Adventures in Non-Euclidean Distance

1. Adventures in Non-Euclidean Distance Ricky Bobinchuck Tashauna Thompson Tosha Pelfrey

2. How would you find the shortest distance between the two points on the two dimensional plane below?

3. Pythagorean Theorem? 5 units The distance between the two points is 5 units.

4. The shortest distance between two points is a straight line. The distance formula finds the shortest distance between two points when working in a two dimensional plane. The Distance Formula

5. In Euclidean Geometry, can you find another path that is equal to the shortest distance? Nope, in Euclidean Geometry only one path is the length of the shortest distance.

6. When you are traveling in a car the shortest distance between two locations is still a straight line, but this path may not be an option to travel. Get Real!

7. In the 19th century, Hermann Minkowski, proposed a family of metrics where the notion of distance is different depending on the space in question. Minkowski’s ideas helped Albert Einstein to develop his Theory of Relativity. Distance Depending on the Space

8. Karl Menger was the first to use the term “taxicab” to describe Minkowski’s metric in the booklet You Will Like Geometry. In 1952, Menger had an exhibit of taxicab geometry in the Museum of Science and Industry in Chicago. "Taxicab Geometry" First Coined

9. Non-Euclidean Geometry • Route a cab would take • Can’t cut though buildings • Limited to streets – gridlines and one-way traffic flow Taxicab Geometry

10. Shortest path between 2 points least number of blocks a taxi must travel along streets points at intersections (x, y) coordinates are integers Example What’s the shortest taxicab distance from (0, 0) to (1, 2)? Taxicab Distance

11. 4 Taxicab Distance Activity GeoGebra File 2. Taxicab Distance Formula 1.

12. Taxicab Geometry in Middle School

14. The objective of this game is to see who can catch the gangster first. Gangster Grid Game • Rules • Each player chooses a point where their gangster is hiding on their game board (keep it a secret). • Player one should choose the ordered pair that he or she thinks the gangster is located on the opponent’s board. • Player two will tell player one the taxicab distance that he or she is away from the gangster. • Now player two will choose the ordered pair that he or she thinks the gangster is located on the opponent’s board. • Player one will tell the taxicab distance that he or she is away from the gangster. • Repeat steps until the gangster is located. • The first to locate the gangster is the winner. • Replay the game until time is called. • Hint: Keep track of how many blocks you are away from the gangster.

15. Distance The shortest path between any two points. Euclidean Geometry There is one path, a straight line, that describes the shortest distance between two points. Found using Taxi Cab Geometry The shortest path between two points is restricted by gridlines and streets. Found using Is there only “one” shortest path between two points???? Recap…

16. How many Paths can you find?

17. Is the number of Paths related to the distance? How many Paths are there? How many paths can you find between points A(0,0) and B(2,2)?

18. Is there a way that we can find the number of paths? Using combinations, we can find the number of paths with the formula for ; where m is the distance between two points: Think Combinations “!”

19. We defined the distance between two points earlier as the horizontal change (x) plus the vertical change (y). m=distance y k x n

20. Now we can define distance in terms of n and k, where n represents our x coordinate and k our y. If Substituting n+k for the distance m we have: Algebra!

21. How many paths did you say we can find for points A and B? Let’s check it out…

22. Answers

23. Will that always work?

24. Do you notice anything?

25. Find the number of paths!

26. Excel File

27. Number of Shortest Paths

28. Number of Shortest Paths at Each Intersection Notice anything familiar?

29. Taxicab Euclidean Geometry VS. Taxicab Geometry Euclidean Geometry Distance Formula The shortest path(s) are the path(s) where the least number of blocks are traveled to get from one point to the next. The shortest path is a straight line. Shortest Path(s) Coordinate Plane Only one shortest path Points Usually has multiple shortest paths Lines Number of paths found with the formula

30. Foldable

31. Amazing Race • Label the cover of your booklet with your name and group color. • Each group should choose a runner. • The runner must present the correct answer and show work to get the next question. • At the end of the race, the runner will turn in everyone's work. • All booklets must be complete in order to be the winner and receive your prize.