Posted by: Dave Richeson | February 10, 2013

Editing mathematical writing

As I mentioned in a previous post, I’ve been assigning large-scale collaborative writing projects in my mathematics classes. I’ve had my topology students write a textbook for their class, and this semester I’ve been doing the same in my discrete mathematics class. As I mentioned in that post, the approach has been very successful, but one area that needs work is the editing/revising processes. Some  students are very reluctant to edit what has already been written. I think partly they are uneasy about editing the work of others, but partly they do not know how to revise/edit writing. So I’m trying a few new things this semester.

As I have in the past, I will give the students a document that I wrote called “The Nuts and Bolts of Writing Mathematics.” But now I’ve taken that document and turned it into a set of questions that the students should keep in mind while editing the document. (In fact, this checklist should work for any mathematical writing, not just collaborative writing.) You can download the pdf or view the HTML document below. (Note: If you have any suggestions for questions that I should add, please post them in the comments!)

I also plan to model good editing. I will give them a mathematical document (that I wrote) that needs a lot of revising, and we will spend some class time going through it, discussing how to revise it.

Finally, I’ve been reading this book on team writing. It discusses three ways to do team writing: face-to-face (you and I work together on the writing), divided (you do this, I do that), and layered (you look at it first, I look at it next). I think the layered approach will work best for my students. The book breaks up naturally into small parts—theorems and examples. For each problem I will practice the layered approach to writing and revising: Student 1 does the writing, student 2 does the editing, I give feedback, then student 3 does the final editing. (I may add one more layer of student editing before I see it.) I’ve created a Google Docs spreadsheet which gives, for each theorem/example, the names of the student writers and editors and the due dates for their work.


Editing mathematical writing: a checklist

The mathematical argument

  • Is the mathematics correct?
  • Is the argument easy to follow?
  • Is there any extraneous information?
  • Is the argument clear?
  • Would it benefit from reordering the sentences?
  • Is the level of detail appropriate for the audience?
  • Are all proofs free of examples?

Mathematical writing

  • Is the writing in first person plural? (Use “we,” not “I” or “one.”)
  • Is the writing in the present tense? (Write “we find that,” not “we found that.”)
  • Is the writing in the active voice? (Avoid “it was shown that.”)
  • Are there symbols where there should be symbols and words where there should be words? (Use “x≤ 0,” not “x is ≤ zero” or “x is less than or equal to zero.”)
  • Are the connecting words (thus, so, hence, therefore, moreover, furthermore, in addition, consequently, etc.) used appropriately and not repetitively?
  • Is the writing free of flowery, imprecise, descriptive, and vague language?

LaTeX

  • Are all mathematical expressions written in math mode? (We should see x, not x.)
  • Is all text written in text mode? (We should see “such that,” not “suchthat.”)
  • Are mathematical functions typeset correctly? (We should see sin(x), not sin(x).)

Figures

  • Are the figures of high quality?
  • Do they have captions?
  • Does the text refer to them?

Mechanics

  • Is the spelling, grammar, punctuation, capitalization, subject/verb agreement, sentence structure, etc. correct?

Style

  • Does the layout reflect the style of the rest of the document?

Note: do not be afraid to scratch an entire proof and start over. Sometimes completely rewriting an argument is better than trying to fix a poorly written one.

Posted by: Dave Richeson | December 16, 2012

How I teach topology: an inquiry-based learning approach

Recently I’ve had a number of people ask for more information about how I teach topology. I’ve taught it five times using a “modified Moore method” or “inquiry-based learning” approach. I’ve modified it each time, trying to work out the bugs. I think it is pretty successful now.

Context. At our college all math majors are required to take a 400-level class. Typically it is Complex Analysis or Topology (they are taught in alternating years). Thus, while topology is not a required class, many math majors have to take it in order to graduate. Some of them find it extremely challenging. On the other hand, I often have very talented students in the class. In fact, often the juniors (or even sophomores) who take the class are among the best students. Typically I have 8-15 students in the class (depending on whether we offer one or two sections). The other important piece of information is that the college wants the 400-level course to be writing intensive. Many departments have a “senior seminar” that fills this role. We don’t, so that’s an extra reason to teach topology in the way that I do.

