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Monday, 20 September 2010 08:22

Center Off-Center #2

Written by

coc2-1

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


coc2-4

(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.

 

coc2-17

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

Posted by brad at 7:31 PM 0 comments

 

 

 

 

 

 

 

 

 

Monday, 23 August 2010 17:02

Center Off-Center

Written by

coc1

 

Accuracy in touching is seeing with the eyes, with the mind, coordinating with the body. To not see is to be off-center. This is not about measurement, it is about movement through space with purpose. Without awareness towards the Whole can we know the center? Can we know off-center? By starting with the circle as Whole, and through action of folding we will explore center and off-center locations.

 

Axiom for folding circle: Any two of an infinite number of points on the circumference of the circle when touched together and creased will equally divide the circle where the linear distance between points will move spherically at right angle to the diameter diminishing the distance between those points as they join on the circumference, where the linear distance between points becomes half the original distance.

This gives symmetry to the division of the circle

 

The corollary to this: where any one point on the circumference with any other point not on the circumference (or two points not on the circumference nor aligned to a diameter) when touched together and creased will generate a chord less than diameter where the right angle circular movement will diminish from one point to touching the other, and the line segment distance between the two points is reduced to half the original distance. There is no symmetry to this division of the circle.

 

The axiom is the only way to symmetrical fold a circle, assuming it to be the first act of folding. The corollary holds true for the circle and applies to folding any shape. Two circles will be used to demonstrate the same process for finding center and off-center locations. Points are touching or they are not; there is proportional harmony or disharmony to the circle. Center is always sustainable, what is not sustainable has no center. Inaccuracy in putting points together becomes arbitrary, random movement without attention to finite boundaries.

 

(left) axiom; (right) corollary. There is no right or wrong in observation

coc2  cocc3

We see the difference in touching two points on the circumference and touching two points with only one point on the circumference. How does each develop as we apply the same folding process to both?

 

The next two folds are generated by folding the semicircle into thirds around the circumference. Fold one third in front and one third is folded to the back behind the middle third. It is not necessary to measure; use your eyes to know all points are touching by sliding the folds bringing them into alignment. Don't crease until all the edges are lined up and exactly even. Fold the off-center circle into thirds the same way; sliding circumference sections back and forth until the edges are even (points will not match up, you might look for division of central angle.) When edges look even, then crease. This folding process can be seen at the “How to Fold the Circle” page on my website: www.wholemovement.com

(Left) All chords are diameters. (Right) Only one chord is diameter The lines have been traced with a marker to clearly show the creases in each circle.

 

coc4   cocc5

 

 

Both circles show three evenly spaced chords revealing a point of intersection, where each is differently located to the circumference. If the points of intersection were a rotational axis, one would wobble and the other would show steady alignment to concentric movement.

 

 

 

To further develop folding in each circle in the same way, reform each by folding one of the chords to itself. The center is even with two opposite radii touching. With the off-center circle the division of chords is uneven, the end points will not meet. Use a bobby pin to hold the creases together. There are three folds, therefore three possibilities for folding a crease onto itself. Only with the off-center circle does it make a difference in the configuration. This joining a crease to itself is called a “bowtie” configuration.

 

 

.coc6   coc7

 

Fold another circles as before, making another “bowtie” for each. Join together the two individual sets of each, straight edge to straight edge, using bobbie pins or other means to hold them together.

 

coc8   coc9

 

 

This makes obvious the difference between center and off-center forming and joining. Each of these sets of two circle each is one half of a spherical pattern. (In the off-center the circles are creased to one-quarter distance of a diameter of the circle, to hold to some proportional consistency. Otherwise they could be folded in anyway.)

 

 

 

Make a duplicate set of each ( two sets of two circles) center and off-center, and join them together respectively on straight edges. (Again described on my website in the How to Fold the Circle section). The off-center is now getting more difficult to work with and the center system goes easily together.

 


coc10  coc11

 

Each is shown in same orientation so the difference is apparent. The one diameter in each of the four off-center circles has been assembled to reveal the one composite circle for comparison to any of the four composite circles seen in the center joining.

 

Below. Other options in joining off-center circles do not show a composite circle. 


coc12  coc13

 

All center and off-center spherical systems show alternate open triangles and squares. The off-center of each circle displaces the circumferences showing no regularity of a patterned spherical form. The form is deviant, while having an invariant pattern of three equally divided chords within a local center, not obvious in looking at the form.

 

coc14 

 

Here one half of the centered sphere is joined to one half of the off-center sphere. This does not show balance, only that the pattern of developing three equally space chords for both is consistent and allows irregular joining of center and off-center.

 

 

 

There is only one of an infinite number of proportional symmetries in touching two points together in that first fold of the circle. Everything starts form this ratio of 1:2, one Whole two parts. This is the first expression of symmetrical pattern. Anything short of that will eventually cease to generate. The off-centered asymmetrical forms, while they can be interesting and many, are limited and will not sustain generation through multiple joining.

 

 

Below are  examples of off-centered variations of spherical forms.

 

coc15  coc16

coc17  coc18

 

 

  coc20coc19

 

Below; the edges of each, center and off-center, are folded in-between the six points on each circumference before joining, this is more in keeping with the straight edge flat plane truncated look we are traditionally familiar with. The vector equilibrium, traditionally called the cuboctahedron, is pattern for both; clearly identifiable in center folding, not so easily seen in off-center folding.

 

coc21  coc22

 

Where is the center off-center? There is no outer boundary or inner boundary to concentric circles meaning there is no center to the circle; the circle is the center. The circle demonstrates the center is never out of center; there is only awareness about when inaccuracy slides into random movement and becomes misaligned. The mind functions as a balance, a connection, between seeing a finite center in physical form, and possible insight to perfection of infinite center regardless of form. This folding is a demonstration to seeing what works, therefore is sustainable, and what does not work and is unsustainable.

 

Over the next couple of months I will continue this center off-center exploration in folding and joining circles.