Non-Euclidean Geometry
The processes by which spherical surfaces are mapped onto flat planes are known as projections. The most famous of these projections is called ‘Mercator’s Projection’ after its founder, Gerardus Mercator. This is the projection used in the most common world atlas along with most other maps, both on and offline. The main advantage of the Mercator Projection, and the reason it was and is so widely used in navigation, is that a straight line plotted on a map has a constant bearing, allowing navigators to plot straight-line courses. However, this comes with a downside, namely that scale becomes distorted away from the equator. Areas away from the equator begin to appear disproportionally large to the point that Greenland appears to be roughly the same size as Africa, despite the latter being approximately 14 times larger. 23
AMercator projection producing the common world atlas, with notable distortions of scale at the top and bottom
In conclusion, variations of non-Euclidean geometry have been in development since antiquity with interest in the topic blossoming during the early 19 th century. Its ramifications have been pervasive, echoing through many facets of life from abstract mathematics to navigation and travel with the latter two fields in particular greatly impacting all of our lives and demonstrating why non-Euclidean geometry matters.
Bibliography
Beardon, A. T. (2011) https://nrich.maths.org/5654 Consulted: 11/08/20 Chauhan, Y. (1998) https://www.britannica.com/science/Mercator-projection Consulted: 14/08/20 Complex Manifold. Wikipedia (unknown) https://en.wikipedia.org/wiki/Complex_manifold Consulted: 12/08/20 Gardner, M. (2001) The Colossal Book of Mathematics . New York Gowers, T. (2002) Mathematics, A Very Short Introduction . New York Hayter, R. (2008) ‘The hyperbolic Plane ‘A strange new universe’ ’ http://www.maths.dur.ac.uk/Ug/projects/highlights/CM3/Hayter_Hyperbolic_report.pdf Consulted: 11/08/20 Henderson, D. (1999) https://www.britannica.com/science/non-Euclidean-geometry Consulted: 10/08/20 Hosch, W. (1998) https://www.britannica.com/science/curvature Consulted: 11/08/20 Kailasa, S. Lai, J. Khim, J. (unknown) https://brilliant.org/wiki/holomorphic-function/ Consulted: 12/08/20 Krioukov, D. (2010) https://arxiv.org/abs/1006.5169 Consulted: 12/08/20
23 Chauhan 1998.
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