Geometry

I need by Monday around 2:15pm Houston time. Please, read before you ask to do it, thank you.

2) We looked at Triangle numbers in our last class (the
sequence 1, 3, 6, 10, …) – each number is given by the
number of dots in a 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!