In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative, or relaxing the metric requirement. In the former case, one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the , which give rise to that have also been called non-Euclidean 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, which states that, within a two-dimensional plane, for any given line l and a point A, which is not on l, there is exactly one line through A that does not intersect l. In hyperbolic geometry, by contrast, there are infinitely many lines through A not intersecting l, while in elliptic geometry, any line through A intersects l. Another way to describe the differences between these geometries is to consider two straight lines indefinitely extended in a two-dimensional plane that are both perpendicular to a third line (in the same plane): * In Euclidean geometry, the lines remain at a constant distance from each other (meaning that a line drawn perpendicular to one line at any point will intersect the other line and the length of the line segment joining the points of intersection remains constant) and are known as parallels. * In hyperbolic geometry, they "curve away" from each other, increasing in distance as one moves further from the points of intersection with the common perpendicular; these lines are often called ultraparallels. * In elliptic geometry, the lines "curve toward" each other and intersect. (Wikipedia).
Non-Euclidean geometry | Math History | NJ Wildberger
The development of non-Euclidean geometry is often presented as a high point of 19th century mathematics. The real story is more complicated, tinged with sadness, confusion and orthodoxy, that is reflected even the geometry studied today. The important insights of Gauss, Lobachevsky and Bo
From playlist MathHistory: A course in the History of Mathematics
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In the early 19th century, people started to wonder if the Fifth Postulate couldn't be proven at all--meaning that it could be right, but it could also be wrong. Bolyai, Lobachevsky, and Riemann started exploring hyperbolic geometry and other strange realms... Support us on Patreon! http:/
From playlist Extra History: Chronological Order (1700 CE - Present)