Exploration of non-Euclidean Geometries

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**Abstract**

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, our 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? To accomplish this we conducted many tests for each geometry, on their related mode, we explored several things, lines, circles, angles, propositions, Saccheri quadrilaterals. In our exploration we sought to compare each of the objects in the Non-Euclidean geometries to the objects in the Euclidean geometry and see the difference.

**Acknowledgements **

I would like to thank the Frontiers of Science Institution hosted by UNC MAST department, for holding this program.
Lori Ball (middle right) and all FSI staff for instruction and helping make this program possible. Adolph Coors Foundations for allowing me to be here and for funding my research. Samantha Moore (bottom right) for being an amazing mentor. Zabedah Saad (bottom left) for being a great advisor. |