Exploration of non-Euclidean Geometries
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Definition
Taxicab Geometry arose from a change in the way distance is measured, where the distance formula changes to:
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dt=|x2-x1|+|y2-y1|
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Methods
The goal of exploration was to see how Euclid's first 31 propositions change when on a different distance formula was used. To accomplish this we used a computer program called Geogebra. Then two examples of each proposition were tested, purposely trying to find a counter example. If one was not found it was deemed a possible proposition.
Results
From exploration of taxicab geometry using a Geogebra, these are the results found.
Propositions
In exploration counter examples to 10 propositions (4-6,18-21,24-26) were found, proving by counter example that these do not hold in spherical geometry.
Lines
Lines will be drawn as a Euclidean line, but measured in taxicab units. This is due to the fact that lines are an extention of a geodesic, and taxicab geodesics are not straight lines, and are not unique. If they were to be extended, we would get lines that do not make much sense. So instead we will be expressing lines as that in Euclidean geometry.
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Circles
Angles
Angles are very different, and have their own unit, taxicab radians, or t-radians. They are a measurement of angles that are based on a taxicab unit circle. One t-radian is an angle whose vertex is at the center of a unit taxicab circle and intercepts an arc of length 1. In a unit taxicab circle there are 8 t-radians, where 2 t-radians are equivalent to 90, where 4 t-radians is equal to 180. However 1 t-radian is not equal to 45 so a 45 angle in taxicab may not have a t-radian measurement equal to 1. So, this formula is used to find an angle in t-radians using its reference angle:
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