Geometry Project - Writing Mathematics: Proofs with and without words
The goal of this project is to explore some problems in neutral geometry and Euclidean geometry and to practice writing
and communicating mathematics. There are many ways to prove a theorem and many ways to communicate mathematical ideas.
The MAA's Mathematics Magazine routinely has an article titled "Proof without words". Some examples are shared on
their webpage here.
Many of Euclid's theorems lend themselves well to these kind of proofs. In this project you should prove one of Euclid's
theorems and write it two ways: with words and without.
Topic
You may choose from Euclid's constructions. Please choose from the list of propositions on page 31 of your book (except I.47). If you
want to do a problem not on this list you can ask permission of your professor.
Work
To begin your project, work through the mathematics of the problem you chose. You may work with
a partner on this stage of the project (but the write-up will be your own). You should
solve the problem and write a draft by Friday, October 26.
Write-up
Your finished product should include two parts. A proof with words and a proof without words.
Your proof with words will look like a more detailed version of the kinds of proofs you write for your
homework. You should write as complete as a solution as possible to the problem you chose. You should
begin with an introduction to the topic, including relevant context of the problem. You should describe
any necessary background for your proof (any axioms, propositions or definitions you need), with references.
Your proof should be complete and clearly written, and may include proofs of any lemmas necessary to support
your work. It should conclude with some exposition about the value of the discoveries you made. In
particular, you might pose some further questions about your topic that you think would be interesting
to explore. Be sure to include a complete bibliography.
Your proof without words should include no words. It can be a detailed diagram, but may also be done electronically
and include animation. Be sure that it is detailed enough to explain your result without requiring any words. You
are strongly encouraged to use color in thoughtful ways to add depth to the mathematical ideas you are illustrating.
Your proof without words will be graded on its ability to communicate mathematical ideas, not on your artistic ability
(although a certain amount of care is necessary to produce a clear diagram).
LaTeX
When writing mathematics, we often need to use mathematical symbols in our exposition. For this
reason, most mathematicians prefer to use LaTeX to write their math. For this project you should
write your paper in LaTeX. If you have not used LaTeX before, a great resource for learning the basics
is this wikibook. There are lots of other great
resources available online, some of which are listed here.
Logistics
A draft of your project is due October 26 in class. It can be extremely rough on the writing but should have a solution to your
problem and a rough idea of how you will write it by this point. The draft will not be formally graded. Your project
is due Nov 2, by class time. You can work with a partner on the mathematics portion, but you should
each write up separate papers. Your grade will be determined based on quality of mathematics, quality
of exposition, and quality of presentation (effective use of LaTeX).