# The Challenge of Non-Euclidean Geometry

TLDR Non-Euclidean geometry challenges the traditional axioms of Euclidean geometry and has led to the development of alternative geometries such as elliptic and hyperbolic geometry. These new geometries have real-world applications in fields such as aviation, surveying, and the theory of special relativity.

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Non-Euclidean geometry challenges the axioms of Euclidean geometry and changes how we think about geometry.

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Euclid's book, titled Elements, is considered the foundational work in the subject of geometry and is known for its logical development of geometric theorems based on a few simple axioms.

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The fifth axiom, known as the parallel postulate, is more complicated than the other four axioms and mathematicians attempted to derive it as a theorem from the first four axioms but failed.

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In the 18th century, mathematicians began to challenge the assumption of a flat plane in Euclidean geometry and developed alternative axioms, leading to the exploration of non-Euclidean geometry.

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There are two general types of non-Euclidean geometry: elliptic geometry, which has a positive curvature like the surface of a sphere, and hyperbolic geometry, which has a negative curvature like being in the middle of a horse saddle.

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Non-Euclidean geometries, such as hyperbolic and elliptic geometries, have real-world applications in fields such as aviation, surveying, and the theory of special relativity.

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Mathematicians were able to develop entirely new geometries by envisioning what things might look like if Euclid's fifth axiom were false, leading to new fields of mathematics today.