Folding circles appears to have little history. There is no precedent, no references for folding the circle as a modeling tool or as a process for generating information. The only reference I have come across refers to folding circles and squares in Japan in the late 1930's. Somewhere in the history of origami lies the circle, unrecognized and discarded. The square and all other polygons are only a part of the circle, all limited to a specific number of sides. The circle has no sides, it dose not have limits. Mathematics is based on the geometry of drawing pictures of the circle and using parts of the circle for constructing information from these static images. The symbol of the circle is used as metaphor for nothing and for everything, and for all manner of parts in-between. We have accepted the drawn image in place of the actuality of the circle. We do not yet know the nature of what the circle is.
The only other person who has folded the circle with informational intent was Buckminster Fuller. He would fold the vector equilibrium, using folded paper plates and bobby pins. He also explored folding the great circles of the icosadodecahedron and octahedron in spherical pattern. That is as far as he went in folding circles. His work can be found in "Synergetics; Explorations in the Geometry of Thinking", R. Buckminster Fuller in collaboration with E.J. Applewhite, 1975. Fuller is the inspiration for my own exploration into geometry and provided the seed for folding and joining circles. If the vector equilibrium can be folded proportionally into the circle, then everything else must be held within the proportionally folding and joining of the Wholemovement of the circle.