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Picking up on last months entry I went a little further in exploring the hyperbolic surface since having observed for many years that it is inherently within the folding of the circle. First it was necessary to crumple some paper and to look to see what happens in a random act. There is no order to the surface plane when crumpling a piece of paper (in this case a paper plate circle.) The reforming of the surface is arbitrary, unpredictable, and sometimes with interesting results.
The surface is contained by the edge showing a chaotic wave action taking place in the plane as the interior surface moves perpendicular away from the flat plane forcing the edge to move in two opposite directions. There is a warping of mountains and valleys that move through the horizontal surface plane. Other than the random forming it is similar to dropping an object in still water. A tension and compression is created in the movement through the plan that reflects the dynamics in compressing the sphere to a circle disc. The outward tension gives balance to the inner compression.
Below) The hyperbolic surface is mathematically portrayed as a smooth curved saddle form in two directions, a distortion of the square shaped plane. It shows an underlying tetrahedron pattern inherent in opposing right-angled movement.
Below) Looking from a polygon perspective there are three basic modes of planer forming. Starting in the center there is the hexagon surrounded by hexagons (6,) a flat tiling plane. The pentagon surrounded by hexagons (5) shows openings between hexagons where a continuous surface must curve away from the plane in a spherical direction. The heptagon surrounded by hexagons (7) finds overlapping of planes that must buckle off the plane to keep a continuous surface. This demonstrates a hyperbolic plane.
Below) Start by folding three diameters into the circle; that is the hexagon pattern with seven points. There are six equal areas (6.) The circle can be reformed by folding in one radius a pentagon pattern of 5 (6-1), to the square pattern of 4 (6-2,) then to the 3 pattern (6-3 on right.) With each decrease in parameter there is an increase in height, a precessional movement off the circle plane. There are two ways to decrease the circle by half (6-3), one an in/out star-like configuration (left) and the other a tetrahedron (right.)
Above) Each increase in altitude shows a decrease in circumference and when all are in alignment on the flat circle of origin there are concentric rings that correlate to the eight divisions across the circle; a folded 8-frequency diameter circle ( http://wholemovement.com/blog/itemlist/date/2011/2?catid=140)
Below) Each decrease is inscribed onto one circle, then accordion pleated in alternate fashion. This forms a hyperbolic curve to the surface going up and down in perpendicular direction to the plane. The saddle curve is seen in the reformed edges not the continuity of surface.
Above) Three points on the saddle surface form an inward curving triangle. The same three points also show a relationship off the surface to be a flat open triangle plane. By locating a fourth point anywhere on the surface, a tetrahedron pattern of six relationships and four triangle open planes are formed.
Below left) A paper plate circle with five inscribed circles accordion folded forms an extreme saddle configuration. Many people have done this; most notably Jose Albers with his students in the 1920s at the Bauhaus School in Germany and later in the 1940’s at Black Mountain College in North Carolina. There are more current variations by others slicing through the circle and cutting out the center. Here the circle remains whole keeping unity throughout, which I believe was the point of the exercise in the first place.
Below right) Two circles are reformed and joined, one into the other.
Concentric circles give order to the plane by determining where the surface will go. With crumpling the circle the same dynamics randomly occurs without any sense of order.
When three diameters are folded into the circle a similar ordering occurs as seen in the concentric circles. The 8–frequency triangular grid is a straight-line polyhedral reflection of concentric circles. The above illustration using three diameters shows the flat hexagon transformation from 6 into 5 the pentacap, to 4 the square based pyramid, and 3 the tetrahedron; all revealed in concentric circles and formed using straight lines all self-referrenced to the circle through proportional movement.
Above) Using the 8-frequency grid shows a simple folding of two views of reconfiguring the in/out of the circumference.
Above) On the left is arbitrary crumpling to the center where on the right are combined a few of the folds from the 8-frequency grid giving an interesting combination of a semi-controlled crumple.
Below) Two variations where the circumference is more controlled by the regular symmetry towards the interior of the circle using a 3-fold symmetry.
Below) This show basic opposite direction folding where the center is going in one direction and the circumference moving at right angle in the other, much like the saddle only in angular form. A tetrahedron interval is formed at the center by two solid faces and two open faces of triangles; four points and five edges of six relationships.
There is a slight fold variation between the left and right hand pictures.
Above) Two units on the left shown above have been joined bringing the open tetrahedra together forming a double saddle system with a solid tetrahedron center. Two views.
Below) A picture from lost month http://wholemovement.com/blog/itemlist/date/2012/11?catid=140 shows the same hyperbolic organization around the center tetrahedron as above, only this is formed using the 4-frequency folded grid and shows a single saddle-like system.
Above) Four circles were folded to a tetrahedron net of nine creases and crumpled into balls, flattened out to the circle and joined in a tetrahedron pattern. The surface is determined by the random crumpling of the circles and does not affect the patterned arrangement of four circles; the curmpled surface gives interest to the form.
Above) The four-circle sphere from above with four circles added to it, each a reformation of the tetrahedron net forming three pronges each.
Above) Two views of the above system with four more circles added that have been reconfigured from the 8-frequency diameter grid; one to each three pronged extension. Three are a slight variation to the other two and one is in the form of an icosahedrom.
