# Euclidean Geometry vs Non-Euclidean Geometry

Discussion in 'Off-Topic Chat' started by nycmathguy, Jul 15, 2022.

1. ### nycmathguy

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Simply put, what is the main difference between Euclidean Geometry and Non-Euclidean Geometry? Why is Non-Eucludean Geometry not taught at the high school level? Is Euclidean Geometry another name for Plain Geometry?

nycmathguy, Jul 15, 2022
2. ### MathLover1

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Euclidean geometry seeks to understand the geometry of flat, two-dimensional spaces, non-Euclidean geometry studies curved, rather than flat, surfaces.

Euclidean and non-Euclidean geometry are both examples of neutral geometry, that is systems obeying the first four of Euclid’s axioms. Ultimately that completely describes the similarities between the two: both obey any theorem of neutral geometry, and can differ in any other way imaginable.
Non-Euclidean geometry is any geometry that is different from Euclidean geometry. Each Non-Euclidean geometry is a consistent system of definitions, assumptions, and proofs that describe such objects as points, lines and planes.

The two most common non-Euclidean geometries are spherical geometry and hyperbolic geometry.

The essential difference between Euclidean geometry and these two non-Euclidean geometries is the nature of parallel lines:
In Euclidean geometry, given a point and a line, there is exactly one line through the point that is in the same plane as the given line and never intersects it.
In spherical geometry there are no such lines.
In hyperbolic geometry there are at least two distinct lines that pass through the point and are parallel to (in the same plane as and do not intersect) the given line.

MathLover1, Jul 15, 2022
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3. ### nycmathguy

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Understood. Where does Axiomatic Geometric fit in the world of geometry?

nycmathguy, Jul 16, 2022
4. ### MathLover1

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Axiomatic geometry : the study of the axioms of geometry
an axiomatic system is a set of axioms used to derive theorems

MathLover1, Jul 16, 2022
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