MAMT 590-52 Origami Mathematics Revisited

Welcome! If you're looking at this web page, then you're either registered for or planning/thinking of registering for the online class MAMT 590-52 Origami Mathematics Revisited for the Spring 2023 term at Western New England University.

Here you will find some information about the course. More details will be provided in the course syllabus. Please feel free to EMAIL ME if you have any questions!

*Interested in this course, but not an MAMT student?* Then you can apply to enroll in the course as a non-degree student. WNE charges $1275 for our MAMT courses (which is fairly cheap as these things go). Here is the non-degree study application and here is the listing of the MAMT 2023 Spring Term courses.

**Meeting times:** Mondays and Wednesdays from 6:00pm - 7:20pm (Eastern Time Zone), Jan. 23 - May 10, 2023.

**Instructor: **Tom Hull, Associate Professor of Mathematics, WNE. Office: Herman 308-G

**Description:**
Origami, the art of paper folding, has become an active area of scientific and mathematical research in physics and engineering over the past 10 years. It has also been a source of innovative, hands-on techniques for teaching mathematics for even longer. This course will dive deep into the mathematics of origami and the ways it can be used to teach math topics in geometry, algebra, and combinatorics. On one hand, this is a continuation of MATH 574, Origami in Math and Education, but on the other hand we will explore areas of origami-math that the previous course did not touch. Some review will be given, on geometric constructions and flat-foldability, for students who have either not taken MATH 574 or who need a refresher. Then the topics we explore will include modular origami (with an exploration of polyhedral geometry), the general field of origami constructible numbers, simple and advanced techniques for counting the ways to fold a crease pattern, computer simulations of flat and rigid origami using Mathematica, and discovering how origami can fit into current state math curricular frameworks. Core class

**Online course:** This course will be conducted online in a *synchronous* format. (Actually, our whole MAMT program is now 100% online.)

- Classes will be conducted over Zoom at our prescribed class times (see below, MW at 6:00pm-7:50pm Eastern Time Zone), with lectures given in real-time by me and group work conducted in Zoom breakout rooms.
- We will use the online, shared whiteboard platform Limnu for lectures and group work activities.
**Note:**Using online whiteboards is much easier if you have a tablet (like an iPad) or a touch screen and a good stylus. - Students will be required to keep their Zoom cameras on at all times as a way to create classroom community and model an in-person class as much as possible. (Students may choose to use a "background filter" in Zoom to keep their surroundings private.)
- We will also use WNE's course management platform, Kodiak, to handle course communications, uploading of homework assignments, exams, and grading.
- Since the class is about origami, there will be folding activities done during class time. Participants will need paper for this purpose. Students who are within driving distance to Western New England University (in Springfield, MA) may choose to visit the campus and be given suitable origami paper from the instructor, Tom Hull. Otherwise, students will have to get their own paper:
- A pack of 3.5 in square Memo Cube paper from Staples (500 sheets).
- A pack of 6-in origami paper that is colored on one side and white on the other. This can be also be found at Staples (but is kind of expensive there), or can be found at most art supply stores, like Michaels (notice, much cheaper!). But I recommend buying this origami paper (even cheaper!) from OrigamiUSA, the national, not-for-profit origami society.
- You will also need some standard, rectangular paper for folding, like 8.5 x 11 in paper in the USA or A4 paper in other countries.

**Textbooks:** A required text for this course (from which there will be some assigned readings) is:

- Origametry: Mathematical Methods in Paper Folding by Thomas Hull, Cambridge University Press, 2020.

An **optional** book for this course, which many people find useful, is:

- Project Origami: Activities for Exploring Mathematics, 2nd edition by Thomas Hull, A K Peters, 2012.

**About this class in particular:** This is the second course we have in the MAMT program on the mathematics of origami. The first, MAMT 574 Origami in Mathematics and Education, was last offered in the Winter of 2022. Many students in that class stated that they would like to have a second origami-math class, and that's what this is!

