Rectangle folding is the operation of folding a rectangle along a straight line. One optimization problem asks which
fold through the center minimizes the area
covered by both halves. Let the side lengths be 
. The minimizing shape changes at the aspect
ratio 
(Gaichenkov 2023, pp. 1-6). For 
, the minimum overlap is
a triangle of area 
, obtained by a fold at 
to a side. For 
, the minimum overlap
is a pentagon. At 
, both shapes attain the same minimum area,
as illustrated above. For a square, the optimal fold makes
an angle 
or 
with a side and the minimum overlap area
is 
(Gaichenkov 2023, pp. 4-6).
For 
, the pentagonal minimum occurs at an
angle 
or 
to a side of length 
, where 
is the smallest positive root of
the quartic equation (Gaichenkov 2023, pp. 2-6)
 |
(1)
|
The threshold is obtained from the unique real root 
(OEIS A366185)
of the quintic equation
 |
(2)
|
using
 |
(3)
|
where 
.
If the crease crosses the interior of the rectangle but is otherwise unrestricted in position, the overlap area
has infimum 0, which is not attained. Folding an isosceles
right triangle with legs 
, where 
, at a corner of the rectangle
produces overlap area 
. This approaches 0 as 
, while every crease crossing the interior
of the rectangle produces positive overlap area.
The auxiliary root 
admits a neusis construction
(Gaichenkov 2026, pp. 2-3). In Cartesian
coordinates, draw the line 
and the circle with center 
and radius 
. Slide a ruler with
two marks one unit apart through the origin 
so that the nearer mark 
lies on the circle and the farther
mark 
lies on the line, with 
between the origin and 
. Choosing 
below the x-axis gives 
. The formulas above then construct the threshold
aspect ratio 
from 
.