Brief description. In this course the students do not have a textbook; in fact, they are forbidden from using outside sources at any time. Instead, they are given the skeleton of a textbook. It has definitions, statements of theorems, some explanatory text, and some problems. They must prove the theorems, solve the problems, and type their work into the empty textbook. By the end of the semester they have created their own textbook. I have it printed and bound at the college’s print center and they take it home with them. (I like to have 5-10 people contribute to each textbook, so if the class is on the large side I have them create two textbooks in parallel.)

Textbook. The textbook I use came from the topology course that I took as an undergraduate—from Dick Bedient at Hamilton College. I’ve modified the book quite a bit over the years, but I couldn’t have done this without the excellent book I began with. It began as a Word file, but I moved everything over to LaTeX years ago. I maintain two copies of the book—one has space for the students to write in (like a workbook) and the other has no spaces (this is the version that they edit). I have the college’s print center print the workbook one sided (in case they need to write on the backs of the pages), three-hole punch it, put it in a binder, and sell it in the bookstore (for the cost of printing only). Sample pages of both versions are shown below.

Screen Shot 2012-12-15 at 11.19.40 PMScreen Shot 2012-12-15 at 11.19.24 PM

Homework. Each class period I tell the students what theorems/problems they should work on. It is a guessing game on my part—I have to estimate how far we’ll get in the next class period. They do the work in the workbook at home. They are free to work together, they are free to talk with me, but they are forbidden from looking at outside sources.

Presentations. When I first taught the class I had students volunteer to present the proofs/problems in class. The problem was that only the best students volunteered. So I moved to a system in which I assigned (randomly) presentations to the students. Now the students know ahead of time what “their problem” will be. If a student comes to class unprepared or presents an incorrect proof, then I take volunteers to complete the proof (or the student can stay at the board and repair the proof with the help of the rest of the class). This is also a time to talk about how the proof is written. They know that what goes on to the board will ultimately go into the textbook, so the class often gives wording, notation, or organization suggestions. (At the beginning of the semester I talk about how to give appropriate feedback. We don’t want hurt feelings.) Often the weaker students are nervous about presenting a solution that is wrong, so I tell them that they are free to come to my office in advance to “check out” their proof.

Daily sheets. The students are required to participate in class. Each day I bring the sheet below and keep track of who presents and who makes comments. I also leave myself notes, like “great proof!” or “volunteered to present the theorem when Billy was sick” or “came to class 15 minutes late.” (I number each column by the theorem/problem number.) I use these sheets to help assign a class participation grade at the end of the semester. (Excel file)

Screen Shot 2012-12-16 at 9.33.20 AM

Green pen/homework. At the end of class, the students turn in the pages that we just discussed. I grade each problem on a simple V (very good work), G (good work), L (little work), N (no work) scale. Because we talked about each problem at length, I do not write any comments on their work. One great idea I had a few years ago was to require the students to bring a green pen to class. During class they can take as many notes and make as many corrections as they please, BUT they must use the green pen. This makes my grading especially easy. I can see exactly what they wrote at home and what they wrote in class.

Secretary. Each day one student is the “secretary”—this position rotates through the class. (If the class is broken into two groups, there will be two secretaries—one for each textbook.) The secretary is responsible for writing down everything clearly, taking note of all editorial suggestions. The secretary must type all of the new content into the textbook before the start of the next class.

Textbook. When I was a student the textbook was a MS Word file that lived on a 3.5″ floppy disk. The students would pass it around as needed (this was in 1992—before the web, and at a time when I had to go to the basement of the library to check my email). When I first taught the class the textbook was a Word file that students emailed back-and-forth. Later I turned it into LaTeX (which meant there was a whole directory of files). At first I had the secretaries edit the files on a dedicated computer in the math major’s research room. The last two times I taught the class I placed the files in a shared DropBox folder—the students could edit the book at home or on one of the school machines.  That worked very well, but not flawlessly—a few times we had problems with files being forked. But we’d come a long way. Next time I teach the class I plan to use one of the new online LaTeX websites. I think this will be fantastic—collaboration will be a breeze and the students won’t have to install LaTeX to do their work. (Note: I’ve had my students use LaTeX in many of my classes—by and large they seem to really like it. I made this LaTeX cheat sheet which helps flatten the learning curve a little.)

