Syllabus for Math 160: The Mathematics of Origami
M,Tu,Th,F 9:00-11:50 (HKE 104)
Dr. Tamara Veenstra

Contact Information:

Required Materials: There is no text for this course, however you will need the following items:

• A large collection of square paper – you can buy official origami paper, you can cut regular paper into squares, or you can look for memo cubes with square paper inside. I recommend you buy at least some official origami paper – many designs are nicer with the two colored origami paper. You’ll want a variety of sizes as well. You must bring your own origami paper to class everyday.
• Pens of multiple colors – we’ll be marking the crease lines of our origami patterns. This will be much easier to follow if you have a variety of different colors to use.
• A ruler – this is quite frequently useful for making crease lines.
• Any origami book: your group will from time to time present models of your choice to the class. There are also websites with models available. An especially good book for beginners is Origami, Plain and Simple by Robert Neale and Thomas Hull.

Course Goals: In this course we will be making origami models and studying the underlying mathematics of these models. The design of this course is discovery based and open ended . That is, the in-class activities and assignments for this course will consist of open ended problems, so you will have a great deal of input into what topics we will cover in this course depending on what you discover and think about when solving this problems. This will most likely feel very different from other math courses that you’ve had in the past. Hopefully, it will be more fun and exciting this way; however, it may be slightly more frustrating until you get the hang of it. There will also be a strong group work component. 

Course Content: We will study connections between paper folding and topics in number theory, combinatorics, and geometry. In particular, this course will cover selected topics from:

Basic Geometry: How can we use geometry to analyze our models? How do the dimensions of our models relate to the size of paper that we use? How can we form a 30 degree angle? Can we trisect angles? How do we divide a piece of paper into perfect thirds? Topics in geometry include the Pythagorean Theorem, similar triangles, angles, and properties of parallel lines.

Polygons and Polyhedra: How can we use origami to construct polygons and polyhedra of a given number of sides.  This will introduce us to modular origami where we use multiple pieces of paper to form interesting shapes. Mathematical topics include Euler’s formula, coloring theorems, Hamilton cycles,  and Buckyball classification and edge coloring.

Flat Folding: How can we determine from the crease pattern alone if an origami pattern will fold flat? Partial answers include Maekawa's Theorem and Kawasaki's Theorem.

Course Structure:

There will be significant out of class assignments for this course – they will consist of smaller daily individual or group assignments and two larger group assignments.

Grading: Your grade will be based on the following categories: attendance/participation (20%), daily homework (20%),  daily reflection prompt (20%)  and group projects (40%).  I am also happy to do Johnston contracts and evaluations.

Attendance and Participation: We will cover a huge amount of material in each 3-hour class session. Thus attendance is required and will be a part of your grade. If you are sick or have some other dire emergency you may excuse up to two absences by calling or emailing me before 2 pm on the day you miss class. You are still responsible for any material that is covered during your absence and must get and complete the homework assignment for the next class. You must also actively participate in class to receive full credit in this category. Required participation includes discussing homework at the beginning of class, working on in-class activities, and generally behaving in a way that maintains and supports a good learning environment.

Daily Reflection Prompt: This is a new course and new material for me and we are using activities and materials from a colleague of mine who is working on writing a text for a course like this. As such your feedback on how class is going is extremely important. Every day I will ask you to reflect on class and email your responses to me. Full credit will be given to all thoughtfully completed assignments.

Daily Homework: There will be daily assignments. These will include both folding of origami models and analysis of the crease design. We will start class everyday with a discussion of problems. Most days, I will check to make sure that everyone has completed their assignment, but not collect and grade them. In this case, full credit will be given to all thoughtfully completed assignments. However, occasionally I will collect and grade these assignments. You are encourage to work in your groups, but must complete each task individually as well, unless otherwise stated. That is, you may discuss the homework in groups but everyone should individually fold each model and write up the analysis in his or her own words. Occasionally, your group will teach the class how to fold an object of your choice.

