Math Forum / Mathematics / Research / July 2009

I was wondering what is known about hinged quadrilateral panel systems.

If you connect 4 quadrilateral panels, using hinges like this

    A|B
    - -
    C|D

you generally get a "pop-up card" type system, that has one internal
degree of freedom: If you move one hinge, the other hinges all move in a
coupled motion.

If you connect a larger number of quadrilateral panels, you generally
get a rigid surface.
But under some conditions, you get a surfaces with an internal degree of
freedom, like in a "Miura fold":

http://demonstrations.wolfram.com/MiuraMapFoldingAndUnfolding/

I can build a few other surfaces that are slight generalizations Miura
folds. If the quadrilateral pattern is composed of pairs of strips that
are repeated in 1 direction like this:

 ABABABAB...
 CDCDCDCD...
 EFEFEFEF...
 GHGHGHGH...
 ...........

it will also have 1 internal degree of freedom, giving coupled motion of
all hinges.

My question is:
Is there a general criterion that the quadrilaterals must satisfy to
have 1 internal degree of freedom.

Gerard