Taxicab Geometry

Olga Radko took the lead on this activity for the 2012 Circle on the Road festival.

AUDIENCE: 5th grade and up

PLANNING ABSTRACT:
In Euclidean geometry, the shortest distance between two points is along the straight line connecting them. The distance between two points is the length of the shortest line connecting them. Normally, such a path is a straight line. But in a city consisting of a square grid of streets shortest paths between two points are no longer straight lines (as every cab driver knows). We will explore the geometry of this unusual distance and play several related games.

PUBLIC ABSTRACT:
In Euclidean geometry, the shortest distance between two points is along the straight line connecting them. The distance between two points is the length of the shortest line connecting them. Normally, such a path is a straight line. But in a city consisting of a square grid of streets shortest paths between two points are no longer straight lines (as every cab driver knows). We will explore the geometry of this unusual distance and play several related games.

APPRENTICES: Pari Ford, Elisa Young, Kathy O’Hara, Cynthia Davis, Michele Anderson

DISCUSSION: The attached hand out provides a full description.

I just had some fun playing with the problems. Thanks Olga. I’ll start by giving you my thoughts about the answers, and then you guys can let me know if I’m on the right track.

Before I start, did any of you get instructions yet on where and when we are supposed to show up? Are we supposed to be there on Friday?

OK, Problem 3. #1. The two points live on horizontal or vertical line segments.
#2. Any two points where the difference between both coordinates is non-zero.

#3.Can’t do it. Triangle Inequality gets in the way.

Problem 4. #1. The closest point is (1,1) but I had to use Calulus to prove it. I said the points on the line segment joining (2,0) and (0,2) look like (x, 2-x). Then I did the distance formula, but to find a minimum, I took the derivative. Help! We could try to graph the parabola, but that’s a little bit much for 5th graders too. Yikes.

#2. y + x = 2
#3. distance = x + y
#4. We should maybe re-think the wording here. Should we assume the points on the line segment have integer coordinates?

Circles and pi in Taxicab Geometry
#1. I could describe the circle as either a square or diamond. Any preference?
#2. Let’s skip the drawing for email purposes.
#3. I got (1,1) and (1,-1)
#4. I computed the circumference in taxicab geometry – so each edge of the circle is has length 2r. The total circumference is 8r, and if I divide that by the diameter, I get 4.

Game Two: Find the hidden treasure.

This was challenging. I got 5 questions as the max. I plotted the starting point, and its corresponding taxicab circle with radius equal to the distance between it and the treasure. So I was looking at a graph with my starting point as the center, and the circle of taxicab radius r, and the assumption that the treasure was somewhere on the circumference. Knowing that, I thought it a good idea to place the second and third guesses on any two adjacent corners, and to go from there. (That’s three questions now. ) If both distances were twice the first answer, then I knew the treasure was on the side opposite the segment that is defined by the second and third guess. If they weren’t I did some more logical type tests in order to tease out which edge the treasure was located. Once I knew that, I needed at most two questions to figure out its location.

The trick for me was to prove some lemma about the distance function between the treasure and the four corners. But maybe there is a simpler way?

Game Three: Perpendicular Bisectors.

I just graphed them. C was the most curious, and could probably use a follow up question. But maybe that’s what the School Districts’ Boundaries is about?

I’m ready for the Boundaries problem and Lloyd’s game.

See you all soon. Let me know if I messed up.

Best, Kathy

Since Kathy O’Hara posted first, I’ll post anything I have differently.

Problem 3: Same answers.
Problem 4:
1) I got (1,1) as well, but I don’t think it’ll be necessary to have the participants prove it through Calc methods. It’s simple enough to tell that (1,1) is closest by trial and error using Pythagorean, and I think that’s probably sufficient for this activity.
4) I was also confused with the wording of this problem at first, but I think it’s just asking for the closest point to the origin using taxicab. So, that would mean all points on the segment, since the paths to any point, whether the coordinates are integers or not, are drawn with horizontal and vertical lines. (Ex. (0,0) to (1.5,0.5) would be 1.5+0.5=2)
Circle&Pi: Same answers.
Find the Hidden Treasure:
I got 4 moves (as a max), but it seems that Olga is saying that we are trying to find the least number of moves. I was confused with the wording here and the objective, because find the least number of moves could mean that 1 move (being very lucky, of course) could get a player to the treasure. Olga says 2 moves would suffice to win.
What I thought was:
1st move – initial guess; ask the distance
2nd move – draw circle using given distance, pick a point on an edge away from the side’s center which will narrow it down to either 2 points or 1 side; ask distance
3rd move – choose one of these 2 points or a point from the side; ask distance.
4th move – keeping track of all the circles from previous moves, this move should land the player at the treasure.

(Are there any flaws to this?)

However, I’m still wondering whether we are working toward the same objective. Since Olga says that the objective is to find the minimum and not the maximum, there’s a difference between our answers.

Thanks,
Elisa

Attachment
Taxicab geometry session.pdf