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Monday, 08 June 2015 23:33

Regularity and Randomness

Written by

Random crumpling of paper disrupts the plane transforming it towards a compact sphere-like objects. Creasing straight lines gives order through predetermined sequencing towards specific reformations of the plane. This exploration looks at combining a straight-line creased grid with randomly crumpled creases.

The regularity of a self-distributive structurally ordered grid exhibits symmetry in circle-pattern not found elsewhere. It is independent from individual shapes or quality of material. http://wholemovement.com/blog/item/729-order-without-boundary-ii-geocoding

Crumpling appears without order. Yet there is a sense of organization in relationship to how the crumpling is done. The self-organization of a circle-patterned grid and random crumpling seem to compliment each other. In exploring this I have in some cases used printed images to add another layer of information to the reforming process.

Combining the circle-patterned grid and crumpling has parallels in nature as well as how we live our lives. Wrinkles record habits, a consistent tracking about what we have done, where we have been as we interact through space over time. These folds enliven the surface with unique expressions responsive to internal and external forces. A crumbled “mess” when opened suggests some kind of inherent order with a great deal of textural interest. Ordered folding is geometric in style and it lacks the uniqueness found in spontaneous crumpling. 

 

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Crumpled wad of paper                      Unwadded crumpled paper

 

To carefully unfold and then reform the creases of a crumpled surface reveals interesting variations with little possibility of exactly refolding the original wad ball. 

       

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Above left.) Paper folded to the circle-pattern hexagon grid, a 3-6 symmetry. The vertical line is the arbitrary-placed first fold. The two diagonal creases are the next two folds. These are the same proportional folds of division used in folding the circle. The rest of the creased grid lines are informed by the first three equally space creases.

Above right.) The same paper after it has been crumpled. Once the primary creases are established any random crumpling has little effect over the consistency and structural nature of reorganizing the folded grid.

 

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Above left.) Paper reformed using the grid to reconfigure a five-fold symmetry.                                                   

Above right.) Same paper reformed to a square symmetry.                                                           

 

Below left.) This shows a rectangular shaped paper folded into a tetrahedron net; same as it would be in a circle. (The net creases are traced to make them easy to see.) These nine creases for the tetrahedron net are primarily used in the following demonstrations.

Below right.) I have tried to leave the center triangle in the net uncrumpled. Usually we cut off the “extra” paper leaving an isolated polygon. It is difficult to crumple one part of the paper without effecting the entire surface. Every crease is interconnected to all others through the paper itself.

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Below.) Two reformations of the tetrahedron net above.

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Below left.) The net is reformed to a tetrahedron.

Below right.) The same tetrahedron is formed before crumpling. It has the stylized geometric form. The tetrahedron pattern is identical in both.

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Below left.) A regular tetrahedron with the excess paper more tightly crumpled.

Below right.) Three tetrahedra joined forming an open plane tetrahedron cavity.

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Above left.) A fourth tetrahedron is added to the open face of the three on upper right, making a “solid” closed stellated tetrahedron. Each tetrahedron unit is consistent to the same size rectangle, roughly positioned to the same location on the paper, using the same measure to insure congruency of scale.

Above right.) Four tetrahedra rearranged to form a two-frequency tetrahedron revealing the open octahedron.

 

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Above.) Four more tetrahedra added to the open triangle planes forming a stellated octahedron, or cube patterns. These added tetrahedra are without crumpling. There is consistent regularity of pattern not obvious in the random look of the form. Patterns are consistent; forms are a continually changing variable.

 

Below left.) Tracing the lines allows another way to see what is going on. Two papers differently creased are joined. One rectangle has been folded to the circle-pattern hexagon grid and reconfigured to pentagon symmetry. It has been placed on top of the other that has been crumbled.

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Above right.) One paper has been both crumbled and folded to the circle-patterned hexagon grid and reconfigured to the pentagon.  

 

 

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Above.) Four pieces of copy paper folded to circle-pattern grid, partially crumpled and joined. The circle removed before folding is separately folded and crumbled; where folding the empty circle reveals the grid in the surrounding rectangle paper. This references last month’s blog where we saw the circle paper folds the grid in and the empty circle folds the grid out.

Again creases have been traced with a marker making them more pronounced. Glass is placed on the flat surface on top with the reformations below hanging out in 3-dimensions.

