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Vicol
- 14
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I'm wondering what could happen if we remove one axiom from Euclidean geometry. What are the conseqences? For example - how would space without postulate "To describe a cicle with any centre and distance" look like?
Euclidean geometry is a branch of mathematics that deals with the study of points, lines, angles, and shapes in a two-dimensional or three-dimensional space. It is based on the work of the ancient Greek mathematician Euclid and is considered the foundation of modern geometry.
In Euclidean geometry, an axiom is a statement or a principle that is accepted as true without requiring any proof. These axioms serve as the building blocks for the rest of the geometric principles and theorems.
Removing an axiom from Euclidean geometry allows us to explore different geometric systems and see how altering the fundamental assumptions can affect the resulting theorems and proofs. It also helps us understand the limitations and boundaries of traditional Euclidean geometry.
The most commonly removed axiom from Euclidean geometry is the parallel postulate, also known as the fifth postulate. This axiom states that if a line intersects two parallel lines, then the interior angles on the same side of the transversal are supplementary. Removing this axiom leads to the development of non-Euclidean geometries.
Removing an axiom from Euclidean geometry results in the creation of different geometric systems, such as hyperbolic and elliptic geometries, which have different properties and theorems compared to traditional Euclidean geometry. It also challenges our understanding of space and shapes and allows us to explore new concepts and ideas in mathematics.