Is there an exact solution for tan(36) using origami from a unit square?

Question

This question is more for chagrins and curiosity than anything else: Is there a way to use origami to construct the tangent of 36 degrees (~0.7265425)? I've come up with the image below, which is accurate to about 5 decimal places (the exact decimal eludes me at the moment), but I'd like something closer if it exists.
The vertical red line on the right is at x = ~0.7265403.
As I said, the question is mostly for chagrins (as the method above is close enough in practice), but I'd appreciate any more creative solutions.

Answer 1

Form a pentagon base. All ten resulting central angles are 36 degrees. Taking the short radial fold to have length $1$, the half-edge has length $\tan(36 ^\circ)$. (That is, you have ten right triangles with adjacent $1$ and opposite $\tan(36 ^\circ)$.)

So start with this step of the pentagon base.

Now unfold to the beginning. Observe that there is a central vertex with four 36 degree angles to its right. The two right triangles sharing a leg along the first bisection fold are what we want to project to an edge. Extend either hypotenuse all the way across the paper. (This extension almost auto-folds for me. You have the two long legs of the adjacent right triangles to guide this fold.) Taking the extended hypotenuse as a unaxial base, construct a hinge passing through a (either) corner. (The extended hypotenuse, the axis for this hinge, is indicated with pink arrows in the following diagram.)

The hinge meets the edge opposite the corner at the desired point (indicated by the light blue arrow).

Note that the foreground flap is a $36^\circ \text{-} 54^\circ \text{-} 90^\circ$ triangle, as desired. Why? The right triangle we started with has legs parallel to the edges of the paper. Any perpendicular to its hypotenuse meets an edge at either $36^\circ$ or $54^\circ$. We just arranged for the hinge to hit one of the two sides forming a $54^\circ$ angle.