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Visual Mathematics - Mathematical Origami
We will deconstruct origami in order to study the crease pattern. For this part of the assignment, you will fold the
same origami animal twice. One you will leave folded and the other you will unfold, neatly trace the crease pattern,
determine and label all angles in the crease pattern, and label parts of the crease pattern corresponding to
appendages of the folded animal (this last part may be easier to label while animal is still folded).
The hyperbolic paraboloid is a saddle-shaped surface represented by the equation
𝑧
𝑐
𝑦2
𝑥2
= 𝑏2 − 𝑎2. We can
approximate a hyperbolic paraboloid in origami. Construct a hyperbolic paraboloid with at least 8 divisions in the
paper.
Modular origami is origami made from many simple modules that fit together by inserting flaps into pockets to
make a larger, more interesting model. For this project you will build a skeletal octahedron, a sonobe stellated
octahedron, and a sonobe cube. It is suggested that you divide each square of paper you are given into 4 smaller
squares for modular projects. Bonus points will be awarded for sonobe models using a variation of the basic
module (8 different variations are provided to you as options; please indicate which variation you used).
You will be given 12 sheets of origami paper. Any other paper needed by you for this assignment will be up to you
to find/acquire. Get creative – recycle!
Deconstruction/crease pattern study
Folded animal
Crease pattern
Hyperbolic paraboloid
Completed
/2
Creases
/2
Neatness
/3
Angles
/3
Completed
Neatness
Difficulty bonus
(more division)
/5
/5
/0
Completed
Neatness
/5
/5
Difficulty bonus
/0
/10
/10
skeletal octahedron
12-module sonobe stellated octahedron
(or 30-module icosahedron)
/10
Completed
Neatness
Difficulty bonus
(variation;
icosahedron
instead of or in
addition to
octahedron)
/5
/5
/0
Completed
Neatness
Difficulty bonus
(variation; 24module instead of
or in addition to
12-module)
/5
/5
/0
If variation,
indicate which:
or
12-module sonobe cube
(or 24-module cube)
or
If variation,
indicate which:
/10
/10
/50
Origami folding project
Due:
Please make sure each model you submit is clearly labeled with your name, and contain all models in a
bag or box if possible.
Origami Writing Assignment
Due:
Following the same basic guidelines as the previous writing assignment, answer some or all of the
following questions:








What is the mathematical significance of origami?
Are there any important folding algorithms?
What are the constraints? Is it possible to fold anything?
Who are the important people in the history of origami (especially as it relates to mathematics,
and why are they important)?
Who are the current VIPs of origami, and what sort of research are they doing that is so
important?
What sort of applications to science and engineering does origami have?
Are there any contemporary artists working with origami?
What is an origami tessellation and what is its significance?
Textbooks/journal articles available for this topic in S201 for use in Math Lab:
1. Lang, Robert. Origami Design Secrets: Mathematical Methods for an Ancient Art, 2nd ed. Boca Raton: CNC Press,
2012.
2. Peterson, Ivars. Fragments of Infinity: A Kaleidoscope of Math and Art. New York: John Wiley & Sons, 2001.
In the ASMS Library:
1. Gjerde, Eric. Origami Tessellations. Wellesley: AK Peters, 2009.
2. Gould, Vanessa. Between the Folds. Arlington: Green Fuse Films, 2009.
Other particularly useful sources:
1. Demaine, Erik. Geometric Folding Algorithms: Linkages, Origami, Polyhedra. MIT OCW course with video
lectures, notes, and slides. Available at: http://courses.csail.mit.edu/6.849/fall10/
2. Lang, Robert. “Angle Quintisection.” Robert J. Lang Origami. Available at:
http://www.langorigami.com/science/math/quintisection/quintisection.php
3. Lang, Robert. “Huzita-Justin Axioms.” Robert J. Lang Origami. Available at:
http://www.langorigami.com/science/math/hja/hja.php
4. Lang, Robert. “Origami Diagramming Conventions.” Robert J. Lang Origami. Available at:
http://www.langorigami.com/diagramming/diagramming.php
And, of course, the many articles in the Google Drive “Origami” folder:
https://drive.google.com/a/dragons.asms.net/folderview?id=0B26XdI9aXnUQUDBKMlRYVzFxbG8&usp=sharing