Folding gives rise to many mathematical and recreational problems. Origami, the Japanese art of paper folding, is one well-known example.
Folding can change the spatial shape of an opening in paper. Tokieda demonstrates passing a circular disk through a square opening by folding the surrounding sheet out of the plane, even though the disk does not fit through the flat opening (Numberphile 2018).
One rectangle folding problem asks which fold through the center minimizes the area covered by both halves (Gaichenkov 2023).
It is possible to make a surprising variety of shapes by folding a piece of paper multiple times, making one complete straight cut, then unfolding. For example, a five-pointed star can be produced after four folds (Demaine and Demaine 2004, p. 23), as can a polygonal swan, butterfly, and angelfish (Demaine and Demaine 2004, p. 29). Amazingly, every polygonal shape can be produced this way, as can any disconnected combination of polygonal shapes (Demaine and Demaine 2004, p. 25). Furthermore, algorithms for determining the patterns of folds for a given shape have been devised by Bern et al. (2001) and Demaine et al. (1998, 1999).
Wells (1986, p. 37; Wells 1991) and Gurkewitz and Arnstein (2003, pp. 49-59) illustrate the construction of the equilateral triangle, regular pentagon (illustrated above), hexagon, heptagon, octagon, and decagon using paper folding.
The least number of folds needed to map the vertices of a suitable regular polygon to an arbitrary
prescribed set of 
points is not known in general,
but upper bounds are available. In particular, for
, every set of 
points is the image of a suitable
regular 
-gon under at most 
folds, where
 |
(1)
|
The right-hand side takes the values 0, 2, 3, 5, 6, 8, 9, 11, 12, 14, 15, 17, 18, 20, 21, ... for 
, 2, ... (OEIS A007494).
The points accessible from 
by a single fold which leaves 
,
..., 
fixed are exactly those points
in the intersection of the closed disks bounded by
the circles through 
with centers at 
, for 
,
..., 
. Given any three points in
the plane 
, 
,
and 
, there is an equilateral
triangle with polygon vertices 
, 
, and 
for which 
, 
,
and 
are the images of 
, 
, and 
under a single fold.
Given any four points in the plane 
, 
, 
,
and 
, there is some square with
polygon vertices 
, 
, 
, and 
for which 
, 
, 
, and 
are the images of 
, 
, 
, and 
under a sequence of at most three
folds. In addition, any four collinear points
are the images of the polygon vertices of a suitable
square under at most two folds. Every five (six) points
are the images of the polygon vertices of suitable
regular pentagon (hexagon)
under at most five (six) folds.
Assuming it were possible to fold paper without restriction, the height of a piece of folded paper would double in thickness each time it was folded. Since one sheet
of typical 20-pound paper has a thickness of about 0.1 millimeter, folding 50 times
(if this were physically possible, which of course it is not) would produce a wad
of height 
meters, and folding one more time would make the stack higher than the distance between
the Earth and Sun.
The function
 |
(2)
|
gives the loss function for folding paper in half, where 
is the minimum possible length of the material, 
is the thickness, and 
is the possible number of folds in a given direction. This
formula indicates how much "normalized" paper has been lost for 
folds, and thus sets a limit for the number of times things
of finite thickness can be folded in one direction (Pomona Valley Historical Society).
For 
, 1, 2, ... the sequence 
gives 0, 1, 4, 14, 50, 186, 714,
... (OEIS A076024). The formula was derived
by high school student Britney Gallivan in December of 2001. Britney then proceeded
to set a new world record by folding first gold foil and then paper in half a whopping
12 times in January of 2002, thus debunking the assertions of Math@Home and PBS Kids
that paper cannot be folded in half more than eight times. Britney and her feat were
mentioned in the Season 1 episode "Identity
Crisis" (2005) of the television crime drama NUMB3RS.
-Gons by Folding the Plane." Amer. Math. Monthly 102,
620-627, 1995.Sloane, N. J. A. Sequences