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

This is a rough outline of topics covered and HW assigned for the course.

Note: Each day is a 3 hour class session and the HW assignments are correspondingly long.

References:
Most of the material came from activities from Tom Hull’s book due out in March. Anything listed as ___ activity came from that material. Some of the more basic geometry and other modular units came from Unfolding Mathematics with Unit Origami by Betsy Franco (key curriculum press) and Paper Square Geometry: the Mathematics of Origami by Youngs and Lomeli, (AIMS Educational Foundation). I believe the last is only available through the AIMS website. The Fujimoto applied to angles and folding regular n-gons is from Mathematical Reflections: In a room with many mirrors by Hilton, Holton, Pedersen, (Springer) but most of what is in their book is too complicated for this audience. Most of the boxes came from Origami boxes by Tomoko Fuse though there are lots of origami boxes books.

Rough Schedule

Week 1
Day 1: basic geometry and equilateral triangle activity
Folded some origami models and analyze crease paterns, find dimensions of crease lines and angles reviewing necessary geometry (Pythagorean theorem, sin and cos, etc). Folded a box and determined how end dimensions related to starting dimension of paper. Started activity on how to fold an equilateral triangle.
HW: fold other boxes and try to fold a bigger equilateral triangle.

Day 2: more on equilateral triangle activity (maximal part) and fujimoto activity
Worked on folding a larger (largest) equilateral triangle and verifying the triangles they folded were really equilateral (many were almost equilateral) and did Fujimoto approx technique for folding paper into nths. Took a lot of examples for them to fully understanding the Fujimoto technique.
HW: Folding and analyzing dimensions with activities from books by Franco and AIMS.

Day 3: rediscussed how to fold 15 degrees and maximal equilateral triangle (much confusion in HW comments) and did the number theory of fujimoto.
HW: was to turn in write ups explaining how to find all dimensions of edges in the crease pattern for the maximal equilateral triangle from the folding.

Day 4: folding regular n-gons with fugimoto applied to angles from Mathematical Reflections ch. 4.
HW: write up how the error goes down with several examples of Fujimoto and try to find a way to fold a piece of paper into thirds exactly (without approximating). Fold regular pentagons and hexagons in square sheet of paper.

Week 2
Day 5: flat folding activity, part 1
HW: group project topic and analyze one conjecture from class about flat folding.

Day 6: trisecting angle activity and dividing equal thirds activity
HW: write up the argument for why the angle is trisected and generalize the method for dividing a piece of paper into thirds for dividing it into 5ths and 7ths. Do error calculation on Fujimoto one more time.

Day 7: more on flat folding, presentation 1
Discuss conjectures made on day 5 and talked about why they were true (or found counterexamples to false conjectures). Gave example of a single and multi vertex patterns that do not fold flat and had students analyze why (to get more conjectures about when flat folding does work)
HW: work on group projects

Day 8: Group presentations, more flat fold, modular unit folding made tetrahedron
HW: Find all possible ways to assign Mountain and Valley creases to a given crease pattern (using theorems on flat foldability). Pick any crease pattern that folds flat and check that all flat fold theorems are verified. Made cube from modular unit.

Week 3
Day 9: more modular unit folding, made octahedron and hexahedron, euler’s formula, other V,E, F formulas, apply formulas to determine all regular polyhedra with trianglular faces, and PHiZZ unit activitiy
HW: apply formuls to find all regular polyhedra with square faces, use PHiZZ units to construct dodecahedron, make icosahedron

Day 10: coloring question and crane activity
Colored maps to come up with 4 color theorem. Colored flat foldable designs to discover only need two colors.
HW: 3 color the PHiZZ unit dodecahedron, analyze a modular unit – dimensions and angles and how it fits together with other units, and start working on final group project.

Day 11: review, planar graphs activity, buckyballs activity part 1
HW: explore PHiZZ units more and construct planar graph of icosahedron and 3 color it.

Day 12: buckyballs activity part 2
HW: make a soccer ball (90 PHiZZ units) and work on group final projects.

Week 4:
Day 13 -- butterfly bomb activity, fun folds
HW: work on final project

Day 14—fun math folds, parabola, hyperbolic paraboloid, miura map, wave, other fun folds
HW: work on final project

Exam day: present final projects