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).
푧 푦2 푥2 The hyperbolic paraboloid is a saddle-shaped surface represented by the equation = − . We can 푐 푏2 푎2 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 Completed Neatness Folded animal /2 /3 Difficulty bonus Creases Angles Crease pattern /2 /3 /0 /10 Hyperbolic paraboloid
Difficulty bonus Completed Neatness (more division)
/5 /5 /0 /10 skeletal octahedron
Completed Neatness
/5 /5 /10 12-module sonobe stellated octahedron Difficulty bonus (or 30-module icosahedron) If variation, (variation; indicate which: icosahedron instead of or in addition to Completed Neatness octahedron) or /5 /5 /0 /10 12-module sonobe cube (or 24-module cube) If variation, Difficulty bonus indicate which: (variation; 24- module instead of or in addition to Completed Neatness 12-module)
or /5 /5 /0 /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