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Euclidian or Not? What exactly is meant in claiming that Taxicab Geometry is non-Euclidian? The famous mathematician Euclid developed a set of axioms and postulates regarding the world of Euclidian geometry: Any two points can be joined by a straight line. Any straight line segment can be extended indefinitely in a straight line. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. All right angles are congruent. Parallel postulate. If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.  A postulate associated with Euclid's axioms is that two triangles are congruent if and only if they satisfy the SAS, or side angle side, postulate, illustrated above. In Taxicab Geometry, side angle side does not hold. To show that side angle side does not hold we need to show one case where the side, angle, and side are the same but the length of the hypotenuse is different. Consider the points (0,4), (0,0) and (2,0) and the triangle they make. One side of the triangle is 4, the other 2 and the angle is 90 degrees. The distance of the hypotenuse is the distance between (0,4) and (2,0) which is 6 using the taxicab metric. Now consider the points (1,1), (2,0) and (3,3) and the triangle they make. The sides are again 4 and 2 and the angle between them is again 90 degrees. However, now the distance of the hypotenuse is the distance between (1,1) and (3,3) which is 4 using the taxicab metric. Now, we have shown that we can have the same side, angle, side and a different hypotenuse, therefore side angle side does not hold in taxicab geometry. Taxicab Geometry, therefore, cannot be considered actual Euclidian geometry. Although it borrows many of the same concepts, such as the coordinate plane, we have illustrated a fundamental difference, among others. |