However, this MAMT 590 class will be self-contained. There will definitely be students wanting to take this course who did not take the MAMT 574 class. That is OK! Origami-math is a big field, and most of the topics we cover will be introduced from scratch, without prior origami-math know-how.

But a few topics will require students who have not taken MAMT 574 (and perhaps even those who have!) to review things ahead of time. To help with this, I will

- Provide a list of sections of the book
**Origametry**to review (namely: Sections 1.1, 1.3, 1.5, 2.1, 2.3). - Provide PDFs of some sections from the
**Project Origami**book on our Kodiak page that will be helpful to review (namely Activities 20, 21, 22, and 23). - Prepare a video where I explain and summarize exactly what I expect people to know going into this class.

This summary video will be provided on our Kodiak class, so that as soon as you register for the class you will have access to it.

**Topics covered:** So what will we be covering in this class? While I do like to keep things flexible in order to respond to student interest, the topics will include:

**Origami algebra and geometric constructions:**It is strange to think of origami as doing algebra, but it does! This is very much related to using origami to make geometric constructions, like folding equilateral triangles and angle trisections. But we will go beyond that to uncover the full extent of what origami can do.**Modular origami and polyhedral geometry:**In the MAMT 574 class we explored a few modular origami models. We will do more in MAMT 590, including a deep dive into 3D polyhedral geometry.**Rigid origami computer simulations:**We will see how to simulate folding and unfolding on a computer. To do this we use matrix transformations and the computer algebra system Mathematica (as in the animation seen above-right), which will be made available, for free, to all students in this class. (This is through a generous donation arrangement with Wolfram, who create Mathematica as well as WolframAlpha.)**Counting foldings:**It is amazing how many questions we still do not know the answer to when it comes to origami-math. Counting the number of ways in which can fold a given origami crease pattern is one of the big open questions in this field. We'll explore this in depth, to bring you all right up to the current level of understanding.**Origami design secrets:**How do origami artists create ways to fold a square piece of paper into a complicated insect? Is there math involved? You bet there is!**Other stuff!**There are so many other topics we could cover. Too many, in fact! We'll look at the map folding problem, what it might mean to fold in higher dimensions, and the controversial topic of folding fractals!

**Grading:** Your grade for the class will be determined by the following:

- 30% Midterm exam
- 30% Final project or final
- 30% Homework
- 10% Participation

**About Folding:** There will be a fair amount of paper folding done in class and sometimes for homework. Please do NOT be worried about your own personal paper-folding skill. Most of what we will fold will be at the simple level (although they will use math!) and not complex bugs like the one shown above. *However*, we will study the math behind such complex-level origami. Those students who do wish to explore such advanced paper folding will have opportunities to do so.

**About the instructor: ** Thomas Hull has been practicing origami since he was 8 years old. He has co-authored two origami instruction books (Origami, Plain and Simple with Robert Neale (1994), and Russian Origami with Sergei Afonkin (1997), both published by St. Martin's Press) and has invented numerous origami models. His PHiZZ Unit has been especially popular, and his Five Intersecting Tetrahedra model was voted by the British Origami Society as one of the top 10 origami models of all time. From 1994-2007 he was on the board of directors for OrigamiUSA, a national nonprofit organization for the advancement of origami. He is also one of the world experts on the mathematics of origami. He edited the book Origami^{3} (the Proceedings of the Third International Meeting of Origami in Science, Mathematics, and Education) and authored the optional texts for this course, *Project Origami* and *Origametry*. He has been invited to lecture on the mathematics of origami across the USA as well as in Peurto Rico, Europe, and Japan. In other words, this is his specialty and this will be an awesome class!

**Links:** There are a lot of great resources for origami, and origami-math, online. Below are a few links to explore (feel free to search for more) if you want to get a feel for what origami is all about.

- Robert Lang's web page, which contains pictures of fantastic models as well as information on origami applications in math and science.
- My origami math pages, which are some of the oldest origami-math pages on the web! There are instructions for some of my models as well as previews of some things we'll see in class.
- The web page of OrigamiUSA, which has all sorts of resources for matters origami.

Last updated: 1/10/2023