Exploration of non-Euclidean Geometries
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History
Born in Egypt around 300 BC, Euclid wrote 13 books in his lifetime the collection of which is called “The Elements,” which serve as the foundation of Euclidean geometry. This was the only known geometry until the 19th century, when Gauss and Bolyai came up with new geometries by changing the assumption of Euclid’s fifth postulate. Thus, my research is motivated by the question: How does a change in the underlying assumptions of Euclidean geometry impact how objects (such as angles, lines, and shapes) relate?
Euclid
Euclid was born around 300 BC in Egypt, but late became known as a famous Greek mathematician. Throughout his life he complied previous work and his own into the "Elements," which created the most widely known form of geometry and the only one used until the 1800's. From the first book of the "Elements" Euclid included 48 propositions, and 5 postulates.
Parallel PostulateOf Euclid's five postulates, the fifth one stuck out to all. It was much more complex than the other four and many people would have prefered to see it added to the list of propositions instead. Another mathematician by the name of Playfair created an equivalent statement that is more commonly used called Playfairs postulate.
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Given a line and a point not on that line, it is possible to draw one line through the point that is parallel to the first line. |
Non-Euclidean
In the 1800's mathematicians Gauss and Bolyai decided to come up with math that excluded the fifth postulate all together, and in doing so created new forms of geometry that became known as Non-Euclidean geometry. The major ones include Spherical and Hyperbolic geometry which where two of the geometries that I explored in research.