I have looked at various ways of folding the “optimal” pentagon & hexagon (the largest regular hexagon within a square of paper). Additive constraints are a mathematically exact construction, a finite number of operations (no iterative method) and, of course, a folding sequence as simple as possible.

Hexagon

  1. The corner B comes in B’ on the medium vertical line. This allows us to built the intersection F of the fold AE with the diagonal BD. Reverse the model.
  2. Fold D onto F.
  3. Resuming the construction to get the optimal hexagon is also easy.

Pentagon

The goal is now to fold a regular pentagon, as

 

large as possible, within a square of paper. In origami geometry, there exists a lot of techniques to fold an approximate pentagon. Much less are concerned with exact pentagon, and only one about optimal pentagon: R. Morassi, The elusive pentagon, in the proceedings of the First International Meeting of Origami Science and Technology, H. Huzita, editor, Ferrara, pp. 27- 37, 1989. The one proposed herein is much simple.

  1. fold AD onto AB where D is the middle of the edge in order to build C.

   is the golden ratio.

2) bring C on the horizontal mid-crease.
3) bisect the complementary angle.
4) 
bisect again and mark the diagonal BE.
5) 
bisect again. I goes on J.
6) 
half way: B goes on J. Unfold.
7)
& 8 complete the stellated pentagon.

David Dureisseix