Geometry

triangular array. By adding together consecutive numbers you end up with this sequence of numbers (i.e. 1, 1+2 = 3, 1+2+3 = 6, 1+2+3+4 = 10, etc). Now we also know that adding together consecutive odd numbers gives you the Square numbers (i.e. 1, 1+3 = 4, 1+3+5 = 9, 1+3+5+7 = 16, etc.) – we saw different proofs of this result using the picture proofs in class. Using this same approach, figure out what the Pentagonal numbers must be – and make patterns of dots to show each of the first four Pentagonal numbers (hint, sums such as 1+2+3+4+5 give you Triangle numbers, sums where each term goes up by two, such as 1+3+5+7 give Square numbers, so for Pentagonal numbers consider numbers given by sums where each term goes up by …(?)). 3) Now I’d like you all to create another “proof by picture”. We’ll do this one somewhat in reverse by working out the formula first and then creating the picture. First find the formula… consider the Triangle number sequence, 1, 3, 6, 10, 15, 21, 28,… Now take a look at the sum of any two consecutive triangle numbers, for instance 3+6 or 10+15. What kind of number are you getting each time? Is this a coincidence or is something going on – prove it by creating your own proof by picture. Think back to what we saw in class when we put together two identical triangle numbers creating a rectangle N by N+1 (in the picture example we saw, the rectangle was 6 by 7). Now consider what happens to this picture if you combine two consecutive triangle numbers instead… To turn in your work on this problem show this picture result for several distinct cases. Next follow this up by writing down an

algebraic proof of your result as well using the formula we found in class for the nth triangle number. 4) Next, I mentioned an article that included a conversation about definitions of geometric figures – we looked at an example with trapezoids in class. Now please read the article: Conversations in a Geometry Class – you’ll notice that we’ve already had two of the conversations in our class (and we’ll touch on the third a bit later on!) Afterwards, please make a comment on our class discussion board (linked under the “Discussions” tab on the left) about the article, either something that resonated with you, something that you’d like to try in your own geometry class (or better yet, a comment after you’ve tried one of these conversations in one of your classes!) Please feel free to reply back to someone else’s comment as well – and/or write about a similar type of conversation that you’ve had with your students at some point in one of your classes. 5) So far in class we worked through Props 1 and 9, so now it’s time to look at what came in between them! First, start by looking at the inner workings of the mysterious Proposition 2. To remind you of the context for this proposition (and also for proposition 3) – as I mentioned in our last class, Euclid supposes that a compass can be used to draw a circle with a given radius on a page, but that the compass cannot be then be used to transfer distances to another location on the page – with a Euclidean compass it was assumed that as soon as a compass is lifted from the page the arms of the compass became loose and the length of the radius is lost (this is different from most modern compasses that have fixed arms that stay in position until you resize them) Thus it is not possible (using a “floppy” Euclidean compass) to draw two

circles that are exactly the same size – after you draw one circle and lift up the compass from the page the size of the first circle is lost – the compass collapses as soon as you lift it up so it’s not possible to draw another circle with the exact same radius (just for now anyway – Propositions 2 and 3 show you that it can still be done, but boy is it tricky!). So… now try your hand at the construction in Proposition 2 (but don’t read it in Euclid yet!): Given a line segment BC, and another point, A, located a short ways away from the line segment, draw a line segment starting at point A that is the same length as line segment BC (i.e. show how it’s possible to copy a line segment to another location). Note that the whole challenge is to do this construction using a “floppy compass” – i.e. one that cannot transfer distances. Good luck – this is a pretty challenging construction in my opinion – few people are able to get this to work out on their own (but it is possible!) This construction would be trivial if you could use a fixed arm compass to transfer distances – then you’d just draw a random line through point A, and measure off the distance of line segment BC with your compass and mark that on the new line starting at A… but you’re not allowed to do that with Euclid’s “floppy” compass! Count the number of steps required for your construction, then after you’re finished, read Proposition 2 in Euclid and count the number of steps he took (remember that counting steps in a compass and straightedge construction means that every time a circle or a line is drawn counts as one step, but that locating random points, or points that are the intersections of lines or circles is “free” – they don’t count as a step). If you get totally stuck (again this one is quite hard!) then you can always check out Euclid’s construction, but really do try hard to do this on your own first. Please turn in your final construction – if you’re able

to do this without resorting to Euclid, fantastic! – you’ll get a bonus point on the homework. Otherwise at the least, please go through Euclid’s construction in Prop. 2 carefully recreating the steps used there. The result of this proposition (and proposition 3) will be that it really doesn’t matter whether one has a Euclidean compass (a “floppy compass”) or a modern, rigid arm compass – all the propositions from here on will be possible to construct using either type of compass given the results of Props. 2 and 3. Very mild hint for this problem (before you go to Euclid) note that all that’s been done before Proposition 2 is the creation of an equilateral triangle in Proposition 1. So – at some point to do Prop 2 it’s quite likely that you’ll want to consider creating an equilateral triangle somewhere in your diagram, and there are only so many line segments to try building one on given you’ve only got three points at the start, A, B, and C…

7) Next create a list that summarizes each proposition (again, just for the first 12 props) – just a brief one line summary is all that I’m looking for. For instance on your list Prop 6 might read “If the base angles are equal then the triangle is isosceles.” To create this list you’ll need to figure out what each proposition actually does – some of the wording for the propositions is quite flowery and hard to decipher! 8) Counting construction steps: Now, again working with just the first 12 propositions, count the number of construction steps required in each proposition that demonstrates a construction. Note that propositions 4 through 8 don’t really involve constructions, so you can just put down a zero for these non-construction propositions on

your list. If a proposition is used in a subsequent proposition, then you should add its steps in to the total for the proposition that uses it, considering it as including as many steps as it did in the original construction – for example in Prop 9, there is an equilateral triangle construction (Prop 1), and this should show up as requiring 4 steps, not just 1, in the counting of construction steps, given that Prop 1 takes 4 steps. Remember that a step is anytime a new circle or a new line is created (important note – extending an existing line does not count as a new step and locating new intersection points also doesn’t count – points are free!). Note that it is possible to come up with slightly different answers for this question – the point is to try to be as consistent as possible and to count as carefully as you’re able to. This is just one example of a way to study The Elements – we won’t be doing this for the rest of the propositions, so don’t worry about having to do this for all 48 propositions in Book One!

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