Math 361 ACTIVITY 2: Some work with the taxicab

Why The taxicab (Cartesian plane with the taxicab ) provides a useful comparison to the Euclidean plane for checking what our axioms do (or don’t) say. As you will show (below) the taxicab plane is a model for our axioms so far (through D-4, including all the ideas of betweenness), but involves some surprising dif- ferences in the shapes of some geometric objects. Working with this semi-example (it fits parts of the theory, but doesnt match our final goal) can refine your sense of what is really going on in the Euclidean plane

LEARNING OBJECTIVES 1. Work as a team, using the team roles

2. Develop skill in working with non-standard systems 3. Develop inquiry skills in CITERIA

1. Success in completing the exercises. 2. Success in working as a team RESOURCES 1. The Geometer’s Sketchpad software - installed on a computer in the classroom

2. The Geometer’s sketchpad MacIntosh Quick Reference 3. The Sketchpad sketches taxicabd.gsp and taxicircle.gsp [which include the tool TAXIDISTANCE] accessible through the Blackboard site [and on the Public server P:]. 4. Sections 2.4 and 3.2 (on Taxicab geometry) of your text

5. 40 minutes PLAN 1. Select roles, if you have not already done so, and decide how you will carry out steps 2 and 3 (5 minutes)

2. Work through the exercises given here - be sure everyone understands all results (30 minutes) 3. Assess the team’s work and roles performances and prepare the Reflector’s and Recorder’s reports including team grade (5 minutes). 4. Be prepared to discuss your results.

EXERCISE 1. We will first use Sketchpad and the sketch “taxicabd.gsp”to investigate some properties of the taxicab distance. This sketch includes a Sketchpad tool (miniprogram) called “TAXIDISTANCE” which calculates the taxicab distance between two points. (a) Load (from Blackboard or the Public server) the sketch “taxicabd.gsp” and open it with Sketchpad. (b) Select and drag point G - notice that the coordinates of G and m5 (the distance G to F ) and m8 (the distance G to J) change as you move the point. (c) Drag G away from both axes [at a nice, whole-number distance from J, if you can] and construct a through J and G. We will look a the “sum of ” property used in defining betweenness, so we want to look at distances GF (which we have) and JF (which we don’t yet have) so we can compare GF + FJ to GJ. (d) To get distance FJ : i. click on the “Tools” button (triangle with three dots) and select TAXIDISTANCE on the popup menu (Notice the cursor becomes a white pointer with a red dot (a point) on the end) ii. Click on F and then on J and a new table appears showing their coordinates and the taxidistance FJ . Drag the new FJ table down close to the GF and GJ tables to make the next steps easier

1 (e) Select F and move it onto the segment GJ - what happens with GF , FJ and GJ? This is what you would expect from the distances when F is between G and J. (f) Now move F around to various places What happens (with GJ, GF and FJ) when F is near (but not on) segment GJ? - This is why we need the “collinearity” condition to be stated in our definition of “between” – additivity of distances is not enough. ←→ What happens when F is close to the line GJ but out past J? What if F is out past G? ←→ What happens if F moves away from GJ? ←→ (g) Do the results of the prvious step change depending on the slope of GJ — consider particularly a horizontal line, a line with slope 1, a steeper line and note your results. (h) Print a copy of this sketch with F in a position in which no two of the distances (GF, JF, GJ) add up to the third. [is the sum GF + FJ larger or smaller than GJ ?]

2. Now consider another feature of the taxicab distance: (a) Notice that points M, L, A, and I are equidistant from B and H (that is, distance from B = distance from H) — in the Euclidean plane this would put them on the perpendicular bisector of BH. Are they on this line? (Are M, L, A, I even collinear?) (b) Move M to the right and left, and move it up and down — Any pattern to the “equidistant/not equidistant” changes? (c) State some conclusions or conjectures about taxicab distance and some common Eucidean notions/facts such as the , construction of perpendicular bisectors, addition of distances. 3. Now we will look at the taxicab with 3 centered at the origin. [A circle is the set of all points that are a fixed distance — in this case 3 — from a fixed point — in this case, the origin. We’ll see that our axioms so far don’t force to look like Euclidean circles]. (a) Download and open the sketch ‘taxicircle.gsp”. You will see a Cartesian coordinate system and several clusters of points (in all four quadrants) with their distances from the origin (P ). (b) Choose the ”selection” tool (press the esc key or click on the arrow tool) and move the three first quadrant points until all three are at distance 3 from the origin. [Note: you may have to settle for 3.01 or 2.99 - the computer screen is really not continuous but a discrete sample of points] (c) Now repeat the process in the other quadrants, so you have twelve points on the “taxidistance circle of radius 3” - what do you observe? What would you get if you could fill in all points of the circle? (d) Print a copy of this sketch to hand in. (e) What do you conclude about circles in the taxicab plane? Can you prove it? [Use the taxicab metric to write an equation that says “Point (x, y) is at distance r from the origin” and work with it algebraically] SKILL EXERCISES:(hand in - individually - with this week’s assignments) 1. Prove that in R2 with the taxicab metric and the usual definition of “line” (solution set for an equation of the form ax + by + c = 0, a, b not both 0 - matches the Cartesian plane definition), every line given by an equation c with b 6= 0, which can therefore be written in the form y = mx − b t) has a ruler for the taxicab metric, given by f(x, y) = (1 + |m|)x. That is, show (Note that parts a and b duplicate what was claimed for a ruler on a Euclidean line):

(a) f is one-to-one: If A = (xA, yA) and B = (xB, yB) are distinct points on the line, f(A) 6= f(B) [This should be very short and straighforward]

(b) f is onto R: For each r there is a point Xr = (x, y) for which f(X) = r [This should also be algebraic and straightforward]

(c) |f(A) − f(B)| = |xA − xB| + |yA − yB| (the taxicab distance from A to B) [Remember that A and B are points on the line given by y = mx + b] 2. Text p. 136 (section 3.2) # 3, 4 [assume usual angle measurement from Euclidean coordinate geometry]

3. Reread Section 2.4 in the text and read section 2.5

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