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## Folding Hyperbolic Surfaces

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.

## Polygons to Circles

I was given a box of scrap die cut leftovers. Finally decided to explore the possibilities in these truncated hexagon star throw-a-ways. When working with polygons of various shapes there is always the limitation by what has been cut off.

*Below)* the two-sided maroon and blue fractal shapes are limited to a flat “2-D” tessellating unit.

2-D objects are inherently 3-D, which always makes better sense to my mind. So that is the next step.

*Above left)* four units joined together in tetrahedral arrangements. The possibilities are limited in joining these units.

*Above right)* The next obvious thing is to fold unit in half three times and rejoin units in the same tetrahedron arrangement. It became more interesting for the increased possibilities of joining multiples.

*Above)* making three folds across the shortest two opposite points is an equal hexagon division. Folding one crease in half to itself reforms the flat shape into two tetrahedra joined by common edge, a “bow tie” reformation. There are other ways to crease the shape; we will get to them later.

*Below)* four of the above “bow tie” units when joined the same way, edge-to-edge, form a Vector Equilibrium system of three, four, and six (same as with the circle using three diameters discussed in previous blogs.)

*Above)* two more possibilities of joining the same “bow tie” units in different ways. The one on the left has limited rotational movement between the two sets of two. The one on the right is stable.

*Below)* shows variations of reformations and joining going in other directions.

*Above)* two views of a particular development using the reformations above.

*Above)* two views of using the preceding units; twelve sets of two form an edge defined cubic arrangement. The movement in the units shows a distortion from the regularity of the cubic pattern. This is another example of the non-structural nature of the cube without diagonals to structurally define it. This is not a stable model.

*Below left)* four units are arranged in a cubic relationship showing two intersecting tetrahedra forming the diagonals of the cube. This is structurally stable.

*Below right)* a couple of models using one crease and three creases joining only two units for each.

*Below)* using the VE in multiples reveals the closed packing of spheres, origin to the tessellations of polygon and polyhedral arrangements.

In my August blog I talked about joining four circles with only one crease to make a vector equilibrium: http://wholemovement.com/blog/itemlist/date/2012/8?catid=140) . Here we see a very different treatment to do the same thing in a very different arrangement.

*Above left)* VE from three creases as seen before.

*Above right)* the same arrangement formed with only one crease in each unit. This time the creased shapes are positioned differently forming two opposite square planes out of six, showing none of the eight triangle planes. It is a collapsible system that becomes stable when standing on the square opening.

*Below)* sequence of vertical closed position opening to VE arrangement collapsing down to horizontal closed position. There is a 90 degree twisting in the movement. The system becomes fixed when the two square planes are stabilized as tension is applied to each end of the four folds.

This will be explored more shortly.

There are two more ways to symmetrically divide by folding this truncated star.

*Below)* shows three ways to crease symmetrical divisions consistent to the shape.

The last two ways of divisions are much more limited in what can be developed than what can be generated from using the first divisions. They become more specialized because of the greater complexity of division.

*Below)* are two VE systems folded from the first and second divisions above to see what differences occur.

*Above)* the second division does not join together reflecting spherical packing. The will form a planer hexagon ring of six units, of which three joined are shown. The conformation of edges makes a difference in how they join. There was not enough interest to continue this direction

*Above)* an interesting systems using four truncated star units made from the third divisional symmetry of six creases. It has limited potential in multiple joining, but there is interest in the complexity of form.

*Below)* two collapsible systems are each made from the first and the second ways of dividing the units. The dynamics for both are the same. The configuration of perimeter does not matter; the movement is around the center pivot.

Wanting to see the collapsible system in circle form I first folded three diameters thinking they were needed to get accurate proportions before realizing it is unnecessary; one crease in the circle is consistently self-positioning to all four units.

*Below)* six different positions of reforming the collapsible VE system above made using four of my circle business cards.

*Below)* three different proportional quadrilateral openings, one rectangle and two sizes of squares. Any proportion can easily be folded with only one crease because of the symmetrical self-alignment of circles. The one pictured in the center has square edges equal to the radial length.

*Below)* a six-inch diameter ceramic bowl sitting on top of four 3-inch diameter circles with one crease in each. This demonstrates the load-carrying possibilities of four paper circles stabilized by the compressive pressure on the opposite open planes. This would work with any configuration of units arranged in this manner because of the triangulation of compression of four twisting lines of force passing close to the center pivot.

During the above exploration I was also playing with a single circle to see how it can be folded so that when four are joined forming a tetrahedron they would open the four solid planes. With a slight variation of what I have done before it began to look interesting with possibilities.

*Below)* a single paper plate circle reformed to show the triangle plane collapsed towards the center with the ends opening out.

*Above)* a plan and edge view of four of the above units arranged in a tetrahedron pattern. There is a nice opening from the edge perspective but the collapsed triangle plane has opened to a solid plane again. This movement is necessary to allowed the system to come together forming the tetrahedron. The four open end points with the six open edges makes ten openings in the tetrahedron. Now this is interesting given the number describing the tetrahedron is ten (four points and six relationships between them.) The four open ends can also be closed leaving six openings that in the solid form are the edges.

*Above)* two tetrahedra, six units are joined sharing an open interior plane making a bi-tetrahedron. Each tetrahedron is formed using only three units. Here the solid triangle plane is collapsing to accommodate the change of added units. There are now fourteen openings.

*Below)* three views showing a left handed twisting tetrahelix consistent to the development from the bi-tetrahedron formed by adding two more tetrahedra in the same way, open-face-to-open-face. It retains the seven vertex points joining 3, 4, and 5 planes, only here the vertex points are open; they can also be closed slightly collapsing the system. This is a skeletal formation that conforms to the basic helix pattern but having twenty open spaces. The solid planes of the tetrahedron are now collapsed even more as it grows into a four-segmented tetrahelix, starting with four and growing to a ten-circle system, reflecting the number pattern of the single tetrahedra. This is what I was looking for and find delight in the possibilities.

It is not often I go back working with shapes and forms without the circle context, as with these truncated star shapes. When I do it always leads back to the circle. Only through understands circle/sphere origin is there information that allow a greater realization of the inherent potential that transcends the parts. The larger context reveals information otherwise unseen.

We now know there are at least two ways to form the VE pattern with four one-creased circles. This opens the question about other possibility in forming the VE using one crease in each circle. There is also the question of other ways to further open planes of the tetrahelix to maximize spatial flow without disrupting structural pattern; example the DNA helix.

The circle is always a circle no matter what polygon or polyhedral transformations it goes through. The circle Whole is never less than Whole, unless it starts out as a polygon, in which case potential has been diminished. Wholeness of the circle, unity does not change through reformations; what does change is the growth and self-understanding of the folder through a growing awareness of expanded potential working comprehensively within circle unity.