Mathematical Journeys - Saint Ann's School

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  5  EXPLORATIONS INTO EUCLIDEAN AND TAXICAB MINIMUM DISTANCES 3 Figure 2. Creating stair-steps keeps the total Taxicab distance constant. point B. Or it could take a right and then a left and so on, or it could take an infinite number of alternating rights and lefts until it becomes a straight line between the two points. So therefore a 2 + b 2 ≠ c 2 , but a + b = c. This theorem, although false in Euclidean Geometry, is very interesting. It is called the Lythagorean theorem. The idea behind the theorem, not the theorem itself, is essential to solving more complex problems, with the idea that a variety of paths in Taxicab geometry will have the same distance as long as they reach the same end point. 3. <strong>School</strong> Districts Problem A first optimization of distances problem is one that comes up in many cities—to determine which school to place a child based on where they live. If one has an undetermined amount of dots (i.e., schools) placed upon a plane, then the mathematical problem is this: if you place another dot (i.e., a home) upon this plane, which school dot is it closest to? For a small number of homes, you can just do all of the relevant measurements. But if there are a lot of homes, then we need boundaries to clarify the different regions, or school districts. In Euclidean geometry, this is a matter of drawing a line equally in between the three or more points. First, find a point that is the same distance from the three points. Then separate the line into three separate lines, each line equally in between the two points it separates. You must connect pairs of points, then draw in a perpendicular bisector for each of these lines. Then you find the point where all the perpendicular bisections meet. If there are more than three points and more that one point where the perpendicular bisections meet, then you                        
6    EXPLORATIONS INTO EUCLIDEAN AND TAXICAB MINIMUM DISTANCES 4 Figure 3. Four schools; lines and perpendicular bisectors; the boundaries defined. just connect the two meeting places. In order to get the finished result you get rid of everything except the bisectors. 4. Problems with Obstructions Another interesting problem in this theme is finding the shortest distance between two points with obstructions in the way of the otherwise shortest distance. This type of problem is actually more practical than it appears. It is not just for robots trying to find the way through a maze, but what we do every day. Not just in Taxicab geometry, but also in Euclidean as well. We, as humans, almost always find the shortest path. Either when we walk home using Taxicab geometry, or when we get up to get a glass of water using Euclidean. We don’t walk out of our way by three feet or by three blocks; students even crawl under and over desks to avoid walking around a classroom. One simple case is to make the obstacle a circle; in particular, make its radius 1 and place A and B at distance 1 from the circle on opposite sides. Below are an examples non-minimal paths from A to B in Taxicab (dotted) and Euclidean (bold) geometry. Figure 4. These paths are clearly non-optimal.                        
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