Geometry

1) Work through propositions 27 through 34 (this is the second part of Book of Euclid the elements I which introduces results involving the parallel postulate). This time create a dependency map This will help you get a sense of how the propositions build on one another. Don't worry about being completely accurate (but do your best!) - the point is to see how the propositions support each new result, and to see why Euclid chose to introduce the results in the order he selected. 2) Here are a few more "rusty compass" constructions to take care of. These can be useful to give to students who are already used to "regular constructions" and as a way of changing perspectives a bit. Suppose you've got a straightedge and a compass, but your compass has rusted in position and now has a permanent one inch radius (i.e. you can only draw circles with a radius equal to one inch). The straightedge is "normal" in that you can draw line segments of any length. First, given a line L, and a point A on the line, you're asked to construct a line perpendicular to L that goes through A. As part of your answer please be sure to describe your construction so someone else could do the same thing (i.e. provide the steps taken instead of just turning in a finished diagram on its own - writing things such as "draw circle at B with radius BC" ... "label new intersection as point D" etc.) 3) And one last rusty compass construction - this one's a bit more involved than the previous ones, and a reasonable answer can take upwards of 12 construction steps (if you can get it down to under 12 steps, great, if not try to get to as close to it as you can - and there are some solutions that are even lower!). So... again, suppose you've got a straightedge

and a compass and that your compass has rusted in position with a one inch radius. Now given a line L and a point A that is around 3 inches away from the line construct a line through A and perpendicular to L. Note that the line might be a bit more or less than 3 inches - you can't just assume that it's exactly 3 inches in your work. When writing up your solution, again be sure to write down the steps you took (and number them) - so that anyone else could recreate your work precisely. 4) with an erroneous theorem that all triangles are isosceles (and so in fact must be equilateral). This is obviously complete baloney! To figure out what's going wrong with this construction "proof" (which is attributable to W.W. Rouse Ball in 1940), please take time to draw a carefully executed diagram of the construction steps to find out where the error occurred (as a hint - it has to do with where the lines that are supposed to be perpendicular to the sides of the triangle should actually end up going - if you have to extend one of the sides of the triangle to create the perpendicular, then go ahead and do that to see how the problem in the final conclusion shows up). As a bit of a hint - this construction problem involves the issue of "betweenness" - where things actually intersect, and how depending on diagrams can end up leading to problems - we saw that show up in the diagram for Proposition 7 as well. Here's a description of the steps: All Triangles Are Isosceles . As part of your answer, you should try to figure out what's up with the (misleading!) equation CF + FA = CG + GB. Note that it is in fact true that CF = CG and that FA = GB, so there's something else wrong with the equation as it’s written here...(?) Make sure that you explain what's going on with the erroneous "proof" in enough detail so that it's clear what this "proof" has actually

shown. Big hint - point E won't show up where you drew it in the triangle - but just saying that point E must end up somewhere else isn't the end of the story - create the diagrams, find out where E must be located and then, still, figure out what's wrong with the argument I showed you - if necessary (and it will be!) extend the sides (CA and CB) of the triangle so that you can do the construction out completely, finding points F and G (one of which will be forced to lie on one of the extended triangle sides). For bonus credit - dive in and check out this construction using GeoGebra (Links to an external site.) Links to an external site. (it's free and you can download a copy). Print out your construction to show that you were successful at using GeoGebra! We'll continue to see GeoGebra at various times in class, so it's nice to get comfortable with the software. 5) Now it's time to use the power of triangles(!) to prove that the two diagonals of a rectangle are congruent and that they bisect each other. Here use a definition of a rectangle as just being a quadrilateral with four right angles. In your answer you should work solely with the results we've seen so far in Euclid - i.e. you can use results from any of the propositions up through Prop 34 at this point in your proof. 6) Finally, given a line segment, then (using a straightedge and compass as usual) construct a square based on that given line segment (i.e. with the line segment as one of its sides). Note that you have to assume that your straightedge is just a straightedge, not a ruler so it can't be used to measure distances, but you can assume that given the

existence of Proposition 2 (Euclid book 1 the elements), that your compass can in fact transfer distances now (as most modern compasses do anyway). The important part of this question is that you need to do your construction in such a way so that you can prove that your square construction does in fact yield a square (i.e. a quadrilateral with four equal sides and four equal angles). In doing this, please just use the results from any of the first 34 propositions (including, for example, the result from proposition 32 that the sum of the angles in a triangle is 180 degrees - or as Euclid would put it "equal to two right angles"!). This type of construction/proof mirrors many of Euclid's Propositions. It's one thing to construct a particular type of figure, quite another, sometimes, to then prove that the figure actually has the desired properties(!) 7) Back to multidimensional fence building! Suppose you've fallen into a hole in the space time continuum and you've ended up in some weird universe with five dimensions(!) You run across a fence builder who's putting up four dimensional fences splitting up this bizarre five dimensional space. Using what you've learned from the fence problem solutions we looked at in class, write down the maximum number of regions that can be produced by 1, 2, 3, 4, 5, 6, 7 and 8 fences respectively in this five dimensional world. Don't even bother trying to "picture" five dimensional space - you can do all the work just by using the patterns that we've worked out so far (you'll need to extend some of the sequences a bit to nail this down). 8) Now you're really in trouble - you've fallen into another hole in the spacetime continuum and you've ended up in a universe with 2018 dimensions(!) You run across a fence

builder who's putting up fences (each one is 2017 dimensional!) splitting up this bizarre 2018 dimensional space. Using all that you know of these weird situations, write down the maximum number of regions that can be produced if the fence builder builds (a) 2018 of these fences, and then (b) the maximum number produced by 2019 of these fences (hint, your answers will be absolutely huge - you won't be able to write down the actual digits for them). Note, there are no real calculations required to answer this - as we did in class, just take a look at what's happening with all of the earlier fence building information and figure out how the pattern must continue.