Monday, 20 September 2010 08:22

Center Off-Center #2


This month we continue the center off-center investigation. Can a local center be re-centered to spherical alignment? We have seen any point on the circle can be a center point by folding in half or less than half. Folding comes first before centering a location. This is not like drawing a picture of a circle where the compass first sets the center point. Concentrically scale and perspective determines which circle becomes center, usually the smallest in any system gets to be center.


Fold four circles with three diameters each (see last month’s entry.) Do the same with four less than half folded circles. Before joining the circles into the vector equilibrium arrangement, draw out concentric circles from each point of intersection on all circles in both sets. (Below) The circles are drawn on both sides, on one side every other ring was filled in to keep track of different sides of the circles. Be consistent with the intervals. Spacing does not matter.


coc2-2  coc2-3


(Above) concentric circles in a less than half folded circle.

(Below) join each set of four circles using bobby pins to hold them together. Look at the similarities and differences between the two systems.

coc2-5  coc2-6


The concentric circles in both are in alignment to a center location. The off-center folding shows gaps in the planes; parts of circles are missing. If we filled in the gaps, completing each circle, the spherical periphery would then be aligned to the local center in a concentric form. The local center would be realigned to the spherical center showing an increase in the size of the spherical form.


By filling in with parts the sphere would not be Whole. Nothing can be added or taken if it is Whole. This might seem a bit obtuse, but it is an important distinction to make between the Whole and parts that look whole. We don’t want to go around calling things Whole when in fact they are coherent parts in alignment. There is confusion enough figuring out about centers.


To give some perspective to all of this; …“Actuality exist centermost and expands therefrom into peripheral infinity; potentiality comes inward from the infinite periphery and converges at the center of all things.”



There is a center of all things and then the multitudes of local centers that through peripheral or boundary alignment go to the same place. Knowing that concentric centering works for the 3-6 symmetry, the vector equilibrium. Will this also work the same for 4-8 and 5-10 symmetries?


The 4-8 symmetry starts with the same folding a circle in half. Fold the half circle into quarters by touching points and creasing. Then fold one point back to the opposite point; touch and crease. Turn over and do the same to the other side; point over to point and crease. This folds the circle into four diameters, eight equal sections. Fold four circles this way. The dark lines are creases.


coc2-7  coc2-8

Do this with four less than half folded circles in the same way. The proportional division of ½, ¼, will determine the angles for off-center location the same as it does for the half folding. Do not measure, use your eyes, this is about seeing proportional relationships, not measuring.


coc2-9  coc2-10


Overlap two eighths, reforming the circle into a right angle tetrahedron with an open triangle face having three curved edges and three right angle straight edges. Use tape to hold the circle together. Join two on the straight edges going in opposite directions, forming two open right angle tetrahedra between them. Both curved and straight edges make right angle crossings. Bobby pin the circles together.


Now make another set of two units the same way. Join the two sets of two, symmetrically, with two closed planes completing the two open planes of the other with edges touching. Bobby pin together as before. This makes a spherical octahedron with eight equal open right triangle tetrahedra. Do the same folding and joining with the off-center folded set of circles.


coc2-11  coc2-12

(Left) an example of the half folding showing spherical form.

(Right) the same joining using less than in half folding. Both show an octahedron centered pattern. One is spherical; the other is a distorted form. The difference is in peripheral alignment; going back to the first fold.


By filling in around the periphery, spherical potential then moves towards center that is now reflected in the outward form. This demonstrates the possibility of reforming ever expanding distortion of boundary properties to reflect spherical unity. The “converging at the center of all things” brings potential in line with a centered and balanced symmetry. The center is infinitely everywhere when there is alignment of concentricity.


In reforming the circle to a 5-10 symmetry the folding in half and folding less than half circle will be the same. The difference will be in folding five times to a different proportion and using six circles for each set.


coc2-13  coc2-14


1.) Fold circle in half.  2.) Fold one end point of the half circle to the point along the circumference showing a 1:2 division. What is left is one unit and what is folded over is the two units. When it looks correct, lightly crease.  3.) Fold first point back to point of the last fold, reversing the proportions to 2:1, leaving two units with one unit folded over.  4.) Then fold second end point of diameter to the edge of the previous fold. They should look equal with the fold edges down the middle. Lightly crease.    5.) Fold the two sections to the back and together. Everything should line up, if not, go back and make adjustments before giving a strong crease to all folds.


Open the circle to five diameters, ten equal sectors. As with the 3-6 folding, bring one diameter together joining opposite radii with a bobby pin. This forms two open pentagons with two open tetrahedra intervals separating the two pentagons (below left). Three and five are odd numbers making these reformations different that with the 4-8 symmetry.


coc2-15  coc2-16

Above right) Make another unit the same way and join them together. The sides of the pentagons of one unit will close the intervals between pentagons of the other unit.


Fold another circle as before and reform to a double pentagon. Then add that to the two already joined in the same way, completing five open tetrahedra around one of the open pentagons. This will be obvious when your see how the three reconfigured circles fit forming five pentagons around the sixth centered pentagon. As before use bobby pins to join the circles together.



Make two of these sets of three circles each. Join them the same way with edges of open pentagons closing the open sides of the tetrahedra intervals. Putting the two halves together forms an icosidodecahedron sphere.


coc2-19  coc2-20

Above left) This has twelve open pentagons and twenty open triangles.
Above right) Six less than half folded circles reconfigured and joined in the same way showing a distorted boundary. They both have the same pattern center. The potential for distortion is endless; spherical alignment is one.


Demonstrations of the less than half folded circles are all folded about two to three inches off alignment of the circle. You can imagine that even a half a millimeter off will cause misalignment with distortion. This is not about the center or measuring, it is about alignment and symmetry. Symmetry is a quality of spherical formation. Alignment is what locates the center. Within the concentricity of the circle, which one is the center? When exactly is the periphery in alignment to the center? How close is close to be called accurate? Is anything less than spherical a loss of symmetry, or simply a distortion at the periphery of an always there center of everything?


Explore the symmetries and concentric circles of varied intervals. Next month we will continue to explore further the relationships between polyhedra, concentric circles, and the center off-center.


Another view of folding the vector equilibrium sphere can be seen at http://www.wholemovement.com

¹ The Urantia Book, Urantia Foundation, Chicago IL 1955, Paper115: section3, p.1262

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