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.
Cut-the-knot logo.
With a proof of the origami-folklore that this folded-flat overhand
knot forms a regular pentagon.
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.
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.
Tobi Toys
sell the
Vector Flexor, a flexible cuboctahedron skeleton, and
Fold-a-form,
an origami business card that folds to form a tetrahedron that can be
used as the building block for more complex polyhedra.
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?
