Origami metamaterials : design, symmetries, and combinatorics

In the first part of this thesis we study the geometry of folding patterns. Specifically, we focus on crease patterns consisting entirely of four-vertices; these are points where four fold lines come together. A single four-vertex is the simplest example of a foldable crease pattern that can be folded without bending the material in between the folds, and has a remarkable property: despite its single degree of freedom, it has two distinct folding motions. We make use of this property, and show how to design arbitrarily large four-vertex crease patterns, which can fold into two or more shapes. This is in contrast to other design methods, which produce patterns that can only fold into one specific shape. In the second part of this thesis, we study single four-vertices, and show a robust method to obtain four-vertices with three energy minima, which correspond to three different stable folded configurations. This too is in contrast to other experimental methods, which can only generate bistable vertices or patterns. FOM (presently part of NWO-I)Biological and Soft Matter Physic