 

Tracing a folded pattern of creases is easy and straightforward, they are all straight lines; it is more difficult when tracing the crumpled folds. The angle of light changes our perception of where the crease and fold start and stop. Without shadow the crease appears in one place and direction; changing the light shifts perceived orientation. As with all tracings, the lines are a generalization of spatial relationships of movement. Tracing the residue of movement does not give an accurate picture of what has been in-formed. There is delight in the movement and spatial changes as well as coming to rest with it. I value are the unexpected surprises that happen along the way, they are hidden in the image. The limitation in folding any formula shields us from the unexpected, hids the experience of discovering the moment.

 

Below.) Four views of a single arrangement. Two diverse pictures have been joined combining crumpling and the hexagon grid into a single arrangement. The beauty is in experiencing the spatial relationships of images as they form an object.

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           023dsc02601es               024dsc02595es

 

Some of the 2-D images used are from past folded circles models photographed and combined with other imagery that now become material for exploring folding and crumpling. Both 2-D and 3-D are degrees of abstraction removed from the reality of expression.

You can find more about the images used in a previous blog http://wholemovement.com/blog/item/129-exploring-images-of-folding-circles

 

Below.) Two views of integrating three photo prints each folded to the circle-pattern grid and reformed. This is another way to explore the spatial relationships implied in 2-D images.

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Below.) Two sides and one front view of a 3-D object by folding, crumpling, and joining three individual pictures. A transforming takes place from folding circles into 3-D objects, translating the objects to 2-D images, then folding those images back into into objects using the same folded creases used in the original circle folded models. This brings various stages of development together in a single object.

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Above.) An early model using multiple circles folded to the tetrahedron net, where before reforming each circle, they have been crumpled to give an interesting surface and tactile appeal. Random crumpling softens the paper, yet seems to have little effect when reforming the structural net.

 

Below.) Four rectangular pieces of paper were first crumpled then folded in a four-frequency diameter circle-pattern hexagon grid. The units have been reformed to a variation on a truncated tetrahedron. Joined on their end points they form an octahedron with four open and four closed triangle planes. Given the crumpling and shape of paper it does not look like an octahedron, yet there are eight triangle planes in a regular octahedron arrangement.

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Below.) Four more rectangles are folded to tetrahedra without crumpling. They are attached to the inverted triangle faces of the model above. There appears possibility for sustainable “new growth” when there is regularity of pattern. The open triangles are potential for continued growth.

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Below.) A glossy cover stock 2ft square was folded to a randomly placed circle-pattern hexagon (3-6 grid.)  Folding in 1/6 of the grid leaves a 5-10 symmetry formed to a pentagon. The paper outside the pentagon was then crumpled. Following are two variations in reforming the grid keeping to the pentagon symmetry.

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          037dsc02778es        038dsc02783es

          

 

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Above left.) The pentagon is opened to show the hexagon star of the 3-6 symmetry grid.

Above right.) The grid is folded in reformed to a 4-8 symmetry. With each reformation the crumpled rectangle is changed.

 

Below.) Three circles and a scrap of crumpled paper combined to make something that looks like it might be biologically functional; particularly after adding a few twisty ties.

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                      042dsc02668es            042.5dsc02903es

 

 

Below.) Four stages of opening the crumpled material around a solid tetrahedron made from four pieces of paper. The center triangle in each net is left flat and joined to form a solid regular tetrahedron with the rest of the paper around it being crumpled.

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Below.) This series of six images show the tetrahedron from above with each of the four rectangular papers sequentially opened and flattened to show where the triangle is placed on that paper. These images lack the wonderful spatial quality displayed in these changes.

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  Tetrahedron with rectangles compressed.            1st rectangle opened flat.

 

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   2nd rectangle opened flat.                                 3rd rectangle opened flat.

 

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   4th rectangle opened flat.                                  All rectangles opened somewhat equally.

 

 

Below.) Two separate images and one paper circle folded to the circle-pattern grid and combined. One image is crumpled, the other partially crumpled and reformed, with the third forming an icosahedron. Here 2-D images of 3-D objects are folded in 3-D, again using the same patterned grid used for the objects initial folded shown in the images. Each expression is layered into the next into what is now the 2-D images you see. Once this transforming process reached the virtual world it becomes a translations of zeros and ones. This is where this all started, by folding circles and straight line creases.

Two views give you an idea of the dimensionality of what otherwise would look flat.