Above) This is another direction in developing the above tetrahedron sphere. More crumbled circles combined with a creased triangle net have been added. The last four on all four sides have been folded but not crumpled. There are nine layers on each side making a thirty-six circled sphere displaying a tetrahedral design of four open triangles.
Below) Here are a few variations of individual units that can be made form the 8-frequce gird where the circle is reformed to a hyperbolic in/out pattern, in some cases looking totally different than what we associate with the mathematical saddle form. But then that is the advantage of folding circles, it will do the mathematically unexpected.
Following are a couple of systems developed from exploring multiple units from above. There are many possibilities for combining the spherical, the flat, and the hyperbolic systems by folding and joining circles that does not happen with other shapes.
Above) a system made from joining two of one of the reconfigurations, making another the same and joining the two sets of two together.
Above) Two views of an eight-unit circle using one of the units from above. From the side view you can see I decided not to complete the circle but keep it going in a helix form. I may continue to add by sequentially decreasing the diameter of each circle moving it into a conical helix or spiral form.
Above) Two views of two circles folded to a hyperbolic variation and joined in a tetrahedron arrangement.
Above) Another system with four circles folded to hyperbolic configurations and joined in a centered tetrahedral pattern. There are two views showing each end; one is an irregular variation of the other.
I always enjoy seeing the factual beauty in folding and reforming the circle, seeing the underlying structural truth about the nature of pattern as formations emerge, and to reflect on the good that comes from unity where all parts are multifunctional and interconnected allowing endless potential of expression throughout.
A hyperbolic surface warps a 2-D plane perpendicular to that plane in two opposite directions forming a 3-D saddle like shape. This can be demonstrated by adding more than 360° to the circumference of a circle, popularized by Daina Taimina’s crocheting. Given that the circumference is fixed with the paper circle, and one can not add anything to it, folding reveals another approach to forming a hyperbolic surface. Compressing the flat plane perpendicular to the surface towards the center curves the circumference in the same way. This warping of the flat surface is inherent in the nature of folding circles. Rather than going into any explanation about hyperbolic surfaces or the mechanics of folding, let’s just observe for now what happens when we distort the circumference of the circle. We will continue next month going further into folding hyperbolic planes.
Starting with a 8-frequency-diameter grid and reforming by folding around the hexagon center forms a hyperbolic surface.
Above left) This is the reconfiguration I will use in multiples to explore various systems. To the right shows another variation that is more open. The curving of the circumference can vary depending on the design of folding the circle.
Above) A random warping of the circumference coming from a couple of different ways of folding the center into itself. Both of these used a 16-frequency diameter grid (each of the three diameters is folded into 16 equal divisions.)
Below) A more open distortion of the circumference using a simpler in/out folding of a 4-frequency grid..
Below) shows two of a number of ways these two units can be joined; on the left one is nestled into the other and on the right edges are joined.
Below) three views of another kind of surface-to-surface joining. Here four individual units are joined in a tetrahedron arrangement making a very open and rigid system. The open areas are the six edges of the tetrahedron's four vertex points.
Below) two tetrahedral systems are joined one into the other forming a double unit. They are joined on one of the four openings on each system. Joining on the other openings would expand the complexity of the system
Below) three systems joined together on curved edges. This retains the open flow through all units.
Above) two views of combining six circles using the two unit and the four unit systems.
Below) two other variations in rearranging ten circles using the same units. On the left there is an open flow between the open units and on the right the flow is closed off between connections.
Moving on to other directions in thinking about the hyperbolic surface as a natural outgrowth of folding circles I looked to a simpler model.
Below) an example of using the folded net of the tetrahedron in two circles where the circumferences form an interesting hyperbolic system, one fitting into the other, showing a continuous wavy surface around four of the six edges of the tetrahedron.
Below) By using four circles with three diameters each folding two parallel chords connecting the two triangle arrangements of six points, they can be joined to reveal a collapsible transformational system that will fold flat. Three open square planes function in the same way to collapse the system that is similar to the collapsible system of four circles forming two open square planes from last month’s blog.
The collapsing squares brings to mind the collapsibility of the Vector equilibrium using four circles with three diameters each and six open squares, which brings up the off centered systems (from previous Center/Off-Center blogs.)
From here I thought about the collapsing of the vector equilibrium and how that works with the off-center systems and the importance of the center location. Each VE is concentric to the center; the off center is not. Every two points have an implied relationship of a third part connector; without the line relationship they remains two isolated points. By using a sequence of tetrahedra folded to different layers of the triangle grid and connecting them to both centers they reveal a relationship between the two centers that is a uniquely different expression of two points connected by a straight line.
Below) There follows four views show what it looks like. You will see bobby pins holding the circles together that form the off-centered vector equilibrium’s end points. The connecting tetrahedra are glued together.
While making tetrahedra it seemed reasonable to finish another piece by adding four tetrahedra to it.
Below) three views of the complete model where the tetrahedra show an exploded 2-frequency tetrahedron with an elaborated octahedron center.
There are always other directions to be explored. While focused in a particular direction there are always divergent tangents being pursued at the same time that come out of the dialogue between bringing different forms together that otherwise are associations that would go unnoticed. Another one developing over the last few months is shown below. This model uses 12 folded and curved circles.
Next month I will go more into the "ins and outs" if you will, of folding the hyperbolic surface using the circle.