# Biomimetic Origami: Planar Single-Vertex Multi-Crease Mechanism Design and Optimization
## 2 Single-Vertex Multi-Crease Origami Pattern Design
### Dimensional Analysis and Optimization of Single-Vertex Multi-Crease Origami Patterns
It is calculated as:  
\[n=\frac{l_{NB}}{r}=\sqrt{4 sin^{2}\Big{(}\frac{\alpha_{2}}{2}\Big{)}\cos^{2}\Big{(} \frac{\alpha_{3}}{2}\Big{)}+sin^{2}\Big{(}\frac{\alpha_{2}}{2}\Big{)}} \tag{6}\]  
When \(\alpha_{3}<2\alpha_{1}\), the final folded state is shown in Figure 10. Let the radius of the outer circle of the origami pattern be \(l_{OB}=r\), in addition, the long edges of the adjacent facet-2 after folding are coincident, and the short edges points of facet-2 coincide at point E.  
The distance from a vertex on the envelope polygon, formed after folding, to the centre of the structure is  
\[ \begin{array}{ll}n&=\frac{l_{NB}}{r}=\frac{\sqrt{l_{F}^{2}+l_{FB}^{2}}}{r}= \frac{\sqrt{l_{F}^{2}+l_{FB}^{2}}}{r}\\ &=\frac{\sqrt{\big{(}l_{EF}\sin(\angle\text{FEO}-\angle\text{EEO})+l_{EM}\big{)} ^{2}+l_{IB}^{2}}}{r}\\ &=\sqrt{\bigg{(}{\sqrt{{k_{1}}^{2}-{k_{2}}^{2}}\sin\bigg{[}\arccos\bigg{(} \frac{{4{k_{1}}^{2}}}{{2\sqrt{4{k_{1}}^{2}-{k_{2}}^{2}}}}\bigg{)}-\arcsin\big{(} 2\sin\big{(}\frac{\alpha_{2}}{2}\big{)}\big{)}\bigg{]}+2\sin\big{(}\frac{ \alpha_{2}}{2}\big{)}\bigg{)}^{2}+{k_{2}}^{2}}}\\ \end{array}  \tag{7}\]  
Among which,  
\[\text{k}_{1}=\sin\bigg{(}\frac{\alpha_{2}+2\alpha_{3}-\pi/3}{2}\bigg{)},\text{ k}_{2}=\sin\bigg{(}\frac{\alpha_{3}}{2}\bigg{)}\]  
To determine the impact of angle division on the folding ratio of the panels, this paper will follow the calculation and analysis process outlined in Figure 11. First, we designed the initial conditions and input the sector angle ranges for facet-2 and facet-3. For every \(0.25^{\circ}\) increment, a set of sector angle data will be generated, and the corresponding sector angle for facet1 will be calculated. The program first verifies whether the sector angle data sets and the corresponding sector angle for facet1 are within the acceptable range to ensure that the total sector angles for one unit sum to \(60^{\circ}\). Next, it checks for interference by determining if \(\angle FBN\) exceeds \(60^{\circ}\) and if \(l_{NB}\) is greater than \(2l_{\text{BF}}\). The condition \(\angle FBN>60^{\circ}\) signals that adjacent facets might overlap, while \(l_{NB}>2l_{BF}\) confirms that the overlap is significant. These dual checks help ensure a smooth, interference-free folding process. If any of these  
Figure 10: The geometric relationship of the folded state when \(\alpha_{3}<2\alpha_{1}\). constraints are met, the input data set for the sector angles is marked as invalid and the program moves to the next data set. Otherwise, one of the three predefined calculation methods is selected based on the proportional relationships to calculate the folding ratio. Finally, all valid results are output, and a graph is generated to represent the relationship between the input sector angle data sets and the folding ratio. Figure 12 illustrates the relationship between facet angles (\(\alpha_{2}\) on the x-axis, \(\alpha_{3}\) on the y-axis) and folding ratio, represented by color (blue: low, yellow: high). Under the geometric constraint \(2\alpha_{1}+\alpha_{2}+\alpha_{3}=\frac{\pi}{6}\), \(\alpha_{1}\) is determined once \(\alpha_{2}\) and \(\alpha_{3}\) are selected. Two clear trends emerge:  
1. Increasing \(\alpha_{2}\) consistently raises the folding ratio (the colour shifts toward yellow). 2. Increasing \(\alpha_{3}\) initially increases the ratio but eventually lowers it, forming a low value along the line \(\alpha_{3}=2\alpha_{1}\). Figure 11: Diagram of the overall parameter calculation process. Figure 12: The relationship between facet-2 angle \(\alpha_{2}\), facet-3 angle \(\alpha_{3}\), and folding ratio. The darkest blue region (optimal folding ratio) aligns precisely with \(\alpha_{3}=2\alpha_{1}\), where the ratio reaches its minimum. Designs left of this line (\(\alpha_{3}>2\alpha_{1}\)) show higher ratios (blue-green transition), while those to the right (\(\alpha_{3}<2\alpha_{1}\)) also underperform (blue-cyan transition).