In believing the circle is image, we limit the imagination.
Draw a circle anywhere on a piece of paper.
Cut the circle from the paper.
How many circles are there?
Compare properties of both circles.
They are the same circle.
By removing the circle from the paper the other circle is left. One boundary separates the physical circle from the non-physical other circle. We might say one is positive, the other negative. One is where the other is not. The inverse of one is the other. The compliment is so intimate that without separation we miss the dual nature.
Fold both circles in half.
Notice the difference and similarities in each circle.
One circle contains the creases. With the other circle the paper references the unseen crease within the circle boundary. Having folded in half what is not there informs alignment with what is there.
Fold both half-folded circles twice more in ratio of 1:2 http://wholemovement.com/how-to-fold-circles.
Below: Having folded the half-folded circles into thirds, open them to find three diameters in each. Three chords are visible in the one removed; they are not visible in the other circle.
Three diameters, six radii define six areas and seven points. In the other circle the unseen diameters extended outward showing six line segments defining six areas and twelve points. The proportional folding is the same for both circles. Removing the image from the rectangle paper the other self-referencing, self-organizing circle is now a hole defined by context, yet functions in the same way as the folded paper. The creases not seen in the circle extend through the rectangle. The invisible is held with-in the visible, and for that reason often goes unrecognized. The unseen informs through space that which can be seen.
Fold circles to a higher frequency of the same pattern of division, see blog for instructions; http://wholemovement.com/blog/item/97-unity-origami
When folded to a higher frequency each circle reveals six diameters, the organization being in the first three. The second set of three diameters have a different proportioned division functioning as bisecting diameters to the first three.
Below left) Division of the first three diameters show four equal sections revealing a hexagon star and hexagon. With one the hexagon and star are on the inside boundary of the circle, with the other circle the star points and hexagon are creased into the rectangle on the outside of the circle boundary; one goes in the outer goes out.
Above right: When replacing the removed circle back into the other there are two ways to align the six diameters. One aligns the star points so both are in the same orientation; the other (pictured above) is where the star point diameters are in alignment with the in-between diameters of the other, a thirty-degree shift in position. The latter is the complete alignment of the grid showing two levels of grid division using all six diameters.
The creases in the paper have been drawn over for better visibility.
Below: Continuing a higher frequency folding in the rectangle (creasing lines parallel to lines already there) the equilateral triangle grid becomes obvious reflecting the same size grid we see in the folded circle. Coloring the same size and orientation of triangles makes the pattern clear. The removed circle shows more information; each triangle of the hexagon division is bisected in three directions rotating the grid thirty degrees to a smaller scale. This can be described as a fractal like “interference pattern” (upper right.)
Below: The removed circle has been reformed to a tetrahedron. This suggests the other circle will also reform into a tetrahedron. The form will look very different because of the difference in perimeter but the circle-pattern of arrangement will be the same. There are more equilateral triangles in the rectangle, which increases possibilities in multiples tetrahedra on different scales.
Below: The other circle can change symmetries from six diameters (hexagon,) to a five fold symmetry (pentagon,) to four (square,) and the three (triangle.) This is possible with the whole circle, the circle hole, and is demonstrable with any random shaped paper folded to the same circle-pattern. http://wholemovement.com/blog/item/729-order-without-boundary-ii-geocoding
Below: Fold creases into the rectangle reflecting what is already in the removed circle. Color the alternate areas of the second level division revealing another level of design using the creases.
Above: By replacing the removed circle into the rectangle the difference in scale becomes apparent. With consistency to the same alternate coloring of the folded grid the fractal scaling separates the inside and outside of the circle boundary.
Above: Both circles independently have been reformed to a square-based pyramid (half octahedron.) The pyramid from the rectangle is not complete until the removed circle is replaced, as if they had not been separated. The frequency difference is again apparent with the dual reformations aligned to the same symmetry showing consistency in pattern and difference in design.
Above: Left shows both forms folded into a "solid tetrahedron." On the right shows the other circle reformed to a tetrahedron pattern of four points in space. The circle-pattern is indicated by the arrangement of points (any four points in space is a tetrahedron pattern.) There are many possibilities reforming the tetrahedron using this grid-creased rectangle.
