I met Miura one and half year ago, while me, prof. Casale and prof. Valenti were studying surfaces and their applications in Architectural field. The Contemporary architectures are usually characterized by complex surfaces (normally double curved), so the technicians of the facades have to make solutions which turns in a critical composition made of small pieces of not developable surfaces, all different and really expensive. Miura pattern could be one of a set of solutions capable to create a complex shape with all identical flat surfaces.
ALL IDENTICAL FLAT SURFACES.....Do you understand?
Miura pattern is way to fold a planar surface, making mountains and valleys folds. The blu line and the red line allows to recognize the different folds on digital drawings. So the blu one could be an acute angle and the red one could be an obtuse angle, But with these indications, tightening and enlarging the angles, you will obtain only a planar movement.
In the video you will see a non planar movement thanks to the important role of neutral fold, the grey one.
How Do I create the movement? Which is the algorithm? I started from a folded surface and I isolated the blu polylines that are a union of mountain and valley line. As you can see they were not all the polylines, but they were taken every two alignments.
Everyone of these alignments will be forced to move by folding force applied on the lower hinge. At the same time these nodes will be forced to belong to a mathematical surface which change his shape. From plane to a paraboloid or a dome. During the movement you will notice that the area of every face inside the whole polyhedra will not change. Also, all the triangles are equal!!!!!