Editing. The textbook is broken down into chapters. At the start of the semester I create “editorial teams” for each chapter. After we complete a chapter (and the secretaries have typed in the proofs), the editorial team is responsible for “cleaning it up.” They do that, then I take a pass at it, making my corrections and suggestions (hand written on the pages), and then the editorial team implements my changes. If there is a weakness in my system it is at this stage. The editorial team does not make many changes on their own. I think they are uncomfortable editing the work of a classmates. The draft I get generally needs a lot of work. I need to strategize about how to improve this. (Suggestions? Leave them in the comments!) There is also an editorial team that is responsible for the entire book. They coordinate the editing of the full book at the end of the semester.

Images. The book requires many images—open and closed sets in \mathbb{R}^n, the torus, the Klein bottle, the Möbius strip, etc. I have yet to find a good drawing program that is easy to use, powerful enough to produce good images, and free (or cheap). I’ve tried many different pieces of software and have had mixed results with all of them. I’d be happy to hear suggestions for what to use.

Exams. I either give two midterms and a final exam or three midterms and no final exam.

Peer evaluations. I have the students evaluate each other (and themselves) at the end of the semester. I use this information to help me assign their class participation grades. I’ve found that sometimes students who are quiet in class will get rave reviews from their classmates (“she was always extremely helpful with editing,” “he was very good at explaining the concepts to me,” etc.). On the flip side, some of the more “active” members of class could be bad collaborators (“he never showed up for our editing sessions,” “her secretarial work was never done on time,” etc.). Here is an example of the evaluation sheet. (Excel file)

Screen Shot 2012-12-16 at 9.29.32 AM

Final textbook. At the end of the semester, after the textbook has been edited multiple times, I give it a grade. The entire group gets the same grade on the book. Usually the grade is high, but occasionally (rarely) I’ll have a group that does not put in the required effort. When it is complete I quickly send it to our print center to have it printed and bound. The turn-around time is very short, so most students are able to pick it up before they leave for the semester. Typically the students are THRILLED to hold this book. They love to share it with their friends and parents. It is neat to see their pride. However, I used to have a few books weren’t picked up. So (and this also coincided with the recession and subsequent budget cuts) I now have them pay for the book (in advance)—which is usually $5, or so. This way we only print the book for students who want it (usually 75% of the class).

Final thoughts. I love teaching this class and the students (for the most part) enjoy taking it. The material is very challenging and some of the weaker students struggle with it. But I think that a faster-paced traditional class would be even worse. I think that the students really learn the material using this approach. I like that the students spend time presenting mathematics at the board. This is missing from most other mathematics classes. And the emphasis on writing and revising is very valuable. Finally, I love seeing their faces light up when they pick up the finished book at the end of the semester.

Source files. Professors at other institutions have asked me for copies of the blank textbook. Unfortunately, because I was not the original creator of the book I don’t feel comfortable distributing it. However, there is a lot of good IBL content online. For example, look at the Journal of Inquiry-Based Learning in Mathematics and the links on their website.

What’s next? I’ve enjoyed this method of teaching so much that I’ll be teaching my Discrete Math/Intro to Proof class using this technique next semester. I’m very excited about it (and somewhat nervous about it too). I’m using these notes (by Dana Ernst) as a starting point. I’m planning to do things similarly, but slightly differently, in this class than in my topology class. But that’s a topic for another blog post.

