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Taxicab Geometry Presentation

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Tyler Roell presentation on Taxicab Geometry given at Franklin College High School Math Day, November 2009. Culmination of his MAT 490 (Individualized Study) project with Prof. Robert Talbert.

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Taxicab Geometry Presentation

  1. 1. Taxicab Geometry A Study Into Non-Euclidean GeometryUsing Geometer Sketchpad<br />Tyler Roell<br />Studied under Dr. R. Talbert<br />Franklin College Math Day<br />November 21, 2009<br />
  2. 2. The Beginning of Taxicab Geometry?<br /><ul><li>Hermann Minkowski (1864-1909)
  3. 3. Teacher of Einstein
  4. 4. Also known as Manhattan Distance
  5. 5. Karl Menger (1952)</li></li></ul><li>What is Taxicab Geometry?<br /><ul><li>Assumptions made</li></ul>of Taxicab Geometry<br /><ul><li>One Main Difference
  6. 6. Dt=|x1-x2|+|y1-y2|</li></li></ul><li>Differences<br /><ul><li>Dt=|x1-x2|+|y1-y2|
  7. 7. De=
  8. 8. Crows Fly Vs. Urban Geography</li></ul>Dt=|3-1|+|1-4|=5<br />
  9. 9. Redistricting Problem<br />Say you have three schools, K,N, and M. Using the Taxicab Distance, where should the school districts be drawn to keep all the districts the same size in area?<br />
  10. 10. Geometric Figures<br /><ul><li>Circles
  11. 11. The set of all points equidistant from a point p with radius r</li></li></ul><li>Geometric Figures<br /><ul><li>Parabolas
  12. 12. set of all points that are the same distance from a fixed line and a fixed point not on the line</li></li></ul><li>Geometric Figures<br /><ul><li>Parabolas
  13. 13. set of all points that are the same distance from a fixed line and a fixed point not on the line</li></li></ul><li>Bisectors<br /><ul><li>In Euclidean</li></ul>In Taxicab <br />
  14. 14. Now back to the problem<br /><ul><li>What should be done first?
  15. 15. What figures should be </li></ul>used?<br />
  16. 16. Construct circles with the same radius<br />
  17. 17. Construct the bisector between K and N<br />
  18. 18. Bisector between N and M<br />
  19. 19. Bisector between K and M<br />
  20. 20. The Solution<br />
  21. 21. Further Application<br /><ul><li>Ellipses
  22. 22. The set of all points where the sum of the distances from any point on the curve to the two fixed points remain a constant </li></li></ul><li>Applications for Teaching<br />Easier way to measure distance<br />Applicable to the Urban Setting<br />Uses technology in the Classroom<br />
  23. 23. Book Used<br />Eugene F. Krause Taxicab Geometry: An Adventure in Non-Euclidean Geometry<br />

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