 
Summary: Folding the Regular Pentagon Using Bisections and
Perpendiculars
Roger C. Alperin
There are numerous origami constructions for the pentagon, see for example
[3]. Dureisseix [2] provided constructions for optimal polygons inscribed in a square
using origami folds. The optimal pentagon has four vertices on the four edges of
the square and one vertex on the diagonal so that the pentagon has that diagonal
as an axis of symmetry.
However, the construction is known to be a `Pythagorean' construction [1], that
is, we can achieve it using only the origami bisection of angles or the construction of
a perpendicular to a line through a given point. We accomplish this for a maximal
inscribed pentagon in the square by modifying Dureisseix' construction (notably in
his second step).
Suppose the side length of our square OPQR is one. Fold the hoizontal mid
line MN of the square parallel to RO. From the vertex P construct the diagonal
PM and the angle bisector PA of MPO with A on the edge RO. Let = APO,
x = tan() = OA, then 2 = MPO and tan(2) = 2 = MN
P N ; hence x satisfies
2x
1x2 = 2 so x =