Posted by: Dave Richeson | December 9, 2012

The Pigpen Cipher in Latex

Recently my son and his friends have been enjoying sending secret messages back-and-forth using the pigpen cipher (also called the masonic cipher or Freemason’s cipher). It produces codes that look like:

The pigpen cipher is a simple substitution cipher—there is a 1-1 correspondence between these special symbols and letters of the alphabet. The correspondence is illustrated in the chart below. Each letter is replaced by the boundaries around the letter and a dot if there is one. So, for example,

I, being a lover of Latex, wondered if it was possible to create pigpen ciphers in Latex. It is! The cipher text text above was created using Oliver Corff’s Pigpen Cipher package. After downloading the package and including it in the Latex file you must simply type

{\pigpenfont MATH IS FUN}

to obtain

(Pigpen cipher key (picture) from Wikipedia)

Posted by: Dave Richeson | December 5, 2012

Online LaTeX editors

For the last 10+ years I’ve taught topology using a modified Moore method, also known as inquiry-based learning (IBL). The students are given the skeleton of a textbook; then they must prove all the theorems and solve all of the problems. They are forbidden from looking at outside sources. The class types up their work as they go. At the end of the semester they have a textbook that they wrote. It is a great way to learn, and at the end of the semester the student are thrilled to hold a bound copy of the textbook that they created.

When I did this first (as a student) it was a Word file shared on a floppy disk. When I started teaching the class this way it was a Word file emailed between participants. Later I rewrote the textbook as a LaTeX file. I’ve experimented with various means of collaboration. Most recently I used a shared DropBox folder to house the file. This way all of the students could collaborate on the ever-growing document.

This approach worked pretty well, but there are a few downsides. The document occasionally got forked. This happened when two or more students edited the document at the same time. It would take a while to merge the content back together. Also, a student must have LaTeX installed on his or her computer or must be willing to work on one of the computers in our building.

This spring I’m considering trying an online LaTeX environment. (And, in fact, I’m also going to try using this IBL approach in my Discrete Math class—our “intro to proof” course). I wanted a robust, easy-to-use online solution that would house the LaTeX files and any extra files (images, etc.), would allow the students to compile the document online, would allow us (me) to download the files at any time, would allow collaboration by up to 16 people (with 0-3 people editing at any one time), and would allow access to previous versions (in case someone deletes the entire document, for example).

After a little investigating I found these great sites. So I thought I’d share them with you.

Most of them have free and pay versions. It is likely I’ll have to pay for an account so that I can be the “owner” of the file (the textbook), then the students can get free accounts and I can add them as collaborators.

Not all of these sites fulfill my complete wish list—especially the version history requirement—but it looks like they’re all being actively developed and that new features (version history, DropBox integration, etc.) are right around the corner. I’m thrilled that the technology has come so far. I haven’t decided which one I’m going to use. If you have any thoughts/preferences, post them in the comments. Likewise, if I’m missing any sites, suggest them in the comments too.

If I could add some items to my wish list it would be:

  • Point-and-click menus containing the common LaTeX symbols. This isn’t a key feature for me, but the students would love it. (In the meantime I’ll have to point my students to my quick guide for LaTeX and Detexify.)
  • Table and array editors in which you enter the contents of a table in a grid (like this), then the editor would insert the code into the document
  • A BibTeX interface to help create a .bib file. At a minimum something like this, but even better, something that would tie in to databases such as Google Scholar, MathSciNet, Zentralblatt, and JSTOR and scrape the bibliographic content for you.
Posted by: Dave Richeson | October 18, 2012

How do you place incoming mathematics students?

Our department is looking for a better method of placing incoming students in mathematics courses.

Currently we have a placement exam that determines whether a student should begin in a calculus I course or in a calculus/precalculus hybrid course (our lowest-level math class). The exam consists of 25 precalculus questions. It does a pretty good job.

Recently we’ve had an increase in the number of students who have taken a calculus class in high school. If it is an AP class, then we can use their AP score to determine if they should skip the first semester (or more) of calculus. However, students who have taken a good non-AP course slip through the cracks. This is especially true for international students. If those students identify themselves, then we use ad-hoc methods to determine their placement (talking to them, and looking at the syllabus from their course, their exams, and their textbook—which is sometimes not in English).

