MATHEMATICS AND ORIGAMI

Mathematics and Origami

21.3 HYPERBOLOID

The quadric we are going to construct now is a warped hyperboloid of revolution, a one-

sheet ruled surface.

Fig. 1 is its folding diagram. The whole figure is triangulated; both, upper and lower

trapeziums serve to self-pocket the hyperboloid.

Note that this yperboloid structure reminds so much that of the pentagonal prismoid

seen in Point 18.5.3. Now, though, we shall tend to augment the number of sides toward in-

finity.

Paper materialisation of this hyperboloid is not something easy or spontaneously stable,

but yields an attractive result when achieved.

The optimal solution (self-stable) asks for these requirements:

• A paper both, resistant and docile, to guarantee that the generatrices will not col-
  lapse, whereas the folds may be easily produced.


• To fix with an adhesive tape, mountain as well as valley fold ́s settlements in the vi-
  cinity of upper and lower polygons.


• 
  
  
  
  • To pocket both polygons within the trapeziums to fasten the figure and hide the
    adhesive tape. That pocketing should apply to the perfect coupling of every four
    mountain / valley lines that coincide in the upper and lower polygonal vertices.
  
  
  
  


• Also to pocket the figure laterally: that constrains to have an extra pair of triangles
  (an extra parallelogram).

Most likely, it will be rather difficult to meet all those requirements (skill and patience

to be added); that ́s why I propose a practical advice:

• To use normal paper.


• Start with Fig. 1; the trapeziums will only serve to the purpose of reinforcing the up-
  per and lower perimeter.


• To glue both extreme rhomboids to close the hyperboloid.

1 B

C

A