8_31

Lab 1

We were asked to come to class prepared to discuss this. The assignment is due Wednesday, 9/2. We began our discussion by showing how we created an **equilateral triangle**. First take points A and B. Draw a line segment between these two points. Draw a circle centered at point A whose **radius** is the segment from A to B. Similarly, draw a circle whose center is B and whose radius extends out to A. These two circles **intersect** at two points (C and D in the picture). To complete the equilateral triangle, create a line segment from C to A and another from C to B. See picture:
 * Construct equilateral triangle using Geogebra. Discuss all the things you have done that may require a proof. Discuss the proofs. Is every equilateral triangle also equiangular?
 * Use this construction to construct a regular hexagon.
 * Tile the plane with regular triangles (solid lines) and regular hexagons (dashed lines). Basically reproduce the picture:
 * Construct a square.



How do we know this triangle is indeed equilateral? We constructed a simple proof:

Since C is contained in c1, we know that d(C,A) = d(A,B). Since C is contained in c2, d(C,B) = d(A,B). Therefore d(C,B) = d(C,A) = d(A,B). □

In coming this far, we have used many words we have not yet defined: **triangle, equilateral, intersection, radius, distance, length**.

Additionally, is our proof really complete? For instance, how do we know that the intersection of c1 and c2 is not an empty set ?

We discussed how one might continue using this same process to create many equilateral triangles around point A. In fact, one could construct exactly 6 of these **congruent** equilateral triangles that all share a point with A and are enclosed in c1.

Are these triangles all **equiangular**? How would we show that? What is the relationship between angles opposite to sides of a triangle? How do we know that these six triangles are **congruent**? What //is// congruent? Here we ran into a bit of trouble. Our working definition of congruent became: two objects are **congruent** if when you ‘lay them on top of each other’ they are the same. This definition is somewhat vague. Who says we can pick these objects up and lay them on top of each other? Why would we want to?

Mostly we spent this time creating questions and not necessarily answering them. Hopefully we will be able to reach some answers on our own and put them in our Lab 1 write-up.

On to Ideal City:

The ideal city is a lot like Salt Lake City, except the construction crews always made the streets follow a grid. Salt Lake, unfortunately for those who don’t know where it happens, has a few streets that mysteriously disappear or make wild turns.

** Problem 1 ** A dispatcher for Ideal City Police Department receives a report of an accident at X = (-1,4). There are two police cars located in the area. Car C is at (2,1) and car D is at (-1,-1). Which car should be sent? ** Problem 2 ** There are three high schools in Ideal City. Roosevelt at (2,1), Franklin at (-3,-3) and Jefferson at (-6,-1). Draw in school district boundaries so that each student in Ideal City attends the school closest to them.

Problem 1 posed no problems for most groups. At first some applied the distance formula, which puts car D 5 blocks from X and car C only 4.24 blocks away. However, it was quickly remembered that police cars cannot drive through yards and homes and buildings to reach their destination, but must drive on the street. Ultimately C must go up three blocks and then left for three blocks to reach the accident. This is a total of six blocks in contrast to car D’s five. Therefore car D should be sent.

Problem 2 was a bit more challenging. In searching for a solution, most students drew line segments between the three schools and went from there. Some created quadrants of using the midpoints of these segments (although this method proved to create four school districts rather than the desired three), others created circles centered at the schools expanding outward. In either case, students were ultimately attempting to create boundaries based on how the crow flies. In other words, it was not being taken into account the students would likely be driving or walking to school rather than flying. The one group that did remember this fact was only able to create a solution by manually checking every single grid point in the school district. But this very important fact that we were all so keen to ignore because it seriously complicated our graphs led to some interesting questions:

What is a straight line in Ideal City? Is it different than a straight line in Salt Lake City? If a ‘straight line’ is defined as the shortest distance between two points, and there are multiple paths that are equally short between two points in Ideal City, does that mean that there are many different ‘straight lines’ between two points in Ideal City? Or are there none?

This discussion of what straight is leads us to the can, ball and paper experiment. How do we test for straightness? It is not sufficient to use a ruler because you can’t just assume a ruler is straight. Creating a straight line on a sphere might prove even more difficult.

In the last thirty seconds of class, we were asked to write our response to the following question and hand it in:

What is an axiom?

These notes are really thorough. I wouldn't change anything!