Because this is time consuming and because it requires the students to come forward to ask for it, it is not ideal. We are now thinking about constructing a placement exam that would determine if a student should place out of first semester calculus. I thought it would be interesting and informative to hear how you do mathematics placement at your school. As I said, we’re especially interested in the Calc I vs. Calc II placement, but other readers may be interested in other math placement issues. Please share your placement procedures in the comments below. 

If you have a calculus placement exam and are willing to share it with us, please email me. That would be fantastic. We would keep it confidential and would not post it where it would be publicly available.

If you are interested in how we place students, see our online placement guide.

Note: I know many schools allow students to place themselves. We would rather not go that route. We’ve found that students are not always good at judging their mathematical ability. And we’ve found that it is problematic for the students, for the professor, and for the students’ classmates if a student ends up in the wrong class (either too high or too low). Our placement exam forces the students who do well to go into Calculus I and students who do poorly to go into the lower-level class. Students with scores in the middle can place themselves.

Posted by: Dave Richeson | August 30, 2012

Ancient number systems in XeTeX

I am teaching a history of mathematics class this semester. We are beginning with a brief discussion of ancient number systems: Egyptian, Babylonian, Mayan, Chinese, Incan, GreekRoman, and Hindu-Arabic. As I was writing up the first homework assignment it occured to me that I should investigate whether these numbers could be typeset using LaTeX.

It quickly became apparent that, because fonts are involved, I would have to use XeTeX rather than LaTeX. It was a fun (although time consuming) exercise. In the end I was able to typeset Egyptian hieroglyphics, Babylonian cuneiform, and Chinese rod numerals. Because the syntax was often messy, I spent a while burying the complicated TeX in the headers so that the numbers would be easy to work with in the document.

For example, to generate the Egyptian hieroglyphics for 123 I write

\Ehun\Eten\Eten\Eone\Eone\Eone.

The fraction 1/123 is

\Efrac{\Ehun\Eten\Eten\Eone\Eone\Eone}.

To express 123 in cuneiform all I have to write is

\Bnum{123}.

To create a number board with the Chinese counting rods representing 123 I type

\Cnum{|x|x|x|}{\Cvone & \Chtwo & \Cvthree}.

If you would like to give this a try, download my .tex files:

egyptian.tex and egyptian.pdf

babylonian.tex and babylonian.pdf

chinese.tex and chinese.pdf

I’d love to be able to do something similar with the Mayan numbers. I tried for a while, but couldn’t get them to work.

Disclaimer: I know my way around TeX pretty well, but I’m not a power user. It took me quite a while to get all this to work. I’m not sure I can offer much trouble-shooting advice if you can’t get this to work on your computer.

Posted by: Dave Richeson | August 16, 2012

Mathematics departments at liberal arts colleges

I’m often curious about how other mathematics departments do things—how they structure their curriculum, run the Putnam Exam, handle research projects, etc. This invariably leads to a lot of web searching. So I decided to put together a collection of links to mathematics departments at schools like mine (a small liberal arts college). Because I thought others might like this resource, I decided to share the list here.

Here are links to the mathematics departments of top liberal arts colleges in the US (I used the top 60 schools as reported by US News in 2011).

Amherst College
Bard College
Barnard College
Bates College
Beloit College
Bowdoin College
Bryn Mawr College
Bucknell University
Carleton College
Centre College
Claremont McKenna College
Colby College
Colgate University
College of the Holy Cross
Colorado College
Connecticut College
Davidson College
Denison University
DePauw University
Dickinson College
Franklin & Marshall College
Furman University
Gettysburg College
Grinnell College
Hamilton College
Harvey Mudd College
Haverford College
Kenyon College
Lafayette College
Lawrence University
Macalester College
Middlebury College
Mount Holyoke College
Oberlin College
Occidental College
Pitzer College
Pomona College
Reed College
Rhodes College
St. Lawrence University
St. Olaf College
Scripps College
Sewanee University
Skidmore College
Smith College
Swarthmore College
Trinity College
Union College
United States Air Force Academy
United States Military Academy
United States Naval Academy
University of Richmond
Vassar College
Wabash College
Washington and Lee University
Wellesley College
Wesleyan University
Wheaton College
Whitman College
Williams College
Willamette University

Posted by: Dave Richeson | June 20, 2012

Plato’s approximation of pi?

