# Biomimetic Origami: Planar Single-Vertex Multi-Crease Mechanism Design and Optimization
## 2 Single-Vertex Multi-Crease Origami Pattern Design  
### Identification of Origami Unit Types  
Research on single-vertex multi-crease origami (SVMCO) patterns focuses on the distribution of mountain and valley folds, as they directly affect the final folded shape. In traditional biomimicry of single-vertex multi-crease structures, researchers often simplify the creases in biological structures into Z-shaped folds. This simplification arises for two reasons. First, the biological structure under study often resembles a thin film with negligible thickness. Second, researchers tend to focus on the behaviour of these structures during transitions between flat planes, treating them as thickened panels regardless of folding or unfolding.  
In the simple Z-shaped folding, mountain and valley folds alternate, and the folded state appears as shown in Figure 2. However, applying this folding method directly to curved surfaces introduces vertical offsets \(\Delta\)h at the crease vertices, resulting in upward spiral displacements \(\Delta\)S at the connections, as shown in Figure 3. These displacements prevent the origami pattern from forming a closed loop at crease rotation angles between \(0^{\circ}\) and \(180^{\circ}\). The total offset equals the sum of the individual offsets \(\Delta s_{i}\) from each origami unit. To achieve a closed-loop structure, axial displacements must be incorporated into the design of Z-shaped folds. This adjustment adds complexity to the structure.  
Figure 2: Deployment sequence of the Z-shaped crease.  
In nature, structures with significant thickness can also fold and unfold curved surfaces effectively. The frill of the frilled lizard serves as a prime example. Unlike the linear venation in insect wings, the frill's surface features wrinkles with three ridges forming on each lobe of the fringe. Research from [45] indicated that these ridges arise from elastic instabilities during the development of the bronchial ectoderm, a result of natural selection.  
When the frilled lizard folds or unfolds its thick frill, the distribution of mountain and valley folds becomes more complex than a simple Z-shaped folding, which shown in Figure 4. The structure adds more folds, forming a mirrored arrangement of two Z-shaped folds, termed here as the \(\Omega\)-shaped folding unit.  
As shown in Figure 5, this closed deployable structure consists of three types of facets, named facet1, facet-2, and facet-3, arranged according to their increasing facet angle. An \(\Omega\)-shaped folding unit comprises two facet-1s, one facet-2, and one facet-3.  
Figure 4: Biomimetic design of origami creases inspired by the wrinkled distribution [45] on the frill of the frilled lizard (red lines: mountain creases, blue lines: valley creases).  
Figure 3: Upward displacement As during Z-folding prevents crease junction overlapping.  
In the \(\Omega\)-shaped folding unit, the distribution of creases in the sub-unit origami pattern changes from \"mountain-valley-mountain-valley\" to \"mountain-mountain-valley-valley\". When the first Z-shaped unit of the \(\Omega\)-shaped folding unit generates an upward offset of length \(\Delta s_{i}\), the second Z-shaped unit, which is mirror-symmetrical to the first, generates a downward axial offset of length \(\Delta s_{i+1}\). After the offsets of the two Z-shaped sub-units cancel each other out, the total offset of the \(\Omega\)-shaped fold is zero, meeting the requirement for sub-unit connection into a ring.