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Tuesday, 04 March 2014 05:08

Order Without Boundary

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Continuing the exploration of folding the circle pattern without the circle shape ( ) 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 ( 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. ( )

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.)

    01untitled-1es            02img 0184 b cps

 For folding instructions see



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

       1a.  03.img 0703es     b.  04untitled-1aes

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



       2a.  05untitled-2aes     b. 06img 0708ebes

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.  07img 0709aes     b. 08img 0708es

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.  09img 0710aes    b.  10img 0711aes

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.  11img 0712aes     b.  12img 0713aes

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.  13mg 0714aes     b.  14img 0715aes

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.  15img 0720es    b.  16img 0721es

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.  17img 0726es    b.  18img 0731es

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: )

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.  19dsc00587es     b. 20dsc00588es

     c.  21dsc00586es



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. 22.5dsc00598es   b. 23dsc00608es

    c.  24dsc00604es   d. 25dsc00605es



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.  26untitled-6ls   b.  27dsc00550es

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.  28dsc00583es

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.  29dsc00560es    c.  30dsc00566es

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. 31dsc00539es   b.  32dsc00636es



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. 33dsc00641es   b.  34dsc00645es

     c.  35dsc00657es   d.  36dsc00664es


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

    a.  37dsc00666es    b. 38dsc00675es



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.




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Bradford Hansen-Smith

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