Group Projects: You will work in groups everyday and there will be two large group projects. The groups you sit in on the second day of class will be your group assignment for the entire month. These group projects will consist of a presentation to the class and a paper. The first group project will be due on Friday May 13 and will be on some aspect of the culture or history of origami. The second group project will be the final for the class and will be due on Thursday May 26. In this project you will make two models of your choice, one out of a single sheet of paper and one using modular origami (multiple sheets of paper) and perform a mathematical analysis of the crease patterns and model. More details on these below.

First Group Project Info
By Friday May 6, you should have decided upon a group for your final project. These groups should have 3-4 people and ideally would be the people at your table that you usually work with. However, you will need to meet outside of class for this, and I know lots of you work various hours, so you should make sure your schedules allow time for your group to meet before deciding on a group. Then we’ll switch tables around as necessary. You will have the same group for both projects unless some problem arises with schedules.

Mini-Project: Teach the class a fun origami fold. Starting next week each group will pick any origami object they like and present it to the class as our back from break warm-up fold. We will randomly decide which group goes on each of the following dates: Tuesday May 10, Thursday May 12, Monday May 16, and Thursday May 19, and Friday May 20.

Project 1: Pick any area involving the culture and history of origami, write a paper and make a presentation to the class. You should also pick an origami object that relates to your topic (maybe rather loosely) and analyze the crease pattern of this object. This could include any aspect of the history of origami or any or the modern day culture of origami. For example, you could talk about how origami evolved out of Japanese culture or pick a modern day person that designs or studies origami. Each group should pick a different topic.

TOPIC DUE BY MONDAY MAY 9.

Presentation: On Friday May 13, each group will present their project to the class. Your presentation should include a summary of your research area and include teaching the class how to fold your origami object. Your presentation should last about 20-30 minutes. You may include some of your analysis of your crease pattern.

Turn-In: You should turn in a 3-5 page paper describing your topic in the history or culture of origami, a completely folded version of your origami model, and an unfolded version of your origami model with all your crease lines marked. You should also include some analysis of your crease patterns, in addition to the 3-5 pages of history/culture. For example, find some geometry in your crease pattern where you can find dimensions or angles.

Final Group Project Info
Part 1: The 2D model: You should pick an origami model that folds flat and analyze it from a geometrical, flat foldable, and 2-colorable perspective.

You should turn in:

• one crease pattern with all crease lines marked and labeling all vertices, edges, angles, and mountain valley assignments.
• A second crease pattern that only uses the crease lines in the final folded product and this should be 2 colored.
• A third folded model that is 2 colored (using only the crease lines that are used in the final folded object.)

The write-up should contain:

• For the geometric analysis, you should find all the angles and dimensions of your model. Your write-up of this should include detailed information about how you know all the dimensions and angles based on how you folded the paper. That is, you should analyze each step of the folding and discuss what it tells you about the geometry of the crease lines.
• For the flat-foldable analysis: For this analysis you should discuss how each of our theorems about flat foldability are satisfied in your model. In your write-up you should examine each vertex, compute its degree and the number of mountains and valleys that it has. You should check all your angle calculations satisfy the other flat foldable theorems.
• For the 2- colorable analysis: Use the theorem about 2-colorability and your previous calculations to explain why you know your model must be 2 colorable (that is, how you know before you color it.)

Part 2: The 3D model:

Option 1: Find a modular unit that we have not used in class that can construct multiple polyhedral models, analyze the unit, make several smaller models and analyze the models you have made.

Option 2: Use any modular unit (even ones we have used in class), analyze the unit, make a single very large model, and analyze the model you have made. In particular, PhiZZ units can make a very nice torus (doughnut) or large buckyball.

Your analysis of the unit should discuss the dimensions and angles of the unit and how this effects how it can be put together with other units. That is, you should describe what type of faces it can form, how many faces can meet at a vertex, and thus, what type of polyhedra you can make with it.

For each polyhedra you make, you should find the number of faces, vertices, and edges. You should verify that Euler’s formula holds and determine if there are any other relationships that must hold between faces and vertices or edges and vertices or edges and faces and explain how you know this based on how the object is constructed.