A FIELD TRIP IN A NON‐EUCLIDIAN GEOMETRY

From elementary school to high school we have been learning different types of geometry. \nTaxicab geometry is an example of non‐euclidean geometry. Although it has similar points with Euclidian \ngeometry, it differs from Euclidian geometry in an area; in distance functions. I have decided to try a new \ngeometry. So, I have found taxicab geometry. When I researched about taxicab geometry, I have asked \nmyself that “Could I turn this geometry into three dimensions?” \nMy research subject is “Taxicab Geometry in Three Dimensions”. In the first part of the essay, my \nreasons of choosing this subject and an explanation of taxicab geometry are briefly given. After this quick \ninformation in the second part I will be proving Taxicab geometry is a metric*. Third part involves \nEuclidian geometry’s theorems for distance and theorems that are highly related to distance functions in \nthree dimensions are given. Also formulas for distance in two dimensional taxicab geometry are given a. \nFurthermore, these theorems are taken as reference points for converting taxicab geometry to three \ndimensions. Distance function in two dimensions is converted to third dimension in this part. In the \nfourth part, sphere and cylinder which belongs to Euclidian geometry are recreated by me in taxicab \ngeometry using the formulas that are found in the third part. The last part of the essay is about \napplications of taxicab geometry “Taxicab Geometry in Three Dimensions”. The taxicab space’s part in \ndaily life, city planning and in buildings is explained by an example. To conclude, I will try to find answer \nto the question “Is it possible to convert sphere and cylinder from three dimensional Euclidian geometry \nto three dimensional taxicab geometry?”