Euclidean and non-euclidean geometries?

Question

what are some of the similarities and differences? it would be helpful if math examples are given of each

Answer 1

I'm going to focus on the usual non-euclidean geometry due to Lobachevsky. In that geometry, there are infinitely many lines parallel to a given line and through a given point. Squares (i.e. figures with 4 equal sides and 4 equal angles) don't have right angles. There is a pentagon with all right angles. The area of a triangle can be determined by the sum of the angles in the triangle (it is proportional to the difference of 180 degrees and the sum of the angles). There is a 'natural' unit of distance obtained from angles between parallel lines and perpendicular lines.

Answer 2

Euclidean geometry starts with these postulates:

"Euclid gives five postulates (axioms):

Any two points can be joined by a straight line.

Any straight line segment can be extended indefinitely in a straight line.

Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.

All right angles are congruent.

Parallel postulate. If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. " [See source.]

Non Euclidean geometry is one where one or more of these five postulates are not true. For example, if our universe is curved (spherical or saddle shaped), the first postulate (about the straight line between two points) would be false. The line between two points would not be straight; it would curve. So, for example, a Euclidean equation like the hypotenuse of a right triangle, z = sqrt(x^2 + y^2), would not be true in a curved, non-Euclidean universe.

As another example, the sum of interior angles in a triangle is 180 degrees (always) in Euclidean space. Not so in other frameworks. For example, the sum of interior angles of a triangle drawn on the surface of a sphere is not necessarily 180 degrees.

Answer 3

One of the big differences is that triangles in Euclidean geometry has 180 degrees but a triangle in Non-Euclidean geometry can have more. This is caused by Euclidean geometry on works on a plane. ie. flat surface.

Answer 4

countless solutions right here, yet positioned only, Euclidean geometry assumes a flat airplane. So case in point triangles upload as much as one hundred eighty tiers. Non-Euclidean geometry assumes different shapes, like spheres, on which triangles do no longer unavoidably upload as much as one hundred eighty tiers.