This is a continuation of exploring center off-center folding and polyhedral development. It appears polyhedra can only be folded from circle alignment first to a centered circle.

There are two primary ways to see the circle. Traditionally where the circle has a center and concentric circles radiate out from the center; defined by using a compass. The second is where the circle has no center, being itself center, where concentric circles go infinitely into and out from themselves without boundary in both directions. One seems practical because we are familiar with it and the other is conceptual because we are not familiar with it.

Folding the circle demonstrates both ideas. The unit circle has a describable boundary. Through alignment of folding a center is reveled, through more folding more local centers are revealed; eventually the entire circle can be filled with centers. The circle is both unit and unity, one center and infinite centers.

Below left) Concentric circles show one point center, like one point perspective in drawing, it is a perceptual illusion that happens on a flat plane.

Below right) Folding the tetrahedron net pattern reveals six center points through a principled folding process (this is not counting the nine tangent points.) The circle is the quintessential fractal pattern of self-similarity infinitely revealed throughout. Remembering the circle is a compressed sphere; by folding we are decompressing spherical information that represents interference patterns of energy radiation.

Above left) a tetrahedron folded from a one center circle shows one of four sides with a center and the other three sides with partial rings of concentric circles. Reforming the centered circle to a tetrahedron does not consistently accommodate the 2-D image.

Above right) each of the six points of the tetrahedron net generated by the circle are each a vertex and center point. There is an equally distributed, wrap around organization that reflects the order of spherical packing in a cut away polyhedral form.

The folding for both tetrahedra is the same and comes from the pattern of alignment generated from folding the circle in half. Drawing in the concentric rings shows a different spherical organization for the same reconfiguration; the difference between the circle with one center and with multiple, or local centers.

Below left) tetrahedron arrangement of four tetrahedra made from circles with a single center bias all facing in the same direction. This arrangement is consistent to the tetrahedron in a single orientation. The concentric rings are on parallel planes without a common center and do not reflect any consistency to spherical arrangement.

Above left) a tetrahedron folded from a one center circle shows one of four sides with a center and the other three sides with partial rings of concentric circles. Reforming the centered circle to a tetrahedron does not consistently accommodate the 2-D image.

Above right) each of the six points of the tetrahedron net generated by the circle are each a vertex and center point. There is an equally distributed, wrap around organization that reflects the order of spherical packing in a cut away polyhedral form.

The folding for both tetrahedra is the same and comes from the pattern of alignment generated from folding the circle in half. Drawing in the concentric rings shows a different spherical organization for the same reconfiguration; the difference between the circle with one center and with multiple, or local centers.

Below left) tetrahedron arrangement of four tetrahedra made from circles with a single center bias all facing in the same direction. This arrangement is consistent to the tetrahedron in a single orientation. The concentric rings are on parallel planes without a common center and do not reflect any consistency to spherical arrangement.

Below left) Drawing concentric rings around each of the six center points on the net and reforming into the tetrahedron in the same tetrahedron arrangement shows ten spherical centers indicating concentric spherical shells as a slice through spherical packing. Each edge length is divided into 16 equal segments. The edge division depends on the intervals of wave frequency used between center points. These ten spheres reflect the four spheres and six touching points seen in the alignment of the first fold by touching any tow points.

Above right) one more tetrahedron and two octahedra have been added to show how filling in tetrahedra and octahedra further reveals the closest packed order of spheres of the same size. Of course each sphere is a movement into and out from itself creating complex interference patterns.

All observable spherical systems seem to develop from a local center within a universe that itself seem to be centered within others of centered universes all filled with countless moving local centers within what can only be called Whole. The center is everywhere of endless scale; unity containing everything down to the smallest single unit. Alignment of any size circle will demonstrate a similar process of inner-relationship of center points.

Only a circle that is concentrically in alignment into and out from itself can demonstrate something called “true” center. All other centers are off-center, just as we find with less than half folding. Intension generates movement that brings change as dynamic forces of time and space works towards alignment (accuracy is time with experience.) Movement revolves around pattern, forming generations of multiple centers, off-centered systems.

Folding the circle is a practical demonstration of movement from off-center to center, from planetary to cosmic, from a one to the many, from unit towards unity. Between the concept of the centered circle and the circle as center are truths that lie at the center of this exploration. By calling the smallest visible concentric circle a center point we create a conflict with our observations about spherical movement. The value of any unit lies with the context of unity.

In thinking about this I made a five-fold system using concentric circles with the off-centered folding and joined them in an icosidodecahedron arrangement. As you would expect from previous post, there is boundary distortion. (Center Off Center #2)

Above) Three symmetry views of the off-center folding with the concentric circles colored in where the black ring is the furthest out boundary of the circle that is common to these particular off-centered folds.

Above) making twenty elongated tetrahedra and filling in the tetrahedral openings changes the configuration closer to a more familiar polyhedron form. Each tetrahedra is individually designed to the creased lines necessary for the folding of the circles.

Above) all twenty open tetrahedra are filled in with elongated tetrahedra.

Below) Here pentagon pyramids were folded to fill in the open pentagon spaces leaving the tetrahedron spaces open. Again each pentagon was individually designed to the creases necessary to reform the pentagon pyramids. We can see a nucleus beginning to give form to the distorted formation.

Below) Views of evolving polyhedra where both open pentagon and triangle spaces have been filled; each form designed differently. Were they to again expand to the same level we would have a “solid” form identifiable as the icosidodecahedron.

The initial off-center distortion being absorbed by filling in what was missing forms towards a traditional polyhedron. In uniquely designing each surface to the information of the folded creases there is a proportional consistence through out the complexities within each individual unit and together as individual systems combined with other systems all made from multiples of the same circle., same folding process, differently designed. There are profound implications of this centering process and the transforming from off-center to center aligning of the boundary of the circle that is itself center. There is beauty in the proportional consistency and harmonizing of individual relationships that reflect rightness, appropriate interaction between all parts within the circle that can be viewed by the relationship of each individual part to the Whole of the circle.

These posts are to share some of my exploration into the nature of the circle and what we can learn from observation about the consistency of information that is generated through folding. Feel free to make your own models and add your thoughts and ideas about this.