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Continuing the exploration of folding the circle pattern without the circle shape (http://wholemovement.com/blog/item/121-circles-from-scrap ) brings up questions about boundary and the separation between the inner center and outward centered locations. Folding any irregular shape of paper in a sequence of folding proportional to that first fold generates the same three patterned grids found in folding the circle. They come from the first fold in any plane surface and appear to have little to do with the shape of paper used yet each reveals a centered system of organized symmetry.

Polygons are truncated subsystems of the circle limited to the symmetry from number of sides. Conversely polygons of any shape inherently carry the circle pattern and center that can be revealed through folding. The first fold is arbitrary and can be made anywhere (http://wholemovement.com/blog/item/92-center-off-center) The first fold divides one shape in two parts (a ratio of 1:2.) As seen with the circle there are three principled options for consistently folding this ratio; 3-6, 4-8 and 5-10. (http://wholemovement.com/blog/itemlist/date/2010/9?catid=140 )

Once the first fold is made any of these primary symmetries can be initiated at any point along the first line that can be anywhere, thereby establishing a centered coordinate system through the development of a concentric pattern anywhere on the surface. It is through straight-line creases that we see proportional differences between the triangle/hexagon, triangle/pentagon and triangle/square.

FOLDING THE CIRCLE using the 3:6 ratio, a 4-frequency diameter hexagon star is formed. The ratio is extended to a 3:6:12 grid division of the circle. (Folds are lined in black for visibility.)

For folding instructions see http://wholemovement.com/blog/itemlist/date/2011/2?catid=140

Let’s see what this looks like folding scrap paper with an arbitrary perimeter.

1a. b.

1a & b.) Randomly fold any size paper to a ratio of one piece in two parts.

2a. b.

2a & b.) With folded edge towards you place a finger arbitrarily along the folded edge and fold over until the folded angle looks the same as the resultant adjacent angle. This approximates folding into thirds. Press slightly indicating fold; do not crease. Turn over bringing opposite edges together (one over and one under the middle section.) Slide back and forth until edges on both sides are even, then give a strong crease.

3a. b.

3a & b.) Open to flat paper finding three equally spaced straight lines intersecting at a point resulting from the placement of finger on the first fold. Refold to triangular shape just folded.

4a. b.

4a & b.) Place the center point somewhere on the edge between the folded over part and the center point on one side forming a right triangle, give a strong crease. Open the fold and locate where that crease meets the opposite edge.

5a. b.

5a & b.) Fold the same on the other side, bringing the center point to the point where the last crease intersects the edge. A strong crease will form another right angle triangle opposite in direction. Open to see two intersecting right angle bisectors on each edge showing a symmetrical crossing centered to a partially formed equilateral triangle.

6a. b.

6a & b.) Fold in half, giving a good crease when the two edges are even and open to find three intersecting creases (reflection of #3a.)

7a. b.

7a & b.) Open to flat paper to observe the hexagon star with twelve radians. Refold back to the equilateral triangle. Then fold between the two creased edge points creasing the third side of the equilateral triangle.

8a. b.

8a & b.) Open paper flat to shows six equally spaced bisected triangles in a hexagon arrangement. Refold to single triangle with three bisectors. Between the two end points remove excess paper outside of the triangle or by cutting straight across to make an equilateral triangle, or as this example shows, tear an arc approximating a circle.

For another reference for this folding go to: http://wholemovement.com/blog/itemlist/date/2011/2?catid=140 )

The material outside the polyhedral boundary gets in the way of how we perceive order and construct from regular polygons. Is there another purpose for having “excess” material? What can be learned from material that seems extraneous to traditional construction with polygons?

9a-c.) Below, the three symmetries, (*a*.) 3-6-12, (*b*.) 4-8 and (*c*.) 5-10 have all been folded in the same sequence with an arbitrary first fold across the rectangular paper, each with a different proportioned ratio of 1:2. The difference in symmetry is with the angle of the second fold that determines the following relationships. Looking at one triangular sector there is a proportional scaling out from the local center to the limits of the paper.

9a. b.

c.

10.) a.) Below the 3-6-12 hexagon is folded. b.) show the hexagon reconfigured into the pentagon [6-1=5.] c.) shows the hexagon reformed to the square [6-2=4.] and then (d.) reformed to the triangle [6-3=3.]

10a. b.

c. d.