Today I came across an assertion that Plato used {\sqrt{2}+\sqrt{3}} as an approximation of {\pi}. Indeed, it is not a bad approximation: {3.14626\ldots} (although it is not within Archimedes’s bounds: {223/71<\pi<22/7}).

Not only had I not seen this approximation before, I had not heard of any value of {\pi} attributed to Plato.

I investigated a little further and discovered that there is no direct evidence that Plato knew of this approximation. It was pure speculation by the famous philosopher of science Karl Popper! Here’s what Popper has to say (this is in his notes to Chapter 6 of The Open Society and its Enemies, Vol. 1, pp. 252–253).

It is a curious fact that {\sqrt{2}+\sqrt{3}} very nearly approximates {\pi}… The excess is less than {0.0047}, i.e. less than {1 \frac{1}{2}} pro mille of {\pi}, and we have reason to believe that no better upper boundary for {\pi} had been proved to exist. A kind of explanation of this curious fact is that it follows from the fact that the arithmetical mean of the areas of the circumscribed hexagon and the inscribed octagon is a good approximation of the area of the circle. Now it appears, on the one hand, that Bryson operated with the means of circumscribed and inscribed polygons,… and we know, on the other hand (from the Greater Hippias), that Plato was interested in the adding of irrationals, so that he must have added {\sqrt{2}+\sqrt{3}}. There are thus two ways by which Plato may have found out the approximate equation {\sqrt{2}+\sqrt{3}\approx \pi}, and the second of these ways seems almost inescapable. It seems a plausible hypothesis that Plato knew of this equation, but was unable to prove whether or not it was a strict equality or only an approximation.

Popper then spends a couple of paragraphs tying this into an earlier discussion of Plato’s Timaeus. This is the work in which Plato discusses the four elements (earth, air, fire, and water) and associates them with four of the regular polyhedra (cube, octahedron, tetrahedron, and icosahedron, respectively). The connection between Timaeus and {\pi} is the relation between the values {\sqrt{2}} and {\sqrt{3}}, the 45-45-90 and 30-60-90 triangles which can be used to make the faces of the polyhedra, and the approximations to the area of a circle using these triangles.

Popper ends with the reminder/disclaimer:

I must again emphasize that no direct evidence is known to me to show that this was in Plato’s mind; but if we consider the indirect evidence here marshalled, then the hypothesis does perhaps not seem too far-fetched.

Note: if we take the unit circle and construct a circumscribed hexagon and an inscribed octagon, then the area of the hexagon is {2\sqrt{3}} and the area of the octagon is {2\sqrt{2}}. So, it is true that the average of these areas is {\sqrt{2}+\sqrt{3}}.

Posted by: Dave Richeson | June 18, 2012

Puzzler: a squarable region from Leonardo da Vinci

It is famously impossible to square the circle. That is, given a circle, it is impossible, using only a compass and straightedge, to construct a square having the same area as the circle.

I will let you read elsewhere about the exact rules behind compass and straightedge constructions. The punchline is that if you begin with two points 1 unit apart, then you can construct a line segment segment of length {a} if and only if {a>0} can be created from the integers using the operations {+}, {-}, {\times}, and {\div}, and by taking square roots.

Thus it is possible to construct a line segment of length {(1+\sqrt{5})/2} (the golden ratio), but it is impossible to construct a segment of length {\sqrt[3]{2}} (hence it is impossible to double the cube).

A circle with radius 1 has area {\pi}. A square having this same area would have side-length {\sqrt{\pi}}. Because {\pi} is transcendental, this is not a constructible length. Thus it is impossible to square the circle.

Even though the circle is not squarable, some regions with curved boundaries are. For example, in the fifth century BCE Hippocrates of Chios (no, not that Hippocrates) showed that several lunes are squarable.

