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.