Origami: Mathematics, Art, Science and Engineering

An interdisciplinary seminar that uses origami, the art of paperfolding,
to bring together mathematics, computer science, engineering design and art.

Arizona State University, Spring 2009
HON 394-1010 (SLN 26343), TTh 3:00--4:15PM

Instructor: Goran Konjevod (email: my first name at asu.edu)
Office hours (Fall 2008): TTh 1:30-3:00 and by appointment
(There will be no office hours Tuesday 1/20.)
Office location: Brickyard 450

This page is incomplete. More information, including some
of the submitted homeworks and projects, will be added soon.

The course Blackboard site (restricted to enrolled students).

Origami-related Software links:
Jun Mitani (ORIPA; paper on ORIPA)
Robert Lang (TreeMaker, referencefinder)
Tomohiro Tachi (Origamizer, rigid origami simulator)
Alex Bateman (Tess)
Robert Lang's Curved pot Mathematica notebook.

Class topics with links to additional readings and resources available online:

Class 1 (January 20): Syllabus.
Basic history by David Lister. Other writings by David Lister.
More essays on the history of origami.
Fold: Leaf by Maarten van Gelder.
Student work: a new version of four leaves by Jane Kissler.

Class 2 (January 22): Homework 1.
Fold: cube (by Shuzo Fujimoto). Handout: a few other cube designs.
Short discussion about diagrams and crease patterns.
ORIPA by Jun Mitani. (We'll use this later in the course.)

Class 3 (January 27)
Fold: binary folding into 5ths (Fujimoto approximation);
Handout: Robert Lang's Geometric constructions in origami.

Class 4 (January 29)
Fold: hyperbolic paraboloid.
Links: Erik Demaine's pages on hypars and on curved origami sculpture;
Tom Hull's page with a hexagonal version of the hypar;
David Dureisseix on how to fold an optimal hexagon and pentagon from a square.
Handouts: hyperbolic paraboloid diagrams; curved variations.
Homework 2.
Student work: a complete proof written up by Kevin McMillin (local pdf) of correctness for the hexagon construction.

Class 5 (February 3)
Question: can we fold a square so that the perimeter of the folded model is larger than that of the original sheet?
How about the area?
Fold: traditional bases (kite, fish, bird)
Bases, flaps, corner, edge, center. Blintz fold.
Handouts: on traditional origami bases.
Links: Hans Birkeland's diagrams for Meguro's 145-point sea urchin.

Class 6 (February 5)
Fold: knotting a strip of paper gives a regular pentagon.
A heptagon can be folded, too. What about a square?
(No folds anywhere other than at the edges of the polygon.)
What works? What doesn't? Why?
Discussion on crease patterns--what can be flattened?
Proofs and verifications.
Handouts: square twist crease pattern.
Reminder: Phoenix Art Museum trip tomorrow (Friday).
Homework 3.

Class 7 (February 10)
Fold: pine cone (fractal; spread-sink practice; Additional diagram; YouTube video for a very similar model).
Handouts: pine cone diagrams; partial rose diagrams.
Discussion: square twist. Example of ORIPA use.
How many different square twists? ORIPA as a checker (not reliable yet...)
Fold the preliminary steps for the rose. We'll finish it on Thursday.

Class 8 (February 12)
Fold: the Kawasaki rose. (Step folds; tutorial with links to YouTube videos)
(If you're feeling powerful: diagrams for the new rose.)
Handouts: Theorems on flat vertex folds; Diagrams for square and hexagon twist tessellations.
Question: Pevsner's developable surface: can it be flattened? What shape will it be?
Is that a saddle point in there?
Announcement: Bridges Conference Math Art Exhibition.
Homework 4.

Class 9 (February 17)
Fold: simple curved container (folded from the crease pattern)
Folding from crease patterns: which are the first creases to make? What to leave for the end?
Flat-foldable vertex conditions: examples.
Summary of material so far. Future: art, design.
Examples of origami-related programs: ORIPA, Rigid origami simulator, TreeMaker.

Class 10 (February 19)
Fold: pureland pleated pyramid (folded from the crease pattern)
Pureland folding, especially pleat intersections; examples, discussion.
Robert Lang's talk at TED 2008.
Homework 5.

Class 11 (February 24)
Fold: Mount Fuji and the sea.
Project ideas discussion.
Class 12 (February 26)
Fold: Human figure.
Discussion of and questions about midterm projects.
Homework 6.
Class 13 (March 3)
Fold: Hands (fingers).
Discussion: art. Some examples: can we tell between work by famous artists and unknowns? Can we tell the age of the artist (child/adult) (examples used: works by Paul Klee)?
Some classes missing (to be added)

Links for HW 9:
Symmetry groups: which one is it? (crystallographic notation)
Symmetry groups: correspondence between Conway's and crystallographic notation
Some photos of Pevsner's works

Some random data visualization links:
Use R. Examples here.
Junk Charts (check also its blogroll)!
Edward Tufte

Course format

Each class will begin with students being guided through
the folding of a simple but important origami model. The model and/or
the instruction in the folding exercise will motivate the main topic
of the class by illustrating ideas and principles from mathematics
(especially geometry), computer science, art and engineering.
The final part of each class will be an open discussion.

Examples of topics to be covered

Course materials

No single textbook covers all the material planned for this course, so
a set of handouts will be distributed over the course of the semester.
The handouts will come from textbooks, monographs and popular sources
dealing with material relevant to the course.


The students will participate in class discussions, complete homework assignments
and two projects. Homework assignments will include exercises in folding, design,
mathematics, programming and art. Midterm and final projects will be done
either individually or in groups of two or three students (depending on the
chosen topic and scope).