Let’s give a brief proof that the shaded lune shown below is squarable. Suppose the large circle has radius 1. Then triangle {ACD} has area {1/2} and sector {ADCE} has area {\pi/4}. Thus segment {ACE} has area {\pi/4-1/2}. The smaller circle has radius {\sqrt{2}/2}. So the semicircle {ACB} has area {\pi(\sqrt{2}/2)^{2}/2=\pi/4}. It follows that the lune has area {\pi/4-(\pi/4-1/2)=1/2}. A square with the same area as the lune would have side-length {\sqrt{2}/2}, which is constructible. Thus the lune is squarable.

It turns out Leonardo da Vinci played around with squarable figures, and he discovered many beautiful examples. Below I’ve included one of Leonardo’s figures (the on the left). I’ve included the center and right-hand figures to give more information on how Leonardo’s design is created.

So here’s the puzzle: show that Leonardo’s figure is squarable. 

(Hint: assume that the radius of the large circle is 1. Then find the total area of the shaded regions. Show that the area, and hence the square root of the area, is a constructible number.)

Have fun!

Posted by: Dave Richeson | June 1, 2012

Angle trisection using origami

It is well known that it is impossible to trisect an arbitrary angle using only a compass and straightedge. However, as we will see in this post, it is possible to trisect an angle using origami. The technique shown here dates back to the 1970s and is due to Hisashi Abe.

Assume, as in the figure below, that we begin with an acute angle {\theta} formed by the bottom edge of the square of origami paper and a line (a fold, presumably), {l_{1}}, meeting at the lower left corner of the square. Create an arbitrary horizontal fold to form the line {l_{2}}, then fold the bottom edge up to {l_{2}} to form the line {l_{3}}. Let {B} be the lower left corner of the square and {A} be the left endpoint of {l_{2}}. Fold the square so that {A} and {B} meet the lines {l_{1}} and {l_{3}}, respectively. (Note: this is the non-Euclidean move—this fold line cannot, in general, be drawn using compass and straightedge.) With the paper still folded, refold along {l_{3}} to create a new fold {l_{4}}. Open the paper and fold it to extend {l_{4}} to a full fold (this fold will extend to the corner of the square, {B}). Finally, fold the lower edge of the square up to {l_{4}} to create the line {l_{5}}. Having accomplished this, the lines {l_{4}} and {l_{5}} trisect the angle {\theta}.

Let us see why this is true. Consider the diagram below. We have drawn in {CD}, which is the location of the segment {AB} after it is folded, {AC}, the fourth side of the isosceles trapezoid {ABDC}, and {AD}, the second diagonal of {ABDC}. We must show that {\theta=3\alpha}, where {\alpha=\angle DBE} and {\theta=\angle CBE}.

Because {BE} and {DF} are parallel, {\angle DBE=\angle BDF}, and because {DF} is the altitude of the isosceles triangle {ABD}, {\angle BDF=\angle ADF}. Thus {\alpha=\angle DBE=\angle BDF=\angle ADF}. Now, {ABDC} is an isosceles trapezoid and {ABD} is an isosceles triangle, so {ABD} and {BCD} are congruent isosceles triangles. Thus {\angle CBD=\angle ADB= \angle BDF+\angle ADF}. It follows that {\theta=\angle CBE=\angle DBE+\angle CBD=\angle DBE+\angle BDF+\angle ADF=3\alpha}.

The geometric properties of origami constructions are quite interesting. Every point that is constructible using a compass and straightedge is constructible using origami. But more is constructible. As we’ve seen, it is possible to trisect any angle using origami (I’ll leave the obtuse angles as an exercise). It is possible to double a cube. It is possible to construct regular heptagons and nonagons. In fact, where the constructability of {n}-gons is related to Fermat primes, the origami-constructibility of {n}-gons is related to Pierpont primes. While the field of constructible numbers is the smallest subfield of {\mathbb{R}} that is closed under square roots, the field of origami-constructible numbers is the smallest subfield that is closed under square roots and cube roots. In fact, it is possible to solve any linear, quadratic, cubic, or quartic equation using origami!

There are quite a few places to read about geometric constructions using origami, but a good starting point is this online article (pdf) by Robert Lang.

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