Geometry

1) We began the day looking at fences - simpler ones at first (in two dimensions, that is!) Recall that we started off with a problem of splitting up a quarter circle into two equal sized areas. Here is a picture of almost all of the regions that we looked at in class, Quarter Circle Problem along with one extra variation (the one marked "A") that I threw in. For this question, please, before you do any calculations with formulas you might know, try to rank order all of these 8 fences in order of length (like we were doing in class). Pay special attention to fences B and F - how do they compare? write down geometric explanations for any two of your comparisons (you can repeat ones that we came up with in class) - for example can you come up with a simple argument for which one is shorter - A or B? or how about D versus B? What about B versus F? Don't worry about getting the actual orders correct - this is just a exercise for you to build up your geometric intuition "muscles"! 2) Next, now find out the true rank order of the fences by actually calculating the exact lengths of each of the fences using any formulas you'd like - you have to take a bit of care to keep track of the appropriate square roots that show up as you do this. Do this for all the fences except the one marked C - this one does in fact turn out to be the shortest one of the eight. Bonus cre dit: if you do figure out the length of the one marked C then we'll count it for bonus credit (it will likely require a bit of calculus to determine, or finding a solution to a non-trivial equation in any case) 3) Off to multidimensional fence building! Try to create a table giving the maximal number of regions one can create in 1, 2, and 3 dimensions, respectively, for 0, 1, 2, 3, 4, 5, 6,

7, 8 fences, respectively (i.e. you're going to end up with a chart with 3 columns and 9 rows. The first column (dimension 1) was remarkably straightforward, and we made good headway on column 2 (dimension 2) as well. Now, I gave some hints as to how to think about column 3 (dimension 3), for instance by looking at any new fence (a plane) that slides into position - we looked at what the intersections with the previous planes looked like, noticing the lines that showed up on the surface of the new plane (look at the class video again for this). This can help answer how many new regions were created when the new fence/plane was brought in. Another hint is to see if there's any connection to the previous dimensions - is it possible to figure out the next number in column 2 by looking over at an entry near it in column 1. Likewise, is there a way to predict the next entry in column 3 by looking at a nearby entry in column 2? This is an exploration problem, so just do your best! Please, however, don't just try to look up an answer on the internet - the point is to look for patterns yourself as you do this. 4) Now, branching out into some completely uncharted territory... if we were to do the same exercise in the fourth dimension(?!) building fences (which are now three dimensions), then using what you've seen so far in class and in the previous question, see if you can come up with a prediction of the maximal number of regions you could create with 0, 1, 2, 3, 4, and 5 fences in four dimensional space - try using the charts for 1, 2, and 3 dimensions to help out with this as it's pretty challenging/impossible(?) to actually "see" the answers - here we're working with a geometric question that we can only theoretically answer!

5) Back to Euclid! Now it's time to read and digest Propositions 13 through 26, finishing up the first section of Book 1 on triangles and congruences. To do this please go ahead and work through each proposition with your compass and straightedge, recreating each construction just as Euclid directs. In some of the propositions you'll just be drawing diagrams to back up the explanations, not really constructing anything new (e.g. proposition 20), but go ahead and create the diagram as given by Euclid in any case. And note that you don't have to recreate earlier propositions to do this - just do the steps as they're indicated in each proposition. To get credit for this assignment please just turn in a proposition summary sheet like we were doing in class for the first 12 propositions (i.e. just a brief descriptive sentence about each proposition) – please don't turn in your construction diagrams. For this set of propositions also please don't bother counting construction steps as you did last week - the numbers get quite large as each proposition relies on a longer and longer sequence of earlier results. 6) And now here’s an odd problem(!) - suppose you're doing some ruler and compass constructions but you've accidentally broken your ruler. Suppose that your broken ruler is now only three inches long. If you are given two points that are about six inches apart (maybe a bit more, maybe a bit less, just not exactly equal to six inches) show how you can construct the line segment joining these two points. This construction is somewhat challenging - you'll have to be quite creative to bridge the gap using just your compass and the short straightedge - good luck!! The actual number of construction steps for a solution is surprisingly low, however - you should be able to do this with fewer than 10 steps.

7) And to close out the week, some more reading! Please read through the following article on teaching geometry: A Case for Euclidean Geometry – this one should seem somewhat familiar to everyone who made it through the first fence problem! There’s nothing particularly involved to turn in for this problem - please just answer the question they pose in the last paragraph of the article on page 417 – the paragraph right before the Bibliography - "can you find another method to trisect a circle?" by thinking of another approach for doing this - you can just draw your approach (you don't need to show how to actually construct it). .Also, please a