2621 words - 10 pages

Taxicab GeometryTaxicab Geometry was first introduced by Hermann Minkowski in the 19th century. Minkowski was born on June 22, 1864 in Aleksotas, Lithuania of German, Polish, and Jewish decent. One of his most famous students was Albert Einstein, known for his theory of relativity. Throughout his life, Minkowski explored the number theory, mathematical physics, and the theory of relativity.Taxicab geometry is quite easy to understand since it is very close to Euclidean geometry in its axiomatic structure. Points, lines and angles are measured the same as in Euclidean geometry, but distances are measured differently. This form of geometry uses the Cartesian coordinate system to measure the distance between two points by the sum of the absolute difference in their coordinates. The taxicab distance is also known as the Manhattan distance and city block distance. The distance is called Manhattan because the city blocks are laid out similar to the Cartesian coordinate system. Distance in taxicab geometry can be represented by dt. To find the distance, the formula used is dt (A,B) = | a1 - b1 | + | a2 - b2 |. Distance in Euclidean geometry can be represented by de. The formula used for Euclidean geometry is de (A, B) = (a1 -b1)2 + (a2-b2)2. Here is an example: An advantage of using taxicab geometry is that it is more useful in real life. In Euclidean geometry the distance between two points is a line, but it must be taken into consideration that there might be buildings or other things blocking you from traveling in a straight line. Assuming that streets run north-south and east-west, they can be paired with the Cartesian grid system. Points can represent buildings and objects. If one block is represented by a segment on the grid then it would be pretty easy to find the shortest route from point A to point B. For example a pizza delivery boy could use taxicab geometry to find the shortest route to deliver pizzas.

A

B

The pizza parlor is point A and point B is the destination where the pizza is to be delivered. The delivery boy can use the formula dt (A, B) = | a1 - b1 | + | a2 - b2 | or he can graph them on a grid like the one shown above. In the diagram, three possible routes are drawn. The route represented by the dotted lines takes 8 blocks. The route represented by the bold line takes 6 blocks. And the route represented by the thin line also takes 6 blocks. There are many other possible routes too, but the ones represented by the bold and thin lines turn out to be the shortest. So the delivery boy can take either route to get to the destination point.A common example of the difference between Euclidean geometry and taxicab geometry is the circle. In Euclidean geometry, if all the points three units away from point A were connected, they would form a circle. But in Taxicab geometry all the points three units away from point A would form a square.Since the...

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