A Beloch fold is an origami fold that simultaneously maps two given points 
and 
onto two given lines 
and 
, respectively. It is the sixth
of Huzita's origami axioms. For each point-line pair,
a crease that performs the required reflection is
a tangent line to the parabola
having the point as its focus and the line as its directrix.
A Beloch fold is therefore a common tangent line
of two such parabolas (Hull 2011).
Two parabolas can have up to three common tangents. Beloch (1936) used this operation to construct roots of general cubic equations, thereby giving origami constructions for angle trisection and cube duplication (Hull 2011).
For example, a Beloch fold gives a geometric construction of the plastic constant 
(Pegg 2025). On a coordinate sheet containing
the unit equilateral triangle with base endpoints
and 
and apex 
, simultaneously
fold 
onto the 
-axis and 
onto the 
-axis. For a crease 
, the two reflection
conditions give 
and 
, so 
. The unique real
root is therefore 
, and the crease is
 |
It intersects the triangle's base at 
, giving 
, 
, and 
. In addition, the distance
from 
to 
is 
. The reflected points
are 
and 
,
respectively.
Pegg, E. Jr. "BelochFold." Wolfram Function Repository.
May 12, 2025.