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Monday, 08 June 2015 23:33

Regularity and Randomness

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.

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.

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.

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.

Below.) Two reformations of the tetrahedron net above.

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.

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

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

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.

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.

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

Tetrahedron with rectangles compressed.            1st rectangle opened flat.

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

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.

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.

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