 

    053dsc02677esb         054dsc02672es2

 

 

               

                    The growing interest in origami expands the possibilities to further explore folding paper. So, when you wad up paper, unwad it and look at the beauty of what just randomly happened that you were going to throw away. If we looked at everything for what is beautiful before it becomes a throw-a-way, then maybe there would be more beauty and less garbage in the world. We have yet to realize the synergistic balance between regular ordering forward and the seemingly random spontaneous leaps that occur.

  

 

 

Saturday, 02 May 2015 01:20

The Other Circle

Written by

In believing the circle is image, we limit the imagination.

Draw a circle anywhere on a piece of paper.      

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Cut the circle from the paper.

How many circles are there?

Compare properties of both circles.

They are the same circle.                                                             

                                               002dsc01962es

By removing the circle from the paper the other circle is left. One boundary separates the physical circle from the non-physical other circle. We might say one is positive, the other negative. One is where the other is not. The inverse of one is the other. The compliment is so intimate that without separation we miss the dual nature.

 

Fold both circles in half.

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Notice the difference and similarities in each circle.

One circle contains the creases. With the other circle the paper references the unseen crease within the circle boundary. Having folded in half what is not there  informs alignment with what is there.

Fold both half-folded circles twice more in ratio of 1:2 http://wholemovement.com/how-to-fold-circles.

 

Below: Having folded the half-folded circles into thirds, open them to find three diameters in each. Three chords are visible in the one removed; they are not visible in the other circle. 

    

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Three diameters, six radii define six areas and seven points. In the other circle the unseen diameters extended outward showing six line segments defining six areas and twelve points. The proportional folding is the same for both circles. Removing the image from the rectangle paper the other self-referencing, self-organizing circle is now a hole defined by context, yet functions in the same way as the folded paper. The creases not seen in the circle extend through the rectangle. The invisible is held with-in the visible, and for that reason often goes unrecognized. The unseen informs through space that which can be seen.

Fold circles to a higher frequency of the same pattern of division, see blog for instructions; http://wholemovement.com/blog/item/97-unity-origami

When folded to a higher frequency each circle reveals six diameters, the organization being in the first three. The second set of three diameters have a different proportioned division functioning as bisecting diameters to the first three.

Below left) Division of the first three diameters show four equal sections revealing a hexagon star and hexagon. With one the hexagon and star are on the inside boundary of the circle, with the other circle the star points and hexagon are creased into the rectangle on the outside of the circle boundary; one goes in the outer goes out.

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Above right: When replacing the removed circle back into the other there are two ways to align the six diameters. One aligns the star points so both are in the same orientation; the other (pictured above) is where the star point diameters are in alignment with the in-between diameters of the other, a thirty-degree shift in position. The latter is the complete alignment of the grid showing two levels of grid division using all six diameters.

The creases in the paper have been drawn over for better visibility.

 

Below: Continuing a higher frequency folding in the rectangle (creasing lines parallel to lines already there) the equilateral triangle grid becomes obvious reflecting the same size grid we see in the folded circle. Coloring the same size and orientation of triangles makes  the pattern clear. The removed circle shows more information; each triangle of the hexagon division is bisected in three directions rotating the grid thirty degrees to a smaller scale. This can be described as a fractal like “interference pattern” (upper right.)

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         010dsc01976es          011dsc01978es

 

Below: The removed circle has been reformed to a tetrahedron. This suggests the other circle will also reform into a tetrahedron. The form will look very different because of the difference in perimeter but the circle-pattern of arrangement will be the same. There are more equilateral triangles in the rectangle, which increases possibilities in multiples tetrahedra on different scales.

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Below: The other circle can change symmetries from six diameters (hexagon,) to a five fold symmetry (pentagon,) to four (square,) and the three (triangle.) This is possible with the whole circle, the circle hole, and is demonstrable with any random shaped paper folded to the same circle-pattern. http://wholemovement.com/blog/item/729-order-without-boundary-ii-geocoding

 

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Below: Fold creases into the rectangle reflecting what is already in the removed circle. Color the alternate areas of the second level division revealing another level of design using the creases.

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Above: By replacing the removed circle into the rectangle the difference in scale becomes apparent. With consistency to the same alternate coloring of the folded grid the fractal scaling separates the inside and outside of the circle boundary.

 

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Above: Both circles independently have been reformed to a square-based pyramid (half octahedron.) The pyramid from the rectangle is not complete until the removed circle is replaced, as if they had not been separated. The frequency difference is again apparent with the dual reformations aligned to the same symmetry showing consistency in pattern and difference in design.

 

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Above: Left shows both forms folded into a "solid tetrahedron." On the right shows the other circle reformed to a tetrahedron pattern of four points in space. The circle-pattern is indicated by the arrangement of points (any four points in space is a tetrahedron pattern.) There are many possibilities reforming the tetrahedron using this grid-creased rectangle.

Below: Here are a few of many possible reconfigurations using the grid in the rectangle folded to the circle hole. There is no proportional relationship between inside circle and outside rectangle because of the initial arbitrary placement of the image. The diversity of forms suggests rich design possibilities in a proportionally aligned relationship of circle to perimeter. You might call this an organized, controlled crumpling of paper.

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      024dsc02174es         025dsc02130es

     026dsc02131es          027dsc02132es

     028dsc02123es          029dsc02199es

     030dsc02156es                     031dsc02167es

Circle-pattern is inherent to all shapes; all shapes are inherent in the circle-pattern. Were this not so we could not construct what we do with compass and straight edge, or remove the image from the plane and fold the circle into what is possible. By separating the circle from the rectangle part, itself a truncated circle, we see a great variety in different forms through consistency in pattern. The part and whole are so intimately bound that without one there is not the other, even in the appearance of separation.

Symmetry apparently has little to do with specific shapes and is more about proportional divisions of unity. Symmetry bound to pattern is therefore inherent to all shapes and forms. Transformation from one symmetry to another happens because they hold circle-pattern whole in common.

Below) This diagram relates sphere-to-circle compression with removing the circle image from the 2-D plane. Both forms carry spherical unity that can only be realized by moving from the illusion of 2-D to experience the dynamics by folding the circle. In this ways the intimate balance of dual compliments can be directly experienced.

 

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Think about black holes as a function generating movement seen only in a spatial context. We are now theorizing about “white holes,” the opposite to black holes. Is not this what we have been folding; a black hole and a white whole? Sophisticated technological tools have expanded our ability to observe and measure what otherwise can not be seen, similarly the other circle stretches the idea of geometry and structural ordering and rearranging of systems observed to those that we do not see but experience the effects. This allows us to see how little we know about the unseen and unobservable aspects of physical reality. Space perceived as empty once again is shown to be occupied with higher-level phenomena that is beyond detection from lower levels of perception. The folds in the other circle hole are generated by the circle pattern not from the rectangle. There is not enough information within the shape of the rectangle to direct the circle-pattern movement in creasing.

The circumference informs both circle going in and going out. In this dual form there are two visible and two invisible circle planes, two curved planes between top and bottom separating the circle planes; one on the inside and the other on the outside of the paper, they both have the same volume. There are two circle edges where the three planes join. There are five congruent circle parts between two circles.

The dual circle, seen and unseen, reveal the same circle-pattern of organization; one folding into itself and the other folding out from itself. Like wise the concentric nature of the circle goes both endlessly in and out. The circle-pattern is unbounded by context allowing for countless expressions aligned through the formed and unformed concentric nature in unity on all scales.

Below: Two holes of different sizes, arbitrarily placed in a rectangle paper. Keeping the divisional creases in parallel, assures alignment between the two circles. Shown are a few of many reformations possible.

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      036dsc02209es        037dsc02221es

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Below left: Drawing a non-concentric circle in a circle shape and removing it gives us three circles, two are congruent. To the right shows nine folds in the larger circle and three creases in the removed circle.

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Below: A couple of reconfigurations.

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Below: Four arbitrary placed circles cut from four larger circles as before, each with three folded diameters. Three diameters have been folded into the removed four circles. The other circle folding three diameters shows the four larger circles with chords that are not diameters. This makes the larger four circles when joined to a spherical vector equilibrium arrangement irregular and mismatched on the perimeter. The other circles form a regular and symmetrical bubble inside the irregularities of the four larger circles. When the removed circles are joined the same way there is the regular spherical form of the vector equilibrium that is identical to the other unseen bubble. (ref. blog http://wholemovement.com/how-to-fold-circles ) Both models are held together with bobby pins.

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Below: The same four circles from above have been disassembled and rearranged to form a tetrahedron using the four removed circles as hubs to join the ends of the larger circles. The flexibility of the folded struts and removed circles accommodates a variety of angles and the irregularity of the off-centered effect of the arbitrary placement of the other circles. They are all tetrahedra in pattern, yet very different in form. With the model in lower left the circle planes are pushed in, the others show planes opened outward.

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With this, I leave you to explore something that is not seen to find something that is.

 

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