein's]] theory of [[special relativity]] involves a four-dimensional [[space-time]], the [[Minkowski space]], which is [[non-Euclidean geometry|non-Euclidean]]. This shows that non-Euclidean geometries, which had been introduced a few years earlier for showing that the [[parallel postulate]] cannot be proved, are also useful for describing the physical world.

However, the three-dimensional "space part" of the Minkowski space remains the space of Euclidean geometry. This is not the case with [[general relativity]], for which the geometry of the space part of space-time is not Euclidean geometry.<ref>Misner, Thorne, and Wheeler (1973), p.&nbsp;191.</ref> For example, if a triangle is constructed out of three rays of light, then in general the interior angles do not add up to 180 degrees due to gravity. A relatively weak gravitational field, such as the Earth's or the Sun's, is represented by a metric that is approximately, but not exactly, Euclidean. Until the 20th century, there was no technology capable of detecting these deviations in rays of light from Euclidean geometry, but Einstein predicted that such deviations would exist. They were later verified by observations such as the slight bending of starlight by the Sun during a solar eclipse in 1919, and such considerations are now an integral part of the software that runs the [[Global Positioning System|GPS]] system.<ref>Rizos, Chris. [[University of New South Wales]]. [http://www.gmat.unsw.edu.au/snap/gps/gps_survey/chap3/312.htm GPS Satellite Signals] {{Webarchive|url=https://web.archive.org/web/20100612004027/http://www.gmat.unsw.edu.au/snap/gps/gps_survey/chap3/312.htm |date=2010-06-12 }}. 1999.</ref>