Last month I used images of some models to explore 2-D expressions of the 3-D objects I have folded. One of the 2-D images is printed here in black and white, then folded back to a 3-D object completing the transformation from 3-D to 2-D and back to 3-D.

11a. b.

11a. & b.) A black/white square image printed on a rectangular 8 ½" x 11" paper. It has been folded and reconfigured into a pentagon arrangement.

12a.

12a.) Here the paper is folded down to a 4-8 symmetry in a square arrangement that is accordion pleated.

b.) 5 square units are joined forming a cube with one open plane to view the inside cubic space.

b. c.

c.) All six sides form an enclosed cube. Each folded unit has a different orientation leaving an irregular outer boundary to the cube form. The printed image visually disrupts recognizing the cubic form.

13a & b.) Below, the same cubic configuration without the printed image shows an arbitrary and irregular boundary. With the “excess” material cut away the twelve edges and six stellated-truncated square planes of the cube are more easily seen.

a. b.

14a-d.) Below four rectangular pieces of paper have been printed and folded to a tetrahedron configuration and joined in a tetrahedron pattern showing the open octahedron. There are many ways the material beyond the tetrahedron can be manipulated, reformed, and modified to change the boundaries without changing the opened center or tetrahedron symmetry. (Hairpins are holding the tetrahedra together.)

14a. b.

c. d.

15a & b.) Below the “excess” material has been partially eliminated and then cut down to the edges leaving the two-frequency tetrahedron.

a. b.

How much do we lose when constructing a polyhedral object from polygons where the pattern context supporting that object has been eliminated? How far out do boundaries go beyond our perception and to what extent do they affect what we see? It is reasonable to construct and assemble polygons, even folding polygons/polyhedra using circles. What value is there in seeing what seems extraneous or excessive material beyond the ordering of forms we are familiar with? Why deal with material that seems to get in the way? Folding/unfolding is an important life-forming function and an important part of geometry with renewed interest in origami and paper folding.

Expanding the perimeter of any shape does not take away from the order that comes from any individually centered point within an infinite matrix of points on a given plane. Every point reveals a centeredness that gives access to the matrix that generates that point of intersection that at once is everywhere.

The pattern of concentric circles, PCC, seems to prevail throughout being revealed through construction methods, truncation of the circle, reforming by folding the circle, and by folding any shape to show forth a polygonal system of symmetry.

These are all static pictures that represent what takes place prior to and beyond the object we experience.

### Exploring Images Through Folding Circles

Written by Bradford Hansen-SmithA few months of no posting brings another direction in exploring the circle. Using images of folded circles that show the structural nature of geometry and combining them with NASA images of the observable universe allows me to image what would otherwise not be possible to see by bringing together vastly differing scales in one frame.

Over the years I have used photo images of my sculpture collaging them together and with other images giving greater breadth and depth to the ideas that were embodied in the original work. These current images are a continuation of that process.

*Above*: the helix is made into the image of a cosmic flower with the center spiral derived from the same helix image. The second image shows the helix to be a dynamic cosmic force experienceable through local consciousness.

*Below*: is a spiral model with a design printed on the circles. The image has been modified to show a cosmic hot spot that appears to move structurally out forming a galactic cloud. Possibly the cloud is from itself forming a cosmic tendril of local significants. In the lower left is another space-star circle model.

*Above*: this image also uses the space-star model seen imediately above.

*Above*: an image combining two different folded circle spheres gives visualization to the dispersal of regulated energy.

*Below*: another spiral is in process of gathering cosmic debris and at the same time distributing as it moves both in and out of itself. The tape holding the model together is a nice touch.

*Above*: this single model has taken two different directions in these fanciful treatments suggesting familar life forms.

*Below*: A 3-D folded circle wall piece showing a variety of spatial layers. In keeping with the layering of the image, Photoshop was the right tool. The second image departs from the textural layering of the first and is pretty straight forward.

* *

*Below*: an old model is used to show a coming together, a collision or just attraction, of an object to itself where the image of the cosmos has moved to the foreground. The model is from an early exploration of folded circles where bobby pins, vinyl tubing, and string had been attached.

*Below*: is a model with an open interior space that appears to be a floating cube. This image suggest one possibility where floating is somewhere between gravity and illusion.

*Above*: a radiant image derived from the picture of a spherical model.

*Below*: another image using models, both previously used, where the forms take on some semblance of human consciousness.

Below: Two different models spanning 15 years have come together in a computer version of a layered projection through a checkered grid, which in Photoshop indcates an empty or transparent layer.

bove: