If you look at a quadrilateral lattice with planar faces you can ask the question: what are the conditions that the lattice posseses a (finite) deformation that keeps all the faces in shape (an isometric deformation). You will end up with equations for the normals of the faces only.

Knowing the normals gives you the directions of the edges as well. Thus, if a lattice is deformable all lattices made of quadrilaterals with parallel edges (for corresponding quads) will be deformable as well (they are called Combescure transforms).

It is easy to see that in general such a quadrilateral mesh admits at most a one parameter freedom for deformations (so in a sense the Miura folds are degenerate since in the planar limit you may choose not to fold in the “Miura-direction” but along the straight lines. However, as long as the configuration is not completely planar you will indeed have only one freedom)

The normals are governed by a discrete zero curvature condition.The corresponding linear matrix system has a spectral parameter wich can be viewed as the deformation parameter.

Another formulation would be the normals of a deformable lattic correspond to one-parameter-families of solutions to a discrete nonlinear sigma model in the lightcone of a Minkowski R^5 such that the first three components are constant in the family.

The discrete nonlinear sigma model is a well known discrete integrable system and roughly speaking it describes here the “evolution” of the normals along the lattice and not the deformation (or folding) itself.

The folding “sits” in the spectral parameter and is not itself described as a differential equation.

The difference equations describe more or less the normals that allow for a deformation.

timh

]]>A more interesting question (I hope): can you sketch the sense in which the Miura fold is an integrable system? Do you mean there’s some differential equation describing “Miura folds”, which is an integrable system?

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