Origami is the art of paper folding. The word is Japanese, literally meaning **to
fold** (oru) **paper** (kami).

Origami

The Japanese art of paper folding is obviously geometrical in nature. Some
origami masters have looked at constructing geometric figures such as regular
polyhedra from paper. In the other direction, some people have begun using
computers to help fold more traditional origami designs. This idea works best
for tree-like structures, which can be formed by laying out the tree onto a
paper square so that the vertices are well separated from each other, allowing
room to fold up the remaining paper away from the tree. Bern and Hayes (SODA
1996) asked, given a pattern of creases on a square piece of paper, whether one
can find a way of folding the paper along those creases to form a flat origami
shape; they showed this to be NP-complete. Related theoretical questions include
how many different ways a given pattern of creases can be folded, whether
folding a flat polygon from a square always decreases the perimeter, and whether
it is always possible to fold a square piece of paper so that it forms (a small
copy of) a given flat polygon.

Krystyna Burczyk's Origami Gallery - regular polyhedra.

The business card Menger sponge project. Jeannine Mosely wants to build a
fractal cube out of 66048 business cards. The MIT Origami Club has already made
a smaller version of the same shape.

Cranes, planes, and cuckoo clocks. Announcement for a talk on mathematical
origami by Robert Lang.

Crumpling paper: states of an inextensible sheet.

Cut-the-knot logo. With a proof of the origami-folklore that this folded-flat
overhand knot forms a regular pentagon.

Folding geometry. Wheaton college course project on modular origami.

Geometric paper folding. David Huffman.

Rona Gurkewitz' Modular Origami Polyhedra Systems Page. With many nice images
from two modular origami books by Gurkewitz, Simon, and Arnstein.

How to fold a piece of paper in half twelve times. Britney Gallivan took on
this previously-thought-impossible task as a high school science project, worked
out an accurate mathematical model of the requirements, and used that model to
complete the task.

Knotology. How to form regular polyhedra from folded strips of paper?

The Margulis Napkin Problem. Jim Propp asked for a proof that the perimeter
of a flat origami figure must be at most that of the original starting square.
Gregory Sorkin provides a simple example showing that on the contrary, the
perimeter can be arbitrarily large.

Mathematical origami, Helena Verrill. Includes constructions of a shape with
greater perimeter than the original square, tessellations, hyperbolic
paraboloids, and more.

A mathematical theory of origami. R. Alperin defines fields of numbers
constructible by origami folds.

Mostly modular origami. Valerie Vann makes polyhedra out of folded paper.

Number patterns, curves, and topology, J. Britton. Includes sections on the
golden ratio, conics, Moiré patterns, Reuleaux triangles, spirograph curves,
fractals, and flexagons.

Origami: a study in symmetry. M. Johnson and B. Beug, Capital H.S.

Einstein's origami snowflake game. Rick Nordal challenges folders to make a
sequence of geometric shapes with a single sheet of origami paper as quickly as
possible.

Origami & math, Eric Andersen.

Origami mathematics, Tom Hull, Merrimack.

Origami Menger Sponge built from Sonobe modules by K. & W. Burczyk.

Origami polyhedra. Jim Plank makes geometric constructions by folding paper
squares.

Origami tessellations, Alex Bateman.

Ozzigami tessellations, papercraft, unfolded peel-n-stick glitter Platonic
solids, and more.

Paper folding a 30-60-90 triangle. From the geometry.puzzles archives.

Paperfolding and the dragon curve. David Wright discusses the connections
between the dragon fractal, symbolic dynamics, folded pieces of paper, and
trigonometric sums.

Paper models of polyhedra.

Polyhedra plaited with paper strips, H. B. Meyer. See also Jim Blowers'
collection of plaited polyhedra.

Rabbit style object on geometrical solid. Complete and detailed instructions
for this origami construction, in 3 easy steps and one difficult step.

The RUG FTP origami archive contains several papers on mathematical origami.

Spidron, a triangulated double spiral shape tiles the plane and various other
surfaces. With photos of related paperfolding experiments.

Spring into action. Dynamic origami. Ben Trumbore, based on a model by Jeff
Beynon from Tomoko Fuse's book Spirals.

Studio modular origami, geometric paper art.

The tea bag problem. How big a volume can you enclose by two square sheets of
paper joined at the edges? See also Andrew Kepert's teabag problem page.

Unfolding polyhedra. A common way of making models of polyhedra is to unfold
the faces into a planar pattern, cut the pattern out of paper, and fold it back
up. Is this always possible?

Vegreville, Alberta, home of the world's largest easter egg. Designed by Ron
Resch, based on a technique he patented for folding paper or other flat
construction materials into flexible surfaces. See also William Chow's page on
the Vegrevill easter egg.

Joseph Wu's origami page contains many pointers to origami in general.

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