Sunday, August 7, 2022

Euclidean & Non-Euclidean Geometry

euclidean and non-euclidean geometry to understand the theories of the universe and space-time curvature in a better way we must have an idea about the different types of geometry and about their uses euclidean geometry is a mathematical system named after greek mathematician.

Euclid it is described in his textbook of geometry the elements this is the plane geometry still taught in secondary school euclidean geometry is mainly a systematic and logical representation of earlier knowledge of geometry we all know according to this geometry if a straight line and a point not on that.

Line given then there is only one line you can draw that passes through that point and is parallel to the first line it is euclid's parallel postulate so you can see this geometry is flat and its curvature is a zero but the main problem is this geometry defines the situations of the plane only when we are dealing with different surfaces we are faced.

Many drawbacks of solving problems through the same geometry this geometry is also called parabolic geometry if you try to apply the same geometry on the surface of a football then you will face problems but we always need those theorems which can be applied to every situation we saw that our universe is not so flat space and time can be curved.

So now we need some better geometry to explain the laws of the universe carl friedrich gauss and ferdinand karl schweikart had the ideas of non-euclidean geometry at the beginning of the 19th century but they did not publish their thought then in 1830 yanosh boyai and nikolai ivanovich lobochevsky separately published.

Treatises on hyperbolic geometry it is also called boyalogevsky geometry in this geometry the surface is negatively curved this leads to a variation of euclid's parallel postulate according to this for a given straight line and a point not on the line there exist an infinite number of straight lines through the point parallel to the.

Original line then in 1854 bernhard riemann founded a special type of geometry called riemannian geometry it discusses the ideas of manifolds riemannian metric and riemannian curvature it is also called spherical or elliptical geometry here the surface is positively curved in this geometry euclid's parallel postulate has.

A form like this for a given straight line and the point not on the line there are no straight lines through the point parallel to the original line we know that the sum of angles of a triangle is 180 degrees in euclidean geometry we all study it from school but it is very interesting that for hyperbolic geometry the sum of the.

Angles of a triangle is less than 180 degrees and for riemannian or elliptical geometry the sum of angles is always greater than 180 degrees on a sphere the sum of the angles of a triangle is not equal to 180 degree the surface of a sphere is not a euclidean space but locally the laws of the euclidean geometry are good.

Approximations in a small triangle on the face of the earth the sum of the angles is very nearly 180 degree if you like this short video then please subscribe to this channel for more interesting content you can also share this video to your physics lover friends thank you.


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