Below: Here are a few of many possible reconfigurations using the grid in the rectangle folded to the circle hole. There is no proportional relationship between inside circle and outside rectangle because of the initial arbitrary placement of the image. The diversity of forms suggests rich design possibilities in a proportionally aligned relationship of circle to perimeter. You might call this an organized, controlled crumpling of paper.
Circle-pattern is inherent to all shapes; all shapes are inherent in the circle-pattern. Were this not so we could not construct what we do with compass and straight edge, or remove the image from the plane and fold the circle into what is possible. By separating the circle from the rectangle part, itself a truncated circle, we see a great variety in different forms through consistency in pattern. The part and whole are so intimately bound that without one there is not the other, even in the appearance of separation.
Symmetry apparently has little to do with specific shapes and is more about proportional divisions of unity. Symmetry bound to pattern is therefore inherent to all shapes and forms. Transformation from one symmetry to another happens because they hold circle-pattern whole in common.
Below) This diagram relates sphere-to-circle compression with removing the circle image from the 2-D plane. Both forms carry spherical unity that can only be realized by moving from the illusion of 2-D to experience the dynamics by folding the circle. In this ways the intimate balance of dual compliments can be directly experienced.
Think about black holes as a function generating movement seen only in a spatial context. We are now theorizing about “white holes,” the opposite to black holes. Is not this what we have been folding; a black hole and a white whole? Sophisticated technological tools have expanded our ability to observe and measure what otherwise can not be seen, similarly the other circle stretches the idea of geometry and structural ordering and rearranging of systems observed to those that we do not see but experience the effects. This allows us to see how little we know about the unseen and unobservable aspects of physical reality. Space perceived as empty once again is shown to be occupied with higher-level phenomena that is beyond detection from lower levels of perception. The folds in the other circle hole are generated by the circle pattern not from the rectangle. There is not enough information within the shape of the rectangle to direct the circle-pattern movement in creasing.
The circumference informs both circle going in and going out. In this dual form there are two visible and two invisible circle planes, two curved planes between top and bottom separating the circle planes; one on the inside and the other on the outside of the paper, they both have the same volume. There are two circle edges where the three planes join. There are five congruent circle parts between two circles.
The dual circle, seen and unseen, reveal the same circle-pattern of organization; one folding into itself and the other folding out from itself. Like wise the concentric nature of the circle goes both endlessly in and out. The circle-pattern is unbounded by context allowing for countless expressions aligned through the formed and unformed concentric nature in unity on all scales.
Below: Two holes of different sizes, arbitrarily placed in a rectangle paper. Keeping the divisional creases in parallel, assures alignment between the two circles. Shown are a few of many reformations possible.
Below left: Drawing a non-concentric circle in a circle shape and removing it gives us three circles, two are congruent. To the right shows nine folds in the larger circle and three creases in the removed circle.
Below: A couple of reconfigurations.
Below: Four arbitrary placed circles cut from four larger circles as before, each with three folded diameters. Three diameters have been folded into the removed four circles. The other circle folding three diameters shows the four larger circles with chords that are not diameters. This makes the larger four circles when joined to a spherical vector equilibrium arrangement irregular and mismatched on the perimeter. The other circles form a regular and symmetrical bubble inside the irregularities of the four larger circles. When the removed circles are joined the same way there is the regular spherical form of the vector equilibrium that is identical to the other unseen bubble. (ref. blog http://wholemovement.com/how-to-fold-circles ) Both models are held together with bobby pins.
Below: The same four circles from above have been disassembled and rearranged to form a tetrahedron using the four removed circles as hubs to join the ends of the larger circles. The flexibility of the folded struts and removed circles accommodates a variety of angles and the irregularity of the off-centered effect of the arbitrary placement of the other circles. They are all tetrahedra in pattern, yet very different in form. With the model in lower left the circle planes are pushed in, the others show planes opened outward.
With this, I leave you to explore something that is not seen to find something that is.