Print this page
Monday, 26 November 2012 09:16

Folding Hyperbolic Surfaces

Written by 
Rate this item
(2 votes)

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.

 

001img 5765es                      002mg 5786es

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.

 

003img 5871es        004img 5850es

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..

005img 5876es         006img 5872es

 

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.

007img 5755es              008img 5801es

 

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.

009img 5710es           010img 5711es

 

                      011img 5906es

 

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

014img 5724es

Below) three systems joined together on curved edges. This retains the open flow through all units.

015img 5742es


016img 5805es       017img 5809es

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.

018img 5811es      019img 5816es

 

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.       

020img 5900es 

                

 

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.

021img 6034es            022img 6014es

 

      
023img 6027es            024.1img 6039es

 

                                                                                024img 6017es

 

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.

              025img 5933es

 

              026img 5924es

 

             027img 5936es

 

           

 

 

 

 

 

              028img 5928es

 

 

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.

029img 5984es

 

                     030img 5991es

 

                                             031img 5983es

 

 

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. 

                   032img 5631es

 

                   033img 5630es

 

                   034img 5637es

 

Next month I will go more into the "ins and outs" if you will, of folding the hyperbolic surface using the circle.

 

 

 

      

 

 

 

Read 285481 times Last modified on Monday, 26 November 2012 18:43
Bradford Hansen-Smith

Latest from Bradford Hansen-Smith