### Archive

- June 2015 (1)
- May 2015 (1)
- March 2015 (1)
- January 2015 (1)
- December 2014 (1)
- July 2014 (1)
- June 2014 (1)
- April 2014 (1)
- March 2014 (1)
- January 2014 (1)
- August 2013 (1)
- July 2013 (1)
- June 2013 (1)
- May 2013 (1)
- April 2013 (1)
- March 2013 (1)
- February 2013 (2)
- December 2012 (1)
- November 2012 (2)
- October 2012 (1)
- September 2012 (1)
- August 2012 (1)
- July 2012 (2)
- June 2012 (1)
- May 2012 (1)
- April 2012 (2)
- March 2012 (1)
- February 2012 (1)
- September 2011 (2)
- July 2011 (2)
- June 2011 (1)
- May 2011 (1)
- April 2011 (2)
- February 2011 (1)
- December 2010 (1)
- November 2010 (1)

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.

This is the third post exploring the alignment of concentric circles as spherical shells starting with folding the circle in half. Folding less than half of the circle causing misalignment that will locate an off-center position and peripheral distortion. We will continue with the center off-center concentric circles and how they relate to polyhedra.

The touching points are not connected except in relationship to the circle as a dynamic system. Aligning two points on the circle boundary will provide consistency in continued development of folding and combining circles with a common spherical center. We see this in the symmetries of the three primary spherical systems: 3-6, 4-8, and 5-10 (below.)

Concentric circles are self-organizing and self-generating structural pattern. The circumference and center are simply the largest and smallest definable circle units. Movement into and out from each circle boundary is a right angle movement. This is reflected in movement by touching any two points forming a crease at right angle to the movement between points. Centering of the circle comes from this alignment of the circle to itself.

Below left) are equally spaced concentric circles. The intervals are arbitrarily ½ " apart; consistency of intervals is important. This will give an idea of spherical shells as the circle is reformed.

Below right) shows two arbitrarily placed points on the circumference showing concentric circles from each point. The kite shape is straight-line connecting of intervals between the circles as they intersect. This is the same as folding the circle in half. An interference pattern is inherent between two locations of concentric circles in a right angle relationship.

Above) Combining the touching points and the center point of the circle show the interference pattern of three center locations. Three is structural, two by themselves are not. Alternate concentric rings are shaded to make them easier to see. The right angle intersection of the kite shape comes from the intervals inherently patterned to circle movement. Folding the circle in half shows alignment of the circle/sphere context.

Below left) concentric circles with intersection of three folded diameters.

Below right) Using the same 1/2” intervals and shading alternate rings shows the interference pattern with three diameters. Each of the six points is a local center point with concentric circles. Only two rings of each have been shaded.

Above left) Six primary points of the tetrahedron net (a two-frequency triangle) where the center points have two concentric rings each. Expanding the circles to all fifteen points would have been too dense to clearly see the net.

Above right) This wave pattern shows the 13 primary points of intersection forming the hexagon star pattern. Each diameter is divided into four equal sections where each point is a center point. The level of complexity is determined by establishing design constraints, in this case two shaded rings around each local center point.

Above) Three diameters have been divided into eight equal sections, a consistent development in folding the circle. There are twenty-four creases with two concentric rings around each of the nineteen primary points forming a limited interference pattern. The intervals and shading of rings is consistently 1/2" to keep it simple. Each small white triangle interval is a point of three intersecting creases that coincide with circle intersections. This is a fractal design, often called a pattern.

The interference patterns of concentric circles reveal the polyhedral nets that are inherent in the dynamic ordering of circle division. This is where the polyhedral forms relate directly to circles and to spherical packing.

Below left) A tetrahedron unit folded from the tetrahedron net (above with six centers) indicates four spheres in closest packed order. In this case the edge length is an eight-frequency division of a two frequency tetrahedron. A single tetrahedron unit does not exist in spherical order, only as an organization of four spheres that have been truncated.

Below right) A two-frequency polyhedral form of four tetrahedral units showing ten spheres with the open octahedron space. The edges shows a sixteen-frequency division.

Above) The octahedron relationship is a formed unit of two open and joined tetrahedra. It shows six vertex points, six tangent spheres in spherical packing. The concentric circles defined on each face wrap around showing the spherical pattern in a polyhedral form.

Below left) Adding eight tetrahedra and an octahedron to the vector equilibrium sphere we see spherical packing by filling intervals between the thirteen points with concentric rings. There is an unseen interference pattern created between the concentric circles of the centered vector equilibrium sphere and the thirteen local-centered spheres in the closest packed order.

Above right) A polyhedral representation of spherical packing of the tetrahedron/octahedron matrix showing eight equal divisions of each unit edge. This association of polyhedra shows patterned spherical origin in the form of truncated spheres.

Above left) The octahedron is opened on three edges with two points joined forming a dual pentagon cap arrangement of four triangle faces and one open triangle plane; five triangle planes around the two opposite vertex points, ten triangle planes. Seven spheres are tangent on the plane surface, but unlike the matrix above, there is interior spherical distortion by reduction of the radial measure.

(Above right) By adding two open tetrahedra to the octahedron net in a tetrahedron pattern and bringing edges together an icosahedron is formed: sixteen triangle faces and four open triangle planes. This is a non-centered arrangement of twelve tangent circles on the surface planes, but again spherical distortion occurs on the inside. It is not an expression of spherical order.

Below) Another reconfiguration showing two circle each reformed to 1/2 a tetrahedron using the 12 creases (shown above.) Each reformed circle reveals the square dividing through the octahedron within the tetrahedron.

Above) The two units joined on open square faces form the tetrahedron. The concentric circles surface design shows a higher frequency division of each edge. More creases generate more complex reformations, still keeping wrap around consistency of surface design.

(Below left) This model of the off-center folding of the vector equilibrium from last month will be used to look at how the center off-center come together.

(Below right) Eight tetrahedra, from above, fill each of the open tetrahedron spaces. The tetrahedra come together at the center point showing a consistency of outward facing triangles that form the open square relationships. The completion of spherical packing has been drawn to indicate an interference pattern of interpenetrating spherical shells.

Above) Two views of octahedra filling in the six open square spaces This forms a large regular octahedron that begins to approach the off-center periphery. There is a lot of information about the interrelationships between spheres, circles and polyhedra in this system.

Continuing higher frequency development suggest the off-centered periphery will eventually be absorbed and become aligned to the circle center in a polyhedral form. Every location of local phenomena within circle/sphere unity is center, albeit local. It is then a matter of consistent higher frequency development towards bringing the center and off-center into alignment as one.

We will go more into that next month